Analytic Geometry With Introduction to Vector Analysis

BENEDICK A. GANZOStructural EngineerASAS-Omrania Architecture & Engineering ConsultantsKingdom of Saudi Arabia

Brian Ganzo Publishing CompanyPhase 6, V&G, Tacloban CityPhilippines

Copyright © 2009by Brian Ganzo Publishing Company, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or by any means, without permission in writing from the publisher.

Printed and distributedbyB. A. GANZO Printers, Inc.Tacloban City, Philippines

iii

Contents

Preface ix

PLANE ANALYTIC GEOMETRY

Chapter 1 Rectangular Coordinates 11.1 Analytic Geometry Defined1.2 Rectangular Coordinates1.3 Distance Between Two Points1.4 Division of a Line Segment. Midpoint1.5 Inclination. Slope1.6 Slopes of Parallel and Perpendicular Lines1.7 Angle Between Two Lines. Intersection1.8 Area by Coordinates

Chapter 2 Polar Coordinates 92.1 Polar Coordinates2.2 Distance Between Two Points2.3 Relations Between Polar and Rectangular

Coordinates

Chapter 3 Functions and Curves 133.1 Functions. Degree of an Algebraic Equation3.2 Locus of an Equation. Intersection of Two Curves3.3 Intercepts3.4 Symmetry3.5 Asymptotes. Extent of the Curve3.6 Tracing the Curve of an Algebraic Equation

and a Polar Equation 3.7 Equation of a Given Locus

Chapter 4 The Straight Line 254.1 A Line Parallel to a Coordinate Axis4.2 General Equation of a Line4.3 Point-Slope Form4.4 Two-Point Form

4.5 Slope-Intercept Form4.6 Parallel and Perpendicular Lines

iv CONTENTS

4.7 Concurrence of Three Lines4.8 Intercept Form4.9 Normal Form4.10 Polar Equation of a Straight Line4.11 Directed Perpendicular Distance of a Line to a Point

and Between Two Parallel Lines4.12 Two Conditions Determine a Line

Chapter 5 The Circle 335.1 Circles5.2 General Equation of a Circle5.3 Standard Equation of a Circle5.4 Radical Axis5.5 Polar Equation of a Circle5.6 Three Conditions Determine a Circle

Chapter 6 Special QuadraticEquations in Two Variables.Conic Sections 37

6.1 Conic Sections6.2 Parabolas6.3 General Equation of a Parabola6.4 Standard Equation of a Parabola6.5 Ellipses6.6 General Equation of an Ellipse6.7 Standard Equation of an Ellipse6.8 Hyperbolas6.9 General Equation of a Hyperbola6.10 Standard Equation of a Hyperbola6.11 Asymptotes of a Hyperbola6.12 Conditions Describing a Conic Section6.13 Polar Equation of a Given Conic Section6.14 Tracing a Conic Section

Chapter 7 Transformation of Coordinates. The General Quadratic in Two Variables 51

7.1 Translation of Axes in a Plane7.2 Rotation of Axes in a Plane7.3 The General Quadratic in Two Variables7.4 Tracing the Curve of a General Quadratic7.5 Discriminant of a Conic

Chapter 8 Tangents and Normalsto Conics 55

8.1 Tangents and Normals

CONTENTS v8.2 Tangent and Normal Through a Given

Point on the Conic8.3 Poles and Polars of a Conic8.4 Tangent to a Conic Through a Given

External Point8.5 Tangent of Given Slope

Chapter 9 Parametric Equations 619.1 Parametric Equations9.2 A Set of Parametric Equations of

Some Plane Curves9.3 Tracing a Given Set of Parametric Equations

Chapter 10 Transcendental Functions 6310.1 Trigonometric Functions10.2 Congruence and Shifting10.3 Tracing by Composition of Ordinates10.4 Exponential Functions10.5 Hyperbolic Functions10.6 Logarithms10.7 Inverse Functions

Chapter 11 Families of Plane Curves.Curve Fitting 73

11.1 A Family of Curves11.2 A Family of Curves Through an Intersection11.3 Curve Fitting11.4 Line of Best Fit. Method of Least Squares11.5 Nonlinear Curves of Best Fit

SOLID ANALYTIC GEOMETRY

Chapter 12 Rectangular Coordinatesin Space 79

12.1 Space Rectangular Coordinates12.2 Distance Between Two Space Points12.3 Division of a Line Segment in Space. Midpoint12.4 Direction Angles and Direction Cosines12.5 Angle Between Two Space Lines12.6 Parallel and Perpendicular Space Lines

vi CONTENTS

Chapter 13 Cylindrical and SphericalCoordinates 85

13.1 Cylindrical Coordinates13.2 Spherical Coordinates13.3 Relations Between Rectangular, Cylindrical,

and Spherical Coordinates

Chapter 14 Surfaces and Space Curves 8914.1 Locus of an Equation in Three Variables14.2 Symmetry of Surfaces14.3 Intercepts of a Surface. Sections and Traces14.4 Tracing Surfaces by Parallel Plane Sections14.5 A Surface of Revolution14.6 Cylindrical and Conical Surfaces14.7 Intersection of Two Surfaces14.8 Projections, Projecting Lines,

and Projecting Cylinders14.9 Tracing Space Curves by Its

Projecting Cylinders14.10 Sketching Solids Bounded by Surfaces14.11 Equation of a Given Surface

Chapter 15 The Plane 9915.1 A Plane Parallel to a Coordinate Plane15.2 General Equation of a Plane15.3 Three-Point form15.4 Parallel and Perpendicular Planes15.5 Intercept Form15.6 Normal Form15.7 Directed Perpendicular Distance of

a Plane to a Point

15.8 Three Conditions Determine a Plane

Chapter 16 The Straight Line in Space 10316.1 General Equation of a Line in Space16.2 A Family of Planes Through a

Given Space Line16.3 Parametric Equations of a Space Line16.4 Symmetric Equation of a Line16.5 Lines Parallel and Perpendicular to a Plane

CONTENTS viiChapter 17 Special Quadratic Equations

in Three Variables. QuadricSurfaces. Transformation ofCoordinates in Space 107

17.1 Quadric Surfaces17.2 Ellipsoids. Spheres17.3 Hyperboloids of One Sheet17.4 Hyperboloids of Two Sheets17.5 Elliptic Paraboloids17.6 Hyperbolic Paraboloids17.7 Quadric Cylinders17.8 Elliptic Cones17.9 Ruled Surfaces17.10 Translation of Axes in Space17.11 Rotation of Axes in Space

INTRODUCTORY VECTOR ANALYSIS

Chapter 18 Vector Operations 12118.1 Vectors18.2 Equality of Vectors. Negative of a Vector18.3 Sum of Vectors. Difference18.4 Product of a Scalar and a Vector.

The Unit Vector18.5 Dot Product of Two Vectors18.6 Cross Product of Two Vectors

Chapter 19 Vectors in CartesianCoordinates 129

19.1 Cartesian Unit Vectors19.2 Cartesian Representation of a Vector19.3 Operations on Two Vectors with

Cartesian Representations19.4 Products Involving Three Vectors.

The Lagrange’s Identity

Chapter 20 Vector Analysis ofPlanes and Lines 137

20.1 The Equation of a Plane20.2 The Parametric Equation of a Line

ix

Preface

It is the hope of the author that this concise book, Analytic Geometry with Introduction to Vector Analysis, will prove valuableand handy to students of engineering, science, and mathematics, taking up analytic geometry as a preparatory course or simultaneously with calculus. It is expected however, that the students had already completed courses in algebra and trigonometry. A working knowledge of elementary geometry and matrices are important.

Each chapter is organized by presenting immediately the basic definitions, principles, theorems, and formulas without their proofs and definitions. The author knows that sometimes students and practicing engineers are only interested in the immediate formulas that are needed to solve a particular problem.

This book is divided into three major parts. The first eleven chapters cover plane analytic geometry. The next six chapters cover solid analytic geometry, the extension of geometric theorems to the three-dimensional case. The last three chapters provide an introduction to vector analysis, with discussions on the application of the subject to the solution of geometric problems. The author believes that vector analysis should now be an essential part of the mathematical background of every engineer, scientist, or mathematician.

Although every effort has been made to keep the presentation clear and accurate, the author would be very happy to receive suggestions or corrections if necessary.

The author gratefully acknowledges his indebtedness to his colleagues, former students, and former teachers who have extendedhelp in the preparation of this book. Their names would form a list that several pages of this book would still be insufficient to contain them.

Benedick A. GanzoStructural Engineer

1

PLANE ANALYTIC GEOMETRY

Chapter 1Rectangular Coordinates

1.1 Analytic Geometry Defined

a. Analytic geometry is the branch of mathematics, dealingwith the behavior and properties of configurations involving points, lines, curves, surfaces, and solids by means of algebraic methods. If the figures are on a plane, the study is called plane analytic geometry. Solid analytic geometry deals with figures in space.

b. Various methods in analytic geometry that are used to prove directly many theorems of classical Euclidean geometry are called analytic proofs. See examples 1.10 and 1.11.

1.2 Rectangular Coordinates

a. The position of a point on a plane may be determined by its distances from two perpendicular lines, in what we call a rectangular (or Cartesian)coordinate system, Fig. 1.1.

b. A rectangular coordinate system is formed by drawing a pair of perpendicular lines X’X and Y’Y, called the coordinate axes (or the X-axis and the Y-axis respectively), intersecting at a point called the origin O. Perpendicular distances measured from the Y-axis to the right (along OX) and from the X-axis upward (along OY) are positive, while their opposites, from the Y-axis to the left (along OX’) and from the X-axis downward (along OY’) are negative

Fig. 1.1

Y

X’

Y’

X…1 2 3 +∞

123

+∞

……

-1-2-3…-∞

-∞

-1-2-3

first quadrantsecond quadrant

third quadrant fourth quadrant

Pa

b

O

2 PLANE ANALYTIC GEOMETRY

distances. The plane is divided into four regions called quadrants.

c. The x-coordinate (or abscissa) of a point is its perpendicular distance from the Y-axis and the y-coordinate(or ordinate) of a point is its perpendicular distance from the X-axis. Together, these rectangular coordinates (the paired x-coordinate and y-coordinate of a point) determine the position of a point in a plane. Point P for example, Fig. 1.1, is located at (a, b).

d. The notation P(x, y) where x and y are variables, means that a point P has an x-coordinate x, and a y-coordinate y in a rectangular coordinate system. Plotting is the process of locating (by drawing or placing) a point on a plane when its coordinates are known.

e. A directed line segment (or directed distance) is a line segment measured in a definite sense or direction (and it is either positive or negative), Fig. 1.2. The tail end P1 of the arrow is called the initial point (or origin), and the head P2

is called the terminal point (or terminus) of the directed line segment. If the directed line segment joining the point P1(x1, y1) to P2(x2, y2), in that direction (written as

P1P2 or d

, an arrow is placed above the letter if only one letter is used to represent the directed line segment), is taken as positive, then the opposite of that direction, from P2 to P1 (or the directed line segment P2P1), is equal to the negative of P1P2. If P1P2

was initially negative, then P2P1 is the positive of P1P2. That is, directed line segments in opposite directions have opposite signs, or

f. The distance (or segment) on the other hand, between the two points P1 and P2 (written as |P1P2|), is always positive whether measured in the opposite direction |P2P1|. It is the

Fig. 1.2

Y

XO

P2(x2, y2)

P1(x1, y1)

P1P2 or d

(1.1)P1P2 = - P2P1

RECTANGULAR COORDINATES 3magnitude or the absolute value of the directed distance, sothat,

1.3 Distance Between Two Points

a. The distance |d| between two points P1(x1, y1) and P2(x2, y2), Fig. 1.3, is given by,

1.4 Division of a Line Segment. Midpoint

a. If P(x, y) is a point on the line segment |P1P2|, joining the points P1(x1, y1) and P2(x2, y2), such that, the ratio of the directed distances P1P and P1P2 is k, or

then the coordinates (x, y) of P, Fig. 1.4, must be given by,

121

121

yykyy

xxkxx

and

(1.5)

Fig. 1.4

Y

XO

P2(x2, y2)

P1(x1,y1)

P(x, y)

(1.4)21

1

PPPP

k

Fig. 1.3

Y

XO

P2(x2, y2)

P1(x1, y1)

|d|

|P1P2| = |P2P1| (1.2)

2122

12 yyxxd (1.3)


Fig. 1.5

Y

XO

P2(x2, y2)

P1(x1, y1)

P(x, y)

(1.7)tanm =

b. If P lies in the extension of |P1P2| in either direction, Fig. 1.5, equation (1.5) still applies.

c. If Pm(xm, ym) is the midpoint (or a point that divides a line segment into two equal parts) of the line segment |P1P2|, equation (1.5) reduces to,

1.5 Inclination. Slope

a. The angle of inclination(or simply inclination) of a line, θ1 and θ2 for the lines L1 and L2 respectively of Fig. 1.6, is the least counterclockwise angle the line makes with the positive X-axis, ranging from 0 ≤ θ <π. If the inclination of a line is taken in a clockwise direction from the X-axis to the line(sometimes called the declination of the line), it is considered negative in value.

b. The slope m of a line is the tangent of the angle of inclination, written as

21m

21m

yy21

y

xx21

x

and

(1.6)

Fig. 1.6

Y

XO

L2, line of slope m2

L1, line of slope m1

α

θ1 θ2

N(xi,yi)

RECTANGULAR COORDINATES 5

12

12

xxyy

m --

= (1.8)

m1 = m2 (1.9)

2m1-1m (1.10)

where m is positive for 0 < θ < 2 (or lines inclined to the

right), and m is

negative for 2 < θ < π

(or lines inclined to the left). When θ = 0 (horizontal lines), m =

0. When θ = 2

(vertical lines), m is undefined.

c. If P1(x1, y1) and P2(x2,y2) are points on a line, the slope m of the line is obtained by,

since 12

12

1

2

xxyy

RPRP

tanm --

=== , Fig. 1.7.

1.6 Slopes of Parallel and Perpendicular Lines

a. Two lines (with slopes m1 and m2) are parallel if they have equal slopes. That is,

b. Two lines are perpendicular if they have slopes in which one is the negative reciprocal of the other. That is,

Fig. 1.7

Y

XO

P2(x2,y2)

P1(x1,y1)

θ

R(x2,y1)P1R =x2– x1

RP2 =y2 – y1

θ


1

2

mm

arctan2

1m1

m-(1.11)

1.7 Angle Between Two Lines. Intersection

a. The angle α , Fig. 1.6, formed by rotating the line L1 to L2, at their point of intersection N(xi, yi), is related to the slopes of each line by the equation,

This angle is negative if taken in a clockwise direction from L1

to L2.

b. The point of intersection N(xi, yi) of two lines, Fig. 1.6, is the point whose coordinates satisfy the two equations of the lines (or it is the point whose coordinates is the solution of the two equations of the lines, taken simultaneously).

1.8 Area By Coordinates

a. The area A of a triangle, Fig. 1.8, with vertices P1(x1, y1), P2(x2, y2), and P3(x3, y3), traced in acounterclockwise direction, is given by,

where the matrix

1321

1321

yyyyxxxx

is defined to have the

value )xyxyxy()yxyxyx( 133221133221 . The area A yields a negative result if the vertices are traced in a clockwise direction.

1yx1yx1yx

21

A

33

22

11

or

1321

1321

yyyyxxxx

21

A

(1.12)

Fig. 1.8

Y

XO

P2(x2,y2)

P1(x1,y1)P3(x3,y3)

RECTANGULAR COORDINATES 7b. The area A of a non-overlapping polygon of n vertices is

written in the form,

where the vertices P1(x1, y1), P2(x2, y2), P3(x3, y3), …, and Pn(xn, yn) are traced in a counterclockwise direction. The

matrix

1n321

1n321

yyyyyxxxxx

is defined to have the

value )xyxyxy()yxyxyx( 1n32211n3221 . The formula for the area A also yields a negative result if the vertices are traced in a clockwise direction.

(1.13)

1n321

1n321

yyyyyxxxxx

21

A

9

Chapter 2Polar Coordinates

2.1 Polar Coordinates

a. The position of a point on a plane may also be described by its distance from a fixed point and its direction from a fixed line through the fixed point, in another system called the polar coordinate system, Fig. 2.1.

b. A polar coordinate system is formed by drawing a reference line OX, called the initial lineor polar axis, in a horizontal direction to the right, starting from a fixed point O, called the pole (or origin).

c. The radius vector of a point is its distance from the pole and the polar angle of the same point is its direction (or angle) from the polar axis. The polar angle is positive when measured counterclockwise from the polar axis, and negative when measured clockwise. The radius vector is positive when measured from the pole to the terminal side of the corresponding polar angle, and negative when taken in the opposite direction. Together, the polar coordinates (the paired radius vector and polar angle of a point) determine the position of a point in a plane. Point P for example, Fig. 2.1, is located at (ρ, α).

d. The notation P(r, θ), where r and θ are variables, means that a point P has a radius vector r, and a polar angle θ, in a polar coordinate system. The same point P(ρ, α), Fig. 2.1, may be described in a variety of ways using polar coordinates, for

Fig. 2.1

O X

…+∞

α32

1

ρ ……

1 2 3 … ρ… +∞……

… P

43

6

12

32 12

72

125

3

4

1211

65

34

45

67

1213

1217

23

1219 3

5

611

47

0

1223


example P(-ρ, ), P(ρ, 2 ), P(-ρ, ), P(ρ, 2 ), and so on. Generalizing, the point P(ρ, α) may also be written as,

2.2 Distance Between Two Points

a. The distance |d| between two points whose polar coordinates are P1(r1, θ1) and P2(r2, θ2) is given by,

2.3 Relations Between Polar and Rectangular Coordinates

a. A coordinate system is just a tool in describing the position of points and is not inherently present in a specific geometric problem. Either polar or rectangular coordinates is used whichever appears to simplify a particular problem.

b. If (x, y) and (r, θ) are the rectangular and polar coordinates describing the same point in a plane, Fig. 2.2, then the equations relating them have the forms,

and

(2.1)P(ρ, k ) when k is even

or P(-ρ, k ) when k is odd

)cos(rr2rrd 12212

22

1 (2.2)

cosrx , sinry (2.3)

22 yxr ,

xy

arctan (2.4)

(x, y)

Fig. 2.2

Y

XO

P

r

x

y

θ

(r, θ)

POLAR COORDINATES 11where the radical for obtaining r in the last equation follows the sign of x. If 0x , it follows the sign of y, and a value of

2 is immediately assigned to θ. These conditions are

imposed to facilitate a unique conversion from rectangular to polar coordinates.

13

(3.1)

y = f(x) for the variables x and y, which is read as “y is a function of x”

or r = f(θ) for the variables r and θ, which is read as “r is a function of θ”

(3.2)f(x, y) = 0 for the variables x and yor f(r, θ) = 0 for the variables r and θ

Chapter 3Functions and Curves

3.1 Functions

a. If two variables x and y are related such that, for every x we obtain one or more real values for y, then y is said to be a function of x. Since y depends on the value of x, y is the dependent variable (or the function), while x is the independent variable. The variable y is a single-valued function of x if only one value of y corresponds to each value of x; otherwise it is double-valued, triple valued or multiple-valued function of x. The set of values of x is called the domain of the given function and the set of corresponding values for y, for each x in the domain, is called the range.

b. An equation is a mathematical expression that relates the independent and the dependent variable. It may be in

explicit form,

implicit form,

parametric form (see Ch. 9),

(3.3)

x = f(t), y = g(t) for the variables xand y, where t is the parameter

or r = f(t), θ = g(t) for the variables r and θ, and t is the parameter


The equation is the law that defines a curve or locus of a moving point. It may also be thought of as the analytical representation of any given curve. An algebraic equation (or Cartesian or rectangular equation) is a polynomial equation in x and y describing a curve in a rectangular coordinate system, while a polar equationdescribes a curve in a polar coordinate system. Note that equations even though involving trigonometric functionsand are not polynomials (see Ch. 10) but uses a rectangular coordinate system are not polar equations.Instead, these non-algebraic equations in rectangular coordinates are called transcendental.

c. The degree of an algebraic equation is the highest power or sum of powers in any one term of a given algebraic equation. For example, the equations,

01xyx3yx2 22 , 1x2y , and 2x3x

4xy

22

are of third, first, and fourth degree respectively.

3.2 Locus of an Equation. Intersection of Two Curves

a. The locus (curve or graph) of an equation is a curve containing those points, and only those points, whose coordinates satisfy the equation. It may be thought of, on the other hand, as the geometrical representation of a given equation (see Sec. 3.1b)

b. To find whether a point satisfies the equation of a given curve, substitute its coordinates for x and y in the equation of the curve and note whether the equation holds.

c. The points of intersection of two curves are found by solving the equations of the curves simultaneously. The number of intersections of two curves is at most the product of the degrees of their equations.

3.3 Intercepts

a. The x-intercept and the y-intercept of any given curve are the directed distances (Sec. 1.2e) from the origin to the point where the curve intersects the X-axis and the Y-axis

FUNCTIONS AND CURVES 15respectively, Fig. 3.1. In other words, the x-intercept a is the abscissa of the point of intersection P(a, 0) of the curve with the X-axis, while the y-intercept –b is the ordinate of the point of intersection Q(0, -b) of the curve with the Y-axis.To find the

x-intercept,solve for x in the equation y=f(x),with f(x) in factored form if possible and y is set to zero.

y-intercept,solve for y in the equation y=f(x), with x set to zero.

3.4 Symmetry

a. The center of symmetry of two points P1 and P2, Fig. 3.2, is the point P midway between them. Their axis or line of symmetry is the perpendicular bisector L of the line joining them.

b. A curve is symmetric with respect to a coordinate axis if for every point P of the curve on one side of the axis, there corresponds an image point P’ on the opposite side of the axis, Fig. 3.3. A curve is

Fig. 3.1

Y

XOP(a, 0)

Q(0, -b)

curve of y = f(x)

Fig. 3.2

Y

XO

P2

P1

P, center of symmetry

L, line of symmetry

Fig. 3.3

Y

XO

P1

curve symmetric with Y-axis

curve symmetric with X-axis

P2

P1’

P2’

P3

P4

P3’ P4’


symmetric with respect to the X-axis,if its equation remains unchanged whether y is replaced by –y.

symmetric with respect to the Y-axis,if its equation is unchanged even when x is replaced by –x.

c. A curve is symmetric with respect to a point, if for every point P of the curve there corresponds an image point P’ directly oppositeand at an equal distance from the point. A curve that is symmetric with respect to the origin is shown in Fig. 3.4. A curve is

symmetric with respect to the origin O,if its equation is unchanged whether x and y are replaced simultaneously by –x and –y respectively.

d. For a polar equation, its curve in polar coordinates is

symmetric with respect to the polar axis OX,if the polar equation is unchanged when θ is replaced by –θ or when θ and r are simultaneously replaced by (π – θ) and –r respectively.

symmetric with respect to OY (a line perpendicular to OX and passing through the pole O, or this line is the Y-axis equivalent in rectangular coordinates),

if the polar equation is unchanged when θ is replaced by (π – θ) or when θ and r are simultaneously replaced by –θ and –r respectively.

symmetric with respect to the pole O,if the polar equation remains unchanged when r is replaced by –r or when θ is replaced by (π + θ).

The converses of these tests for symmetry of a curve in polar coordinates are not necessarily true.

Fig. 3.4

X

Y

O

curve symmetric with respect to the origin

P1’

P1

P2

P2’

FUNCTIONS AND CURVES 17

(3.4)x = a

(3.5)y = b

(3.6)0 ≤ x < a

3.5 Asymptotes. Extent of the Curve

a. An asymptote of a curve is a straight line approached by the curve more and more closely but never actually touching it. The line,

of Fig. 3.5 is a vertical asymptote, if there is a point on the curve whose ordinate y increases numerically without limit as the value of its abscissa x approaches a. The line,

of Fig. 3.6 is a horizontal asymptote, if the abscissa x of a point on the curve increases numerically without limit as its ordinate y approaches b.

b. An asymptote to a curve of nth degree may intersect the curve in at most 2n points. To find the vertical and the horizontal asymptotes of an algebraic equation, see Sec. 3.6.

c. The extent of the curve in any chosen direction, say for example from the origin to the right (along OX or in the direction of the positive X-axis), is the totality of real values of x which gives real values for y. If the asymptote ax , Fig. 3.5, does not intersect the curve in any other point (that is to say the curve is of degree n ≤ 2 in y), then the extent of that curve in the OX-direction is,

Fig. 3.5

X

Y

O

P(x, y)

x=a,

ver

tical

asy

mpt

ote

Fig. 3.6

X

Y

O

P(x, y)

y=b, horizontal asymptote


(3.7)0 ≤ y < b

(3.9)

y = )x(g)x(f

= m1m

1m1

m0

n1n1n

1n

0

bxbxbxb

axaxaxa

where (n ≥ 0), (m ≥ 1)

(3.10)y2 = f(x)

= n1n1n

1n

0 axaxaxa

where (n ≥ 1)

Similarly, the extent of the curve shown in Fig. 3.6, from the origin upward (along OY or in the direction of the positive Y-axis), or in the OY-direction is,

provided that the asymptote y = b does not intersect the curve in any other point.

3.6 Tracing the Curve of an Algebraic Equation

a. Algebraic equations of degree n ≤ 2 in the dependent variable yn and containing no product terms (terms containing the product of the variables, xnyn), fall into four types or forms.

Type 1

for n = 0, 1, the curve is a straight line. n = 2, the curve is a parabola with axis vertical.

Type 2

The fraction should be in lowest terms; that is (m ≥ n). If m = 0, the curve is of Type 1.

Type 3

If n = 0, the curve is of Type 1, and it is the horizontal line n0 aay , provided that a0 + an > 0.

(3.8)y = f(x)

= n1n1n

1n

0 axaxaxa

where (n ≥ 0)


(3.11)

y2 = )x(g)x(f

= m1m

1m1

m0

n1n1n

1n

0

bxbxbxb

axaxaxa

where (n ≥ 0), (m ≥ 1)

Type 4

The fraction should be in lowest terms, or (m ≥ n).If m = 0, the curve is of Type 3, provided that (n ≥ 1).

If the algebraic equation is given in implicit form, equation (3.2), 0)y,x(f , the equation could be solved for y or y2

and expressed in one of types given above.

Types 3 and 4 are symmetric with respect to the X-axis. Types 2 and 4 only, the rational functions, could possibly have asymptotes. To find the

vertical asymptotes, Let y approach ± ∞ by setting to zero the

denominator of the rational function,)x(g)x(f

y n ,

where n ≤ 2. This is because any number divided by zero equals ±∞.

Solve for x in the resulting equation, 0)x(g ,with g(x) in factored form if possible. The solutions x = a, b, c… are the vertical asymptotes x = a, x = b, x = c, etc., of the curve.

horizontal asymptotes, Let x approach ±∞ in

)x(g)x(f

y n , where n ≤ 2,

after first dividing each term of f(x) and g(x) by the x-term of highest degree (thereby putting all x in the denominator of each term and noting that any number divided by ±∞ equals zero).

Solve for y. The two solutions y = a, b, are the two horizontal asymptotes, y = a, and y = b, of the curve.

Some curves analogous to Type 2, but where n1m (an exception of the condition in equation 3.9,

m ≥ n, for an algebraic curve to be classified as a Type 2 curve), may have a slant asymptote. These are the


curves represented by the equation

)x(g)x(f

y m1m

1m1

m0

n1n1n

1n

0

bxbxbxb

axaxaxa

where the

degree of the numerator f(x), n, exceeds that of the denominator g(x), m, by unity. Example 3.17 illustrates how to find the slant asymptote of a curve.

b. Point-plotting is a way of tracing or drawing the curve of an equation (whether algebraic or polar), and is done as follows:

Assign a range of values for x (or θ or the independent variable).

Solve for the corresponding value of y (or r or the dependent variable), for each assigned value of x. Each pair of x and y (or r and θ) is a point on the curve. The infinitely many possible pairs of x and y, or r and θ, that can be found are the coordinates of all the points on the curve. The values for the independent variable must be chosen carefully so that the few points are enough to give a general description of the entire curve.

Plot the points. Finally, draw the curve through the plotted points.

c. For a more accurate and effective way to trace the curve of an algebraic equation, follow these steps.

Analyze the equation. Express the equation in one of the forms given

(Types 1, 2, 3, or 4). Test for symmetry, by inspection if possible, with

the X-axis, the Y-axis, and the origin O. Find the intercepts. Determine the asymptotes. The horizontal

asymptotes y = a may intersect the curve in at most (n – 2) points, whose coordinates are (x1, a), (x2, a), …, (xn-2, a) where the abscissas x1, x2, …, xn-2 are the solutions of the equation with y replaced by a. The vertical asymptote will not intersect the curves of this type (or curves of degree n ≤ 2 in the dependent variable.

Determine the extent of the curve in the directions of OX and OY (and the opposites of these directions, OX’ and OY’, if there is no

FUNCTIONS AND CURVES 21symmetry with respect to one or both coordinate axes).

Trace the curve of the equation immediately from the properties of the curve seen from the analysis.

If necessary, plot a few points (or point-plot) in areas that would make the curve more accurate.

3.7 Tracing the Curve of a Polar Equation

a. In polar coordinates, a polar equation is traced as follows:

Test for symmetry. Use point-plotting (See Sec. 3.6b).

Assign values to θ and solve for r.

Each pair of r and θ is a point on the curve, in the polar coordinate system. The range of values of θ should be adequate to present a picture of the whole curve if necessary, or the part of the curve to be used.

b. The following figures are the graphs of some common polar equations.

Fig. 3.7

O X

circle, cosar

(a, 0)

Fig. 3.8

OX

circle, sinar (a,

2

)

Fig. 3.9

O X

cardioid, )cos1(ar

(2a, 0)

(a, 2 )

Fig. 3.10

O X(2a, π)

(a, 2 )

cardioid, )cos1(ar


Fig. 3.11

OX

(a, 0)

(2a, 2 )

cardioid, )sin1(ar

Fig. 3.12

OX

(a, 0)

(2a, 23

)

cardioid, )sin1(ar

Fig. 3.13

OX( ba , π)

(b, 2

)

limacon, )ab( ,cosabr

( ab , 0)

Fig. 3.14

O X( ba , π)

(b, 2

)

limacon, )ab( ,cosabr

( ab , 0)

Fig. 3.15

OX

lemniscate,

2cosar22

(a, 0)

Fig. 3.16

OX

lemniscate,

2sinar22

(a, 4

)


Fig. 3.17

OX

four-leaved rose, 2sinar

(a, 4

)(a, 43

)

(a, 45

) (a, 47

)

Fig. 3.18

OX

three-leaved rose, 3sinar

(a, 6

)(a, 65

)

(a, 23

)

Fig. 3.19

O X

four-leaved rose, 2cosar

(a, 0)

Fig. 3.20

O X

three-leaved rose, 3cosar

(a, 0)

(a, 32

)

(a, 34

)

Fig. 3.21

O X

spiral of Archimedes, ar

Fig. 3.22

OX

hyperbolic spiral, ar

a


3.8 Equation of a Given Locus

a. To determine the equation of a given locus (or curve),

Consider a general point on the curve. That is the point whose coordinates are the variables (x, y) or (r,θ), whichever coordinate system is more convenient.

Express the geometric properties of the curve in an equation relating the variables.

b. The number of points required to find the equation of a curve is equal to the number of independent constants in the equation of the curve.

Fig. 3.23

OX

trumpet, 22

ar

Fig. 3.24

OX

conchoid of Nicomedes, )ba( ,cscbar

a

b

25

Chapter 4The Straight Line

4.1 A Line Parallel to a Coordinate Axis

a. The equation of a line parallel to and at a directed distance a,from the Y-axis, Fig. 4.1, is

b. The equation of a line parallel to and at a directed distance b from the X-axis, Fig. 4.1, is

4.2 General Equation of a Line

a. Every straight line may be represented by a first-degree equation in two variables in the form,

For A = 0, the line is parallel to the X-axis, and forB = 0, the line is parallel to the Y-axis

(4.1)x = a

(4.2)y = b

Fig. 4.1

Y

XO

y = b

x = a

b

a

(4.3)0CByAx


(4.5))xx(xxyy

yy 112

121

(4.6)01yx1yx1yx

22

11

4.3 Point-Slope Form

a. The equation of a line of slope m and passing through a point P1(x1, y1), Fig. 4.2, is written in point-slope form as,

4.4 Two-Point Form

a. If a line is known to pass through two points, P1(x1, y1) and P2(x2, y2), its slope could be determined by equation 1.8,

12

12

xxyy

m

, and the point-slope form could be modified to

the two-point form of the equation of a line,

or in matrix form, the above equation is written as,

Equation 4.6 is derived from the fact that the general (or moving) point P(x, y) forms with the two fixed points P1(x1, y1) and P2(x2, y2) a triangle having an area of zero (see

(4.4))xx(myy 11

P1(x1, y1)

Fig. 4.2

Y

XO

line of slope m

THE STRAIGHT LINE 27

(4.8)0CyBxA

0CyBxA

222

111

and

equation 1.12 of Sec. 1.8) since the three points lie on a straight line.

4.5 Slope-Intercept Form

a. A given y-intercept b means that the line passes through the point Q(0, b), Fig. 4.3. It follows therefore that the equation of a line of slope m and y-intercept b could also be found using the point-slope form (equation 4.4), and simplified to give another form of the equation of a line, known as the slope-intercept form, having the form,

b. To reduce the general equation (equation 4.3) of a line to the slope-intercept form,

Solve for y, resulting in BC

xBA

y . The

coefficient of x is the slope (BA

m ), and the

constant term is the y-intercept (BC

b ).

4.6 Parallel and Perpendicular Lines

a. Two lines represented by the equations,

(4.7)bmxy

Q(0, b)

Fig. 4.3

Y

XO

line of slope m


(4.9)

0BABA

BA

BA

22

11

2

2

1

1

form,matrix inor

are parallel, only if they have equal slopes, such that

or perpendicular, only if the slope of any one of them is equal to the negative reciprocal of the other. That is,

4.7 Concurrence of Three Lines

a. Three lines,

are concurrent (which means that the lines intersect at a common point) if,

4.8 Intercept Form

a. A given x-intercept a, and y-intercept b, means that the line passes through two points, P(a, 0) and Q(0, b), Fig. 4.4. The two-point form (equation 4.5) could be modified to give

(4.10)0BBAAAB

BA

21212

2

1

1 or

(4.11)

0CyBxA

0CyBxA

0CyBxA

333

222

111

and

,

,

(4.12)0cBACBACBA

333

222

111

Q(0, b)

Fig. 4.4

Y

XOP(a, 0)

THE STRAIGHT LINE 29

(4.13)1by

ax

(4.14) sinycosx

another form of the equation of a line called the intercept form,

4.9 Normal Form

a. The equation of a line, located at a perpendicular distance ρfrom the origin to a point R on the line, and angle of inclination β defined as shown in Fig. 4.5, is

The line OR whose length is ρ is called the normal of the line. Depending on the choice of β, a line may be represented by a variety of equations in the normal form.

b. To reduce the general equation of a line to the normal form,

Divide each term by 22 BA , depending on the

sign of B (and noting that 22 BA

Asin

and

22 BA

Bcos

).

Transfer the constant term to the other side of the equation.

R

ρ

line

Fig. 4.5

Y

XOβ


(4.15) )cos(r

(4.16)22

11

BA

CByAxd

4.10 Polar Equation of a Straight Line

a. The polar equation of a straight line, Fig. 4.6, is

where ρ and β are constants defined in the same manner as in Sec. 4.9a.

4.11 Directed Perpendicular Distance of a Line to a Point and Between Two Parallel Lines

a. The directed perpendicular distance d

of a line given in the general form (equation 4.3), 0CByAx , to a point P1(x1, y1), Fig. 4.7, is

where the radical in the denominator takes on the sign of B. If 0B , it follows the sign of A. A positive value of the distance d results if the point lies above the line (or to the right in case of vertical lines), and negative if the point is below the line (or to the left in case of vertical lines).

Fig. 4.6

XO

R(ρ, β)

P(r, θ)

ρ

r

βθ

line in a polar coordinate system

P1(x1, y1)

Fig. 4.7

Y

XO

Ax + By + C = 0

d

THE STRAIGHT LINE 31b. If two lines are parallel, the directed perpendicular distance

d

between them is given by,

where d

is positive if the line with constant term C2 is above the line with constant term C1, and negative if below. See Fig. 4.8.

4.12 Two Conditions Determine a Line

a. A set of two independent conditions is required to find the equation of a line (two points, the slope and a point, the intercepts, etc.) since equation 4.3 or any of the standard forms contain two essential constants that could only be evaluated by two consistent equations.

(4.17)22

21

BA

CCd

Fig. 4.8

Y

XO

Ax + By + C1 = 0

Ax + By + C2 = 0

d

33

(5.1)0FEyDxAyAx 22 (|A|>0)

(5.2)0IHyGxyx 22

(5.3) 222 akyhx

Chapter 5The Circle

5.1 Circles

a. A circle is the locus of a point moving in a plane, in such a way that its distance (called the radius) from a fixed point (called the center) remains constant.

5.2 General Equation of a Circle

a. The general equation of a circle is a special case of the general equation of the second degree (equation 7.6) where A = C and B = 0, having the form,

or alternatively, (after dividing the above equation, through by A),

5.3 Standard Equation of a Circle

a. The standard equation of a circle of radius a, and center at the point C(h, k), Fig. 5.1, is

(x-h)2 + (y-k)2 = a2

C(h, k)

Fig. 5.1

Y

XO

a


(5.4)222 ayx

b. For a circle whose center is at the origin O(0, 0), the standard equation reduces to,

c. To reduce the general equation of a circle to the standard form,

Write the general equation in the form of equation 5.2.

Transpose the constant term to the right. Complete the squares in x and y.

In reducing to the standard form, if the right side is 0a2 ,

the graph is a degenerate circle (a point at (h, k)). If 2a is negative, a graph is impossible.

5.4 Radical Axis

a. The radical axis of two non-concentric circles (or circles having different centers, Fig. 5.2) whose respective equations are,

(5.5)0LKyJxyx

0IHyGxyx22

22

and

x2 + y2 + Gx + Hy + I = 0

C1

Fig. 5.2

X

Y

O C2

P1

P2

P

x2 + y2 + Jx + Ky + L = 0

(G – J)x+(H - K)y+(I - L) = 0, the radical axis

THE CIRCLE 35

(5.6)0)LI(y)KH(x)JG(

(5.7)2cc

2c

2 a)cos(rr2rr

is the straight line represented by the equation,

b. The properties of the radical axis are:

The radical axis of two circles is perpendicular to the line connecting their centers.

Each tangent segment, drawn from a common point on the radical axis of two circles to each of their points of tangency, have equal lengths. From Fig. 5.2, 21 PPPP .

The radical axis contains the common chord of two circles intersecting at two distinct points.

The radical axis is the common tangent of two tangent circles (or two circles intersecting at only one point).

5.5 Polar Equation of a Circle

a. In polar coordinates, a circle is represented by the equation,

where C(rc, θc) is the center, and a is the radius of the circle, Fig. 5.3.

C(rc, θc)

Fig. 5.3

XO

a

r2 + rc2 - 2rrc cos(θc - θ)= a2

rc

θc


5.6 Three Conditions Determine a Circle

a. A set of three independent conditions is required to determine the equation of a circle, whether in the standard or in the general form (the conditions may be three points, three tangents, two points and the radius of the circle, etc.).

37

Chapter 6Special Quadratic Equations in Two Variables. Conic Sections

6.1 Conic Sections

a. Conic sections orconics, Figs. 6.1, 6.2, and 6.3, are defined geometrically as sections made by planes intersecting a right circular cone. It may be a parabola, an ellipse (the circle is a special case), or a hyperbola, depending on the position of the cutting plane. The ellipse and the hyperbola are classified as central conics in contrast to the parabola which has no center, since it only has one vertex (or only one focus).

b. Analytically, a conic section is the locus of a point which moves such that its distance from a fixed point (called the focus) is in constant ratio with its distance from a fixed line (called the directrix), Fig. 6.4.

Fig. 6.2

Ellipse (Cutting plane not parallel to any plane tangent to the cone)

Fig. 6.3

Hyperbola (Cutting plane intersects both

upper and lower nappes)

upper nappe

Fig. 6.1

vertex, V

Parabola (Cutting plane parallel to a plane tangent to

the cone)

lower nappe


(6.1)|LP||FP|

e

The axis of a conic is the line through the focus, perpendicular to the directrix. The latus rectum is the chord through the focus, parallel to the directrix. The vertex is the point where the axis intersects the conic. The focal length(or focal distance) is the distance from the focus to the vertex.

c. The constant ratio mentioned in the preceding section for the analytical definition of the conic, is called the eccentricity e, of the conic. From Fig. 6.4, it is given by,

The conic sections fall into three classes as follows:

If 1e , the conic is a parabola; 1e , the conic as an ellipse; 1e , the conic is a hyperbola.The circle is a special case of the ellipse. That is, as

0e (read as “as e approaches zero”), the ellipse approaches a circle as a limiting shape.

d. Degenerate conics (the point-ellipse, two parallel lines, two coincident lines, and two intersecting lines) are exceptional conic sections, formed when the cutting plane passes through the vertex of the right circular cone.

A

Fig. 6.4

Y

XO

conic section

directrix

focal length, |FV|

latus rectum, |AB|axis of conic

B

L

P(x, y)

focus, F

vertex, V

SPECIAL QUADRATIC EQUATIONS. CONIC SECTIONS 39

(6.2)|LP||FP|

(6.3)FV4AB

6.2 Parabolas

a. A parabola (eccentricity 1e ) is the locus of a point that moves such that its distance from the focus and its distance from the directrix are always equal. That is, from Fig. 6.5,

b. The length of the latus rectum is always four times the focal length, or

6.3 General Equation of a Parabola

a. The general equation of a parabola, a special case of the general equation of the second degree (equation 7.6) which

L

Fig. 6.5

X

Y

OP(x, y)

B

A

vertex, V(h, k)

focus,F

latus rectum, |AB|

directrix

axis of parabola

Parabola with axis parallel to the X-axis and opening to the right


(6.4) 0C0FEyDxCy2

(6.5) 0G0IHyGxy2

(6.7) 0H0IHyGxx2

(6.6) 0A0FEyDxAx2

(6.8) hxa4ky 2

contains no product term (or the xy-term) and only one of the two squared terms, is written as:

If axis is parallel to the X-axis,

or alternatively (after dividing through by the constant of the squared term, C),

If axis is parallel to the Y-axis,

or alternatively (after dividing through by A),

6.4 Standard Equation of a Parabola

a. The standard equation of a parabola with vertex at V(h, k) and focal length aFV , is:


where the right side takes the positive sign if the parabola opens to the right, and negative if it opens to the left.


(6.9) kya4hx 2

(6.10)ax4y2

(6.11)ay4x2

(6.12)sPFPF 21


where the sign of the right side is positive if the parabola opens upward, and negative if it opens downward.

b. For a parabola with vertex at the origin, the standard equation becomes:



c. To reduce the general equation of a parabola to the standard form,

Write the general equation in the alternative forms (equations 6.5 and 6.7).

Transpose the constant term to the right. Complete the square in either y or x.

6.5 Ellipses

a. An ellipse (eccentricity e< 1) is the locus of a point that moves such that the sum of its distances from the two foci(plural of focus), is a constant. That is, from Fig. 6.6,

where s is a constant.


(6.13)sa2VV 21

(6.14)b2WW 21

(6.15)ab2

RRLL2

2121

An ellipse is a closed curve with center at a point on the axis and midway between the foci or between the vertices of the ellipse. The major axis is a segment on the axis bounded by the vertices. Its length is equal to the constant sum s in equation 6.12, or

where a is the length of the semi-major axis. The minor axis is a segment on the line through the center and perpendicular to the major axis, bounded by the points of intersection of this line with the ellipse. Its length is,

where b is the length of the semi-minor axis, and is always less than the length a of the semi-major axis.

b. The length of each latera recta (plural of latus rectum) is,

Fig. 6.6

X

Y

O

a a

bvertex,

V1

P(x, y)

D2D1

L1 R1

W1

vertexV2

center, C(h, k)

Ellipse with horizontal major axis

major axis, |V1V2|

directrices

minor axis, |W1W2|

axis of the ellipse (or principal axis)

latera recta, |L1L2| and |R1R2|

focus, F1 focus, F2


(6.16)ea

CDCD 21

(6.17)aeCFCF 21

(6.18)aWFWFWFWF 22122111

(6.19) CACA,CA0FEyDxCyAx 22

(6.20) ba1b

ky

a

hx2

2

2

2

The distance from the center to each directrix is,

The distance from the center to each focus is,

The distance from a focus to one end of the minor axis is,

6.6 General Equation of an Ellipse

a. The general equation of an ellipse is another special case of the general equation of the second degree, containing no product term and B = 0, having the form,

where CACA means that A and C should have the

same sign. If A = C, the equation becomes the general equation of a circle.

6.7 Standard Equation of an Ellipse

a. The standard equation of an ellipse with center at C(h, k), length of semi-major axis a, and length of semi-minor axis b, is:

For horizontal major axis,


(6.21) ba1b

hx

a

ky2

2

2

2

(6.22) ba1b

y

a

x2

2

2

2

(6.23) ba1b

x

a

y2

2

2

2

(6.24)dPFPF 12

For vertical major axis,

b. If the center of the ellipse is at the origin, the standard equation becomes:

For horizontal major axis (or axis coincident with the X-axis),

For vertical major axis (or axis coincident with the Y-axis),

c. To reduce the general equation of an ellipse to the standard form,


6.8 Hyperbolas

a. A hyperbola (eccentricity e > 1) is the locus of a point that moves such that the absolute value of the difference of its distances from two foci is a constant. That is, from Fig. 6.7,

where d is a constant.


(6.25)da2VV 21

(6.26)b2WW 21

A hyperbola is a curve consisting of two open branches with center also midway between the foci or the vertices of the hyperbola. The transverse axis is a segment of the axis of the hyperbola, bounded by the vertices (analogous to the major axis of an ellipse), with length equal to the constant difference in equation 6.24, or

where a is the length of the semi-transverse axis. The conjugate axis is a segment on the line through the centerand perpendicular to the transverse axis, bounded by the points of intersection of this line with the segments of the same length and parallel to the transverse axis having endpoints on each asymptote of the hyperbola. The length of the conjugate axis is,

where b is the length of the semi-conjugate axis and may be greater than, equal to, or less than that of the transverse

Fig. 6.7

X

Y

Ob

b

a a

L1

L2

focusF1

focusF2

R1

R2

vertexV1

vertexV2

D1 D2

W1

W2

centerC(h, k)

asymptotes

directrices

conjugate axis, |W1W2|

transverse axis, |V1V2|

axis of hyperbola (or principal axis)

Hyperbola with horizontal tranverse axis

latera recta, |L1L2| and |R1R2|

P(x, y)


(6.28)ea

CDCD 21

(6.29)aeCFCF 21

(6.30) CACA,CA0FEyDxCyAx 22

axis. The lines or the prolonged diagonals of the rectangle whose midsections are the transverse and conjugate axes, and passing through the center of the hyperbola are called asymptotes of the hyperbola.

b. The length of each latera recta is,

The distance from the center to each directrix is,

The distance from the center to each focus is,

6.9 General Equation of a Hyperbola

a. The general equation of a hyperbola is also a special case of the general equation of the second degree, containing no product term and B = 0, having the form,

where (-A∙C = A∙C) means that A and C should have unlike signs.

6.10 Standard Equation of a Hyperbola

a. The standard equation of a hyperbola with center at C(h, k), semi-transverse axis a, and semi-conjugate axis b, is:

(6.27)ab2

RRLL2

2121


(6.31)

1b

ky

a

hx2

2

2

2

(6.32) 1

b

hx

a

ky2

2

2

2

(6.33)1b

y

a

x2

2

2

2

(6.34)1b

x

a

y2

2

2

2

For horizontal transverse axis,

For vertical transverse axis,

b. If the center of the hyperbola is at the origin, the standard equation becomes:

For horizontal transverse axis (or axis coincident with the X-axis),

For vertical transverse axis (or axis coincident with the Y-axis),

c. To reduce the general equation of the hyperbola to the standard form,


6.11 Asymptotes of a Hyperbola

a. The equations of the asymptotes of a hyperbola are obtained by changing the right side of the equation of the hyperbola in the standard form (equations 6.31 to 6.34) to zero. For


(6.35)

0b

ky

a

hx2

2

2

2

(6.36)

0b

kya

hx

0b

kya

hx

and

example, the asymptotes of the hyperbola given by equation 6.31 is,

or these are the two lines,

6.12 Conditions Describing a Conic Section

a. A set of three independent conditions are necessary to determine the equation of a parabola.

b. A central conic requires a set of four independent conditions to determine its equation.

6.13 Polar Equation of a Given Conic Section

a. To find the polar equation of a given conic section of known geometric properties:

Choose the pole O as focus, Fig. 6.8. Let P(r, θ) be a general point on the conic. Let R be at one end of the latus rectum and let

OR .

Fig. 6.8

XO

P(r, θ)

S

B

C

R

l

directrix

polar axisfocus,

r

θ

conic section

SPECIAL QUADRATIC EQUATIONS. CONIC SECTIONS 49 Drop perpendiculars to the polar axis and to the

directrix from P and R. From Fig. 6.8

OCRSPB . But we also have PBr

RSe

based on the definition of eccentricity (Sec 6.1c), so that we finally have the polar equation of the given

conic in the form, cosree

r .

Simplify the polar equation by solving for r as

cose1r

.

6.14 Tracing a Conic Section

a. To trace the locus of a conic:

Express the equation in one of the standard forms. Determine the geometric properties necessary to

draw the curve. It may be some or all of the following: the length of the latera recta, the location of the vertices, the foci, and the center, the focal length, the lengths of the major and minor axes for an ellipse or the transverse and conjugate axes for a hyperbola, and etc.

51

(7.1)ky'yhx'x

and

Chapter 7Transformation of Coordinates. The General Quadratic in Two

Variables

7.1 Translation of Axes in a Plane

a. A translation of axes in the XY-plane happens if a new set of coordinate axes X’ and Y’, with its own origin at the point O’(h, k), are drawn parallel to the original X and Y axes respectively, Fig. 7.1.

b. If a point P has the coordinates (x, y) with respect to the original X and Y axes, and coordinates (x’, y’) if referred to the new set of coordinate axes X’ and Y’, then the two sets of coordinates are related by the following set of transformation equations for translation,

7.2 Rotation of Axes in a Plane

a. A rotation of axes in the XY-planehappens if a new set of coordinate axes X’ and Y’ are rotated through an angle θ about the origin, Fig. 7.2.

)y',(x'

y)(x,P

Fig. 7.1

X

Y

O

X’

Y’

x

yx’

y’

O’(h, k)

Fig. 7.2

X

Y

O

X’

Y’

θ

x’

y’x

y

)y',(x'

y)(x,P


(7.2)

cosysinx'y

sinycosx'x and

(7.3)

yx

cossinsincos

'y'x

(7.4)

cossinsincos

T

(7.5)

cossinsincos

TT T1

(7.6)0FEyDxCyBxyAx 22

b. If a point P has the coordinates (x, y) with respect to the original X and Y axes, and coordinates (x’, y’) with respect to the new X’ and Y’ axes, then the two sets of coordinates are related by the following set of transformation equations for rotation,

or in matrix form,

where the matrix,

is called the coordinate transformation matrix for rotation. The matrix T is orthogonal, which means that its transpose TT is at the same time its inverse T-1. That is,

7.3 The General Quadratic in Two Variables

a. The general quadratic equation (or the general equation of the second degree) in x and y, having the form,

TRANSFORMATION OF COORDINATES. THE GENERAL QUADRATIC 53

(7.7)CA

B2tan

represents a conic section whose axes are inclined from the coordinate axes at an angle θ given by the formula,

If 0B C,A , then θ = 45o.B = 0, the equation represents a conic whose axes

are parallel with the coordinate axes.

7.4 Tracing the Curve of a General Quadratic

a. To trace the curve of a general quadratic in two variables:

Determine tan2θ by equation 7.7, then find cosθ and

sinθ. The trigonometric identities, 2cos21

cos

and 2

2cos1sin

, are useful.

Substitute the values of cosθ and sinθ, to equation 7.2 or equation 7.3, and solve for x and y, in terms of the new coordinates x’ and y’. That is,

'y'x

cossinsincos

yx

.

Substitute these expressions for x and y to the original general quadratic.

Simplify, to get an equation referred to the new axes. The resulting equation will not contain the product term x’y’ and could be traced by the methodpresented in Sec. 6.14.

7.5 Discriminant of a Conic

a. The locus of a general quadratic equation could be determined using the discriminant AC4B2 , as follows:

If 0AC4B2 , the locus is a parabola,

0AC4B2 , the locus is an ellipse,

0AC4B2 , the locus is a hyperbola.


b. The general quadratic equation has degenerate cases or exceptional forms also (a point, two parallel lines, two coincident lines, two intersecting lines, or it may have no locus at all).

55

Chapter 8Tangents and Normals to Conics

8.1 Tangents and Normals

a. A tangent to a conic is a line that intersects the conic at one and only one point. The one and only one point of intersection is called the point of tangency, Fig. 8.1.

b. A normal to a conic is a line that intersects a conic at a point where the line is perpendicular to the tangent to the conic at that point.

c. A subtangent of a point on a conic is a segment cut off from the X-axis, bounded by the point of intersection of the tangent through the point and the X-axis, and the foot of the perpendicular to the X-axis dropped from the point. A subnormal of a point on a conic is a segment cut off from the X-axis, bounded by the foot of the perpendicular to the X-axis dropped from the point, and the point of intersection of the normal through the point and the X-axis. From Fig. 8.1, the segments |TA| and |AN| are the subtangent and the subnormal respectively, of the point P1 on the conic.

Fig. 8.1

X

Y

O

point of tangencyP1(x1, y1)

T(x2, 0) A(x, 0) N(x3, 0)

subnormal, |AN|subtangent, |TA|

tangent at P1

normal at P1

the conicAx2 + Bxy + Cy2 + Dx + Ey + F = 0


(8.1) 0F2yyExxDyCy2xyyxBxAx2 111111

8.2 Tangent and Normal Through a Given Point on the Conic

a. The equation of a tangent to a conic represented by the general equation of the second degree (equation 7.6),

0FEyDxCyBxyAx 22 , at the point P1(x1, y1) on the conic, is

b. The equation of a normal to a conic at any point P1(x1, y1) on the conic could be determined using the point-slope form of the equation of a line (equation 4.4), since we already know one point, P1, on the normal and we could find the slope of the normal if we know the slope of the tangent at P1. The slope of the normal to the conic at the point P1 is equal to the negative reciprocal of the slope of the tangent at P1 (see Sec. 1.6b).

8.3 Poles and Polars of a Conic

a. The polar of a point (where the point is called the pole of the polar) exterior to a conic, is the line through the points of tangency of the tangents to the conic drawn through the point (or pole). Conversely, the pole of a straight line secant to a conic is the point of intersection of the tangents at the secant points, Fig. 8.2.

Fig. 8.2

X

Y

O

polar of P0

pole P1(x1, y1)

polar of P1

tangent at A

tangent at B

A(a, b)

B(c, d)

CD

pole P0(x0, y0)

TANGENTS AND NORMALS TO CONICS 57

(8.3))xx(myy 00

(8.2) 0F2yyExxDyCy2xyyxBxAx2 000000

b. The polar of a pole on the polar of a second pole passes through the second pole, as shown also in Fig. 8.2. That is, the polar of P1 (where P1 is a point on the polar of P0), passes through P0. The polar of a pole, which is a point on a conic, is the tangent to the conic at that point.

c. To obtain the equation of the polar of a pole P0(x0, y0):

Replace the coordinates of the given pole in the equation of the general tangent to the conic given by equation 8.1. The linear equation obtained is now the equation of the polar of the given pole P0 having the form,

8.4 Tangent to a Conic Through a Given External Point

a. To find the equation of the tangent to a conic, where the tangent passes through a given point P0(x0, y0) not on the conic:

Find the equation of the polar of the given point considered as pole P0(x0, y0) by equation 8.2.

Find the intersections of this polar with the conic. Find the equation of the tangents through these

intersections using equation 8.1. These intersections are already points on the conic and the tangents through these points are guaranteed to pass through the external point P0, as presented in Sec. 8.3.

Or, an alternative procedure is to,

Write the equation of the family of lines through the given point P0(x0, y0) in the point-slope form (equation 4.4) as,

There is a unique value for the slope m, of a line in the family that is to be tangent to the conic.


(8.4)0cbxax2

(8.5)0ac4b2

(8.6)Bmxy

Solve for y in equation 8.2 and substitute to the equation of the conic resulting in a quadratic equation of the form,

where the coefficients a, b, and c are all functions of the slope m of the lines in the family that intersect the conic (these includes the tangents and the secants).

Solve for m by setting,

The equation above is now the condition to be satisfied to determine the unique value for the slope m of a line in the family that is tangent to the given conic since it requires the quadratic equation (equation 8.4) to have equal roots or for the line to intersect the conic at only one point. Equation 8.5 yields two different slopes, which are the slopes of the tangents to the conic passing through P0(x0, y0).

Find the equation of the tangents of slope m using equation 8.3.

8.5 Tangent of Given Slope

a. To find the equation of the tangent to a conic, where the tangent is known to have a slope m:

Write the equation of the family of lines of slope m in the slope-intercept form (equation 4.7) as,

There is a unique value for the y-intercept B of a line in the family that is to be tangent to the conic.

Substitute y to the equation of the conic resulting in a quadratic equation of the same form as equation 8.4 but the coefficients a, b, and c, are now functions of the y-intercept B.

TANGENTS AND NORMALS TO CONICS 59 Solve for B using the same condition of tangency

given by equation 8.5. Find the equation of the tangents of y-intercept B

using equation 8.6.

61

(9.1))t(gy)t(fx

and

(9.2)btyyatxx

1

1

and

(9.3)

sinrkycosrhx

and

Chapter 9Parametric Equations

9.1 Parametric Equations of a Curve

a. The parametric equations of a curve are the rectangularcoordinates of a general point on the curve, expressed as functions of a common variable called the parameter. It has the form,

where t is the parameter. Eliminating the parameter gives the algebraic equation of the curve.

b. The parametric equations of a given curve are not unique. A curve may be represented by an unlimited number of parametric equations.

9.2 Parametric Equations of Some Curves

a. A set of parametric equations for the straight line (Ch. 4) are of the form,

where (x1, y1), are the coordinates of any point on the line,

ab , is equal to the slope of the line, and

t, is the parameter.

b. A set of parametric equations for a circle (Ch. 5) are of the form,

where Φ is the parameter.


(9.4)

sinbkycosahx

and

(9.5)

tanbkysecahx

and

(9.6)at2ky

athx 2

and

c. A set of parametric equations for an ellipse with horizontal major axis (see Sec. 6.7) have the form,

where β is the parameter.

d. A set of parametric equations for a hyperbola with horizontal transverse axis (see Sec. 6.10) have the form,

where α is the parameter.

e. A set of parametric equations for a parabola opening to the right (see Sec. 6.4) are of the form,

where t is the parameter.

9.3 Tracing a Given Set of Parametric Equations

a. Point-plotting (see Sec. 3.6b) may be used to trace the curve of a given set of parametric equations. Pairs of values for x and y are computed by assigning values to the parameter and these pairs of coordinates are plotted and the curve is drawn through the plotted points.

b. Eliminating the parameter results in the algebraic equation of the curve, which could be graphed by the method presented in Sec.3.6d.

c. A combination of point-plotting and elimination of parameter oftentimes simplify the tracing of a given set of parametric equations.

63

Chapter 10Transcendental Functions

10.1 Trigonometric Functions

a. The trigonometric functions are periodic functionswhich mean that the values of the function are repeated for all integral multiples of a constant increment of the independent variable. Figs. 10.1 to 10.3 are graphs of the trigonometric functions, tanx yand cosx, ysinx,y respectively.

period

amplitude

y = sinxcrest

trough

O1/2

1

π/2 π 3π/2 2π 5π/2

-1/2

-1

-π/2-π-3π/2-2π-5π/2

Fig. 10.1

X

Y

y = cosx

O1/2

1

π/2 π 3π/2 2π 5π/2

-1/2

-1

-π/2-π-3π/2-2π-5π/2

Fig. 10.2

X

Y


(10.1))px(f)x(f

The period of the trigonometric function is the smallestinterval after which the function takes the same values again. That is a constant p such that,

for all values of x. The amplitude is the absolute difference between the maximum value of a periodic function and its mean.

b. The period and amplitude of the six basic trigonometric functions are tabulated below.

Function Period Amplitude

asinkx, acoskx, aseckx, acsckxk2

|a|

atankx, acotkxk

none

Oπ/2 π 3π/2 2π 5π/2-π/2-π-3π/2-2π-5π/2

Fig. 10.3

X

Yy = tanx

asymptotes

TRANSCENDENTAL FUNCTIONS 65

(10.2)kxsinay

10.2 Congruence and Shifting

a. The graphs of,

and

placed on the same coordinate axes, Fig. 10.4, are congruent except for position since both curves have the same general

characteristics (an amplitude of |a| and a period of k2 ), but

different first wave-crest coordinates. That is, the curve of bxksinacy has points shifted by an amount b

horizontally to the right, and by an amount c vertically up, with respect to the curve of kxsinay .

10.3 Tracing by Composition of Ordinates

a. To trace the curve of a function,

(10.3) bxksinacy

(10.4))x(fy

X

Y

Fig. 10.4

first wave-crests of each wave

π/k 3π/2k 2π/kπ/2kO

-π/2k-π/k-2π/k

a

2a

-a

-2a

(π/2k, a)

ca,bk2

y – c = asink(x - b)

y = asinkx

-3π/

Y

X


(10.5))x(f)x(f)x(f)x(f n21

(10.6))x(f)x(f)x(fy n21

(10.7))1a(),0a(ay x

by composition of ordinates:

Write the function of x as,

where the curves of (x)f yand , (x),f y(x),fy n21 , could be drawn

easily. The ordinate of a point in the function f(x)y corresponding to each value of x is given

by,

Plot selected pairs of coordinates and draw the curve through the plotted points.

10.4 Exponential Functions

a. The exponential function is defined by,

The constant a is called the base. Fig. 10.5 shows the graph of the exponential function.

y = ax

Fig. 10.5

X

Y

O

(0,1)


(10.8)718281828.2n11Lime

n

n

(10.9)1a0

(10.10)

1a0

1aaLim x

x if if

(10.11)

1a

1a0aLim x

x if if

(10.12))0a(cay bx

(10.13)2

eexsinhxx

b. An important base for the exponential function is e, also called the Euler number, defined to have the value,

c. The exponential function has the following properties:

xa is always positive.

There is no symmetry.

d. The generalized exponential function has the form,

10.5 Hyperbolic Functions

a. The hyperbolic functions (the hyperbolic sine or sinh, the hyperbolic cosine or cosh, and the hyperbolic tangent or tanh) are special exponential combinations expressed as follows:


(10.14)2

eexcoshxx

(10.15)xx

xx

ee

eexcoshxsinhxtanh

Their respective reciprocals are,

Figs. 10.6 to 10.8 are some graphs of the hyperbolic functions.

(10.16)xsinh1hxcsc

(10.17)xcosh1hxsec

(10.18)xsinhxcosh

xtanh1xcoth

y = sinhx

Fig. 10.6

X

Y

O

y = coshx

Fig. 10.7

X

Y

O

(0,1)


(10.19))0a(),xa(xlogy ya

(10.20)xlogyxlogy 10 simply or

b. If the trigonometric functions are defined on the unit circle

1yx 22 , the hyperbolic functions are defined on the

hyperbola 1yx 22 .

10.6 Logarithms

a. The logarithm of a positive number x, to the base a (written as xloga ), is the

exponent y to be placed on a to yield x. That is,

Fig. 10.9 is the graph of the logarithmic function.

b. Two bases of particular importance are 10 (called the common or Briggsian logarithm), written as,

y = tanhx

Fig. 10.8

X

Y

O-1

1

y = logax

Fig. 10.9

X

Y

O(1,0)


(10.21)xlnyxlogy e simply or

(10.22)01loga

(10.23)0loga

(10.24) x)x(fgx)x(gf and

and e (called the natural or Napierian logarithm), having the form,

c. The logarithmic function has the following properties:

Negative numbers have no logarithms.

Numbers between 0 and 1 have negative logarithms.

Numbers greater than 1 have positive logarithms.

As the number increases, its logarithm also increases.

10.7 Inverse Functions

a. Two functions, f(x) and g(x), are inverse functions if the domain of f(x) is the range of g(x), if the domain of g(x) is the range of f(x), and if for every x in the appropriate domains,

b. The logarithmic function xlogy a and the exponential

function xay are inverse functions since if we let,x

a ag(x)andxlogf(x)

then,

xa)x(logg)x(fg

xalogafg(x)fxalog

a

xa

x

TRANSCENDENTAL FUNCTIONS 71c. Geometrically, two inverse functions are reflections of each

other through the line xy , Fig. 10.10.

y = logax

Fig. 10.10

X

Y

O

(0,1)

y = ax y = x

(1,0)

73

(11.1)0)c,y,x(f

(11.2)0)y,x(cg)y,x(f

Chapter 11Families of Plane Curves.

Curve Fitting

11.1 A Family of Curves

a. If an equation in x and y involving an arbitrary constant c is written as,

one value of c represents one definite curve of the equation. As c assumes all real values, we obtain a one-parameter family or system of curves. The arbitrary constant c is called the parameter.

b. The equation cx32y for example represents a one-

parameter family of straight lines of slope 32 . The equation

222

21 a)cy()cx( on the other hand represents a two-

parameter family of circles of radius a.

c. Exceptional forms in the family are curves of a particular value of the parameter, which are different in kind from the other curves belonging to the family.

d. If equation 12.1 is linear or first degree in the parameter c (which means that there is only one corresponding value of c for every (x, y) pair), then one and only one curve of the family passes through a chosen point on the plane.

11.2 A Family of Curves Through an Intersection

a. If f(x, y) = 0 and g(x, y) = 0 are two intersecting curves, the equation,


(11.3)0)CyBxA(cCyBxA 222111

represents a family of curves passing through all the intersections of the original curves.

b. A family of lines for example, through the intersection of the lines 0CyBxA 111 and 0CyBxA 222 has the form,

11.3 Curve Fitting

a. Curve fitting is the process of finding the equation of a curve that best fit or represent a given set of related data from an observation or experiment conducted. The equation obtained is called an empirical equation or empirical formula and is just an approximation to the true relation between the variables. Fig. 11.1 shows the plotted points of the tabulated data below relating gasoline consumption C in L/km with the speed of the automobile in km/hr, and the possible best-fitting curves (straight lines in this case), passing quite close to each point.

Speed of Automobile, S (km/hr)

20 30 40 50 60 70 80

Gasoline Consumption,

C (L/km)0.020 0.022 0.025 0.026 0.028 0.029 0.031

The vertical distance r, Fig. 11.1, from a plotted point, say (30, 0.022), to a line, say the line L3, is called the residual

(30, 0.022)

(20, 0.02)

(40, 0.025)(50, 0.026)

(60, 0.028)(70, 0.029)

(80, 0.031)

L2

L1

Fig. 11.1

S

C

40 km/hr 60 km/hr 80 km/hr

0.02 L/km

0.03 L/km

L3

r

FAMILIES OF PLANE CURVES. CURVE FITTING 75

(11.4)BMxy

corresponding to the point. Analytically, it is the difference between the observed value and the computed value from the equation of the line L3, if L3 is chosen to be the line that fits or describes the given set of data.

11.4 Line of Best Fit. Method of Least Squares

a. One way of finding the equation of the “best-fitting line”,

is the method of selected points. This method is dependent on the choice and judgment of the inquirer since it is done simply by selecting only two points from a given setof data, and then finding the equation of the line through these two points, using the two-point form of the equation of a line (equation 4.5). For example, choosing the line L2 to be the line that passes through the points (20, 0.02) and (50, 0.026), we find its equation using equation 4.5 as,

02.0004.0S0002.002.0)20S(0002.0C

)20S(2050

02.0026.002.0C

or

From the table, the observed value of gasoline consumption C for a speed S of 40 is 0.025, but equation 11.5 gives a different value of C = 0.024 for the same speed of 40. In this case, the residual corresponding to the point (40, 0.025) is 0.001.

b. The method of averages, unlike the method of selected points, is a way of finding the “best-fitting line” based on a mathematical procedure such that the line determined makes the algebraic sum of the residuals of each point zero. That is, to make the line BMxy fit the n points (x1, y1), (x2, y2), (x3, y3), … (xn, yn), by the method of averages, the constants M and B must be solved from the simultaneous equations,

(11.5)016.0S0002.0C


(11.6)

n

1mi

i

n

1mi

i

m

1i

i

m

1i

i

)mn(BxMy

BmxMy

(11.7)

n

1i

n

1i

i2

i

n

1i

ii

n

1i

i

n

1i

i

xBxMyx

BnxMy

where the first equation is taken from the first group of m points, (x1, y1), (x2, y2), (x3, y3), … (xm, ym), while the second equation is taken from the second group, which is the remaining m-n points, (xm+1, ym+1), (xm+2, ym+2), (xm+3, ym+3), … (xn, yn). If the groups are of equal size, that is

2n

m or nearly so, best results are obtained.

c. The method of least squares, one of the most reliable methods of curve fitting, is used to determine the best-fitting line BMxy that best fits a given set of data composed ofn points (x1, y1), (x2, y2), (x3, y3), … (xn, yn), such that the sum of the squares of the residuals is a minimum. The constants M and B are obtained by solving the simultaneous equations,

Fig. 11.1 shows the line L1,

obtained by the method of least squares.

(11.8)0169286.0S000178571.0C

FAMILIES OF PLANE CURVES. CURVE FITTING 77

(11.9)bxaey

(11.10)alnbxyln

(11.11)BMxz

11.5 Nonlinear Curves of Best Fit

a. If the best-fitting curve is nonlinear, or if the points do not follow a straight line, but rather an equation of some other form, exponential for example,

this equation is made linear first by performing legitimate mathematical operations. In this case, taking the natural logarithms of both sides of the equation resulting in the form,

This equation is now linear in x and lny. If we let lnyz , bM , and lnaB , we would get a linear equation

analogous to equation 11.4 in the form,

where the values of M and B are determined by one of the methods presented in the preceding section and the values of a and b for the best-fitting exponential curve (equation 11.9) follows immediately.

79

SOLID ANALYTIC GEOMETRY

Chapter 12Rectangular Coordinates in Space

12.1 Space Rectangular Coordinates

a. The location of a point in space may be determined by its distances from three mutually perpendicular planes, forming a space (or three-dimensional) rectangular (or Cartesian) coordinate system, Fig. 12.1.

b. A space rectangularcoordinate system is formed bydrawing the Y and Z axes in oneplane, and the X-axis as perpendicular to this plane. These three coordinateaxes also intersect at point O called the origin. The pairs of Y and Z axes, the X and Z axes, and the X and Y axes, respectively form the YOZ or YZ-plane, the XOZ or XZ-plane, and the XOY or XY-plane. These three mutually perpendicular planes are called the coordinate planes. Space is divided by these coordinate planes into eight regions called octants. Since the X-axis is impossible to be drawn perpendicular to the page, it is just drawn approximately 1350 with the Y-axis, and the angle between the Y and Z-axis is made somewhat greater than 900. This is just an aid in visualizing space figures since the X, Y, and Z axes are really defined to be mutually perpendicular. Points in the first octant have coordinates which are all positive.

Fig. 12.1

1

Y

X

Z

O

2

3

…

+∞

1 2 3 …+∞1

23

…+∞

- ∞

- ∞

- ∞

P

a

b

c

80 SOLID ANALYTIC GEOMETRY

(12.1) 2122

122

12 zzyyxxd

c. A right-handed space rectangular coordinate system, such as Fig.12.1, is formed such that if the X-axis is rotated in the direction of the Y-axis, a right-handed screw will move in the direction of the Z-axis. Other examples of right-handed space rectangular coordinate systems are shown in Fig. 12.2.

d. The position of a point in space may now be determined by its space rectangular coordinates (or the combined x, y, and z-coordinate of the point). Its x-coordinate, y-coordinate, and z-coordinate, is its distance from the YZ, XZ, and XY-plane respectively. Point P for example, Fig. 12.1, is located at (a, b, c).

e. The notation P(x, y, z), where x, y, and z are variables, means that a point P in space has an x-coordinate x, a y-coordinate y, and a z-coordinate z, in a space rectangular coordinate system.

12.2 Distance Between Two Space Points

a. The distance |d| between two points in space, P1(x1, y1, z1) and P2(x2, y2, z2), is given by,

12.3 Division of a Line Segment in Space. Midpoint

Fig. 12.2

1

X

Z

Y

O

2

3

…

+∞

1 2 3 …+∞1

23

…+∞

- ∞

- ∞

- ∞

1

Z

Y

X

O

2

3

…

+∞

1 2 3 … +∞12

3…

+∞

- ∞

- ∞

- ∞

RECTANGULAR COORDINATES IN SPACE 81

(12.2) 121

121

121

zzkzz

,yykyy

,xxkxx

and

(12.3)

cosncosm

cosl

and

(12.4)1nml 222

a. If P(x, y, z) is a point that divides the line segment joining P1(x1, y1, z1) and P2(x2, y2, z2) such that 211 PPkPP , from equation 1.4, then the coordinates (x, y, z) of P are,

b. If P lies in the extension of segment P1P2 in either direction, or if P is the midpoint of the segment, equation 12.2 still applies as with the two-dimensional or planar case discussed in Sec. 1.4.

12.4 Direction Angles and Direction Cosines

a. The direction anglesare the angles α, β, and γ, that a directed line segment P1P2 makes with a line parallel to the positive X, Y, and Z axes respectively, Fig. 12.3.

b. The direction cosinesl, m, and n, are the cosines of the direction angles expressed as,

The sum of the squares of the direction cosines is equal to 1. That is,

Fig. 12.3

Y

X

Z

O

P1(x1, y1, z1)

P2(x2, y2, z2)

αβ

γ ρ


(12.5) 2122

122

1221 zzyyxxPP

(12.6)222 NMLk

(12.7)212121 nnmmllcos

If the positive sense of a directed line segment is reversed, the signs of the direction cosines are changed since the original direction angles will be replaced by their supplements.

c. The radius vector ρ of any point P2 from a point P1, Fig. 12.3, is the directed line segment P1P2 whose magnitude |ρ| (or |P1P2|) is given by the distance formula as,

d. Direction numbers are products of the direction cosines by any number k. If direction numbers L, M, and N are given such that nk,N and mk,M ,lkL then k could be solved from,

e. The quantities ),z(z and ),y(y ),x(x 121212 given in equation 12.5, form one set of direction numbers called components (or directed projections) of the radius vector ρ (or P1P2), on the X, Y, and Z axes respectively. Each component along the X, Y, and Z axes is equal to the direction cosine l, m, and n respectively, each multiplied by the radius vector ρ.

12.5 Angle Between Two Space Lines

a. The cosine of the angle θ between two lines in space, L1 with direction cosines l1, m1, n1, and L2 with direction cosines l2, m2, n2, is given by,

If the two lines L1 and L2 does not intersect, the angle θ given above is the angle between two lines passing through the origin and parallel to L1 and L2 respectively.

RECTANGULAR COORDINATES IN SPACE 83

(12.9)0nnmmll 212121

12.6 Parallel and Perpendicular Space Lines

a. Two lines L1 and L2, with direction cosines l1, m1, n1, and l2, m2, n2, respectively, are parallel if and only if,

b. Two lines L1 and L2, with direction cosines l1, m1, n1 and l2, m2, n2, respectively, are perpendicular if and only if,

(12.8)1nnmmll 212121

85

Chapter 13Cylindrical and Spherical

Coordinates

13.1 Cylindrical Coordinates

a. The position of a point P in space may also be described using three variable quantities (r, θ, z) called the cylindrical coordinates of P, where r and θ are the polar coordinates of the projection Q of P on the XY-plane, and z is the distance of P, either above or below the XY-plane (this is the same as the space Cartesian z-coordinate), Fig. 13.1. This is called the cylindrical coordinate system, an extension of the planar coordinate system for the three-dimensional case.

13.2 Spherical Coordinates

a. Another extension of the planar polar coordinate system that is used to locate the position of a point P in space is the spherical coordinate system, Fig.13.2. Every point P isassociated with the three variable quantities (ρ, θ, φ)called the spherical coordinates of P, where ρ is the radius vector OP (from the origin O to the point P), θ is the angle from the positive X-axis to the projection OQ of OP on the XY-plane, and φ isthe angle from the positive Z-axis to OP.

Fig. 13.1

Y

X

Z

O

θ

r

z

P

Q

Fig. 13.2

Y

X

Z

O

θ

P

Q

ρφ


(13.3) cosz ,sinsiny ,cossinx

(13.4)

222

222

zyx

zarccos

,xy

arctan

,zyx

13.3 Relations Between Rectangular, Cylindrical, andSpherical Coordinates

a. If (x, y, z) and (r, θ, z) are the rectangular and cylindrical coordinates respectively of the same point in space, then we have the following relations,

and

where the radical for obtaining r in the last equation follows the sign of x. If 0x , it follows the sign of y, and a value of

2 is immediately assigned to θ.

b. If (x, y, z) and (ρ, θ, φ) are the rectangular and spherical coordinates of the same point in space, then they are related by,

and

c. If (r, θ, z) and (ρ, θ, φ) are the cylindrical and spherical coordinates of the same point in space, then,

(13.1)zz,sinry,cosrx

(13.2)zz ,xy

arctan ,yxr 22

(13.5) cosz,,sinr

CYLINDRICAL AND SPHERICAL COORDINATES 87

(13.6)

zr

arctan,,zr 22

and

89

Chapter 14Surfaces and Space Curves

14.1 Locus of an Equation in Three Variables

a. In the three-dimensional space, the locus of an equation, called a surface, contains those points, and only those points, whose coordinates satisfy the equation.

14.2 Symmetry of Surfaces

a. A surface is symmetric with respect to a coordinate plane if for every point of the surface on one side of the plane, there corresponds an image point on the opposite side of the plane. A surface is

symmetric with respect to the YZ-plane,if its equation remains unchanged whether x is replaced by –x.

symmetric with respect to the XZ-plane,if its equation remains unchanged whether y is replaced by –y.

symmetric with respect to the XY-plane,if its equation remains unchanged whether z is replaced –z.

b. A surface is symmetric with respect to a coordinate axis if for every point of the surface on one side of the axis, there corresponds an image point on the opposite side of the axis. A surface is

symmetric with respect to the X-axis,if its equation remains unchanged whether both y and z are replaced simultaneously by –y and –z respectively.

symmetric with respect to the Y-axis,if its equation remains unchanged whether both xand z are replaced simultaneously by –x and –z respectively.


symmetric with respect to the Z-axis,if its equation remains unchanged whether both x and y are replaced simultaneously by –x and –yrespectively.

14.3 Intercepts of a Surface. Sections and Traces

a. The x, y, and z intercepts of a surface are the x, y, and z-coordinate respectively, of the points of intersection of the surface with the respective X, Y, and Z-axis. To find the

x-intercept, solve for x in the equation of the surface,

0z) y,f(x, , with y and z set to zero.

y-intercept, solve for y in the equation of the surface,

0z) y,f(x, , with x and z set to zero.

z-intercept, solve for z in the equation of the surface,

0z) y,f(x, , with x and y set to zero.

b. A cross-section (or simply section) of a surface is a planecurve at the intersection of the surface with a plane, Fig. 14.1.

Fig. 14.1

Y

X

Z

O

cutting planesurface

section

SURFACES AND SPACE CURVES 91To find the equation of a section of a surface,

cut by a plane parallel to the XY-plane (or the plane kz ), replace z by k in the equation of the surface to

get an equation in the two variables x and y. This is the equation of the section, which is a curve on the plane kz .

cut by a plane parallel to the XZ-plane (or the plane ky ), replace y by k in the equation of the surface to

get an equation in the two variables x and z. This is the equation of the section, which is a curve on the plane ky .

cut by a plane parallel to the YZ-plane (or the plane kx ), replace x by k in the equation of the surface to

get an equation in the two variables y and z. This is the equation of the section, which is a curve on the plane kx .

c. If the plane intersecting a surface is a coordinate plane, the section is called a trace. To find the

XY-trace (or the section of a surface cut by the XY-plane 0z ),

replace z by zero in the equation of the surface resulting in an equation in the variables x and y, which is a curve in the XY-plane.

XZ-trace (or the section of a surface cut by the XZ-plane 0y ), replace y by zero in the equation of the surface

resulting in an equation in the variables x and z, which is a curve in the XZ-plane.

YZ-trace (or the section of a surface cut by the YZ-plane 0x ), replace x by zero in the equation of the surface

resulting in an equation in the variables y and z, which is a curve in the YZ-plane.


14.4 Tracing Surfaces by Parallel Plane Sections

a. To trace (or draw or graph) a surface using some of its sections parallel to the coordinate planes,

Test the surface for symmetry. Find the intercepts on the coordinate axes. Determine the traces. The trace could be drawn

using the methods discussed in the previous chapters, since it is a plane curve.

Examine some sections parallel to each coordinate plane and select a set of sections that is easy to draw.

Sketch the surface.

14.5 A Surface of Revolution

a. A surface of revolution is a surface generated by rotating a curve, called the generating curve, about a straight line, called the axis of revolution, Fig. 14.2.

Right sections (or parallels) are sections of the surface of revolution, made by planes perpendicular to the axis of revolution. The right sections are circles whose centers are on the axis of revolution. Meridians are the sections of the surface of revolution, made by planes through the axis of revolution. The meridians are equal curves since they are just the generating curve in successive positions.

Fig. 14.2

Y

X

Z

O

a surface of revolutiongenerated by revolving a curve about the X-axis

generating curve

right sections or parallels

meridians

SURFACES AND SPACE CURVES 93

(14.2))y(fzx 22

(14.3))z(fyx 22

b. The reduced (or simplified) equations of surfaces of revolution with their axis of revolution coinciding with a coordinate axis are:

axis of revolution coinciding with the X-axis,

axis of revolution coinciding with the Y-axis,

axis of revolution coinciding with the Z-axis,

14.6 Cylindrical and Conical Surfaces

a. A cylinder is a surface generated by a moving line, called the generator (or ruling), that intersects all the points of a fixed plane curve, called the directing curve, but always remaining parallel to its initial position, Fig. 14.3.

If the directing curve has a geometrical center, such as a circle, an ellipse, etc., the line through the center parallel to

Fig. 14.3

YX

Z

O

directing curve

generator or ruling

(14.1))x(fzy 22


(14.4)0)z,y(f

(14.5)0)z,x(f

(14.6)0)y,x(f

the generators is the axis of the cylinder. The right sections (or parallels) of a cylinder are sections on planes perpendicular to the generator. These right sections are equal curves.

b. The reduced equation of cylinders with generator parallel to a coordinate axis and directing curve on a plane perpendicular to the generator are:

generator parallel to the X-axis (or directing curve on a plane parallel to the YZ-coordinate plane),

generator parallel to the Y-axis (or directing curve on a plane parallel to the XZ-coordinate plane),

generator parallel to the Z-axis (or directing curve on a plane parallel to the XY-coordinate plane),

c. A cone is a surface whose generator has one point, called the vertex, fixed in space as it moves intersecting all points of the directing curve, Fig. 14.4. directing curve

vertex

generator

Fig. 14.4

Y

X

Z

O

SURFACES AND SPACE CURVES 95

(14.7)

0)z,y,x(g0)z,y,x(f

(14.8)0)z,y,x(cg)z,y,x(f

The axis of the cone is the line through the vertex and the center of the directing curve. The right sections of a cone are the sections made by planes perpendicular to the axis of the cone.

14.7 Intersection of Two Surfaces

a. The intersection of two surfaces is a curve, Fig. 14.5. The curve may lie on one plane, forming a plane curve, but most of the time it will not, therefore forming a space curve(sometimes called a twisted or skew curve). The two equations of the surfaces, taken simultaneously, represent the curve at the intersection of the surfaces.

b. Any given curve may be represented by an infinite number of different pairs of equations. That is, the equations of two surfaces having the curve as their intersection. So that, if the equations,

together, represent a space curve, then,

Fig. 14.5

Y

X

Z

O

curve at the intersection of the two surfaces


represents a surface containing the curve, where c is a constant.

14.8 Projections, Projecting Lines, and Projecting Cylinders

a. The projection of a point P to a plane, is the foot P’ of the perpendicular dropped from P to the plane. The projecting line of a point to a plane, is the line PP’ connecting the point P to its projection P’ on the plane. The projection of a space curve C to a plane, is the curve C’ on the plane, whose points are projections of all the points on the curve C to the plane. The projecting cylinder of a space curve to a plane is the cylinder whose generators are the projecting lines of all the points on the curve to the plane. See Fig. 14.6.

b. The projecting cylinders of a space curve represented by

equation 14.7, or the simultaneous equations

0)z,y,x(g0)z,y,x(f

,

to each coordinate plane, called the YZ, XZ, and the XY projecting cylinders, are obtained by:

for the YZ-projecting cylinder, eliminate x from the pair of equations

0)z,y,x(g0)z,y,x(f

, yielding 0)z,y(h , which is the

YZ-projecting cylinder.

Fig. 14.6

Y

X

Z

O

space curve C P

Q

P’

Q’

projection of the curve C to the XY-plane, planecurve C’

projecting lines of P and Q to the XY-plane, |PP’| and |QQ’|

projection of the points P and Q to the XY-plane,P’ and Q’

SURFACES AND SPACE CURVES 97for the XZ-projecting cylinder,

eliminate y from the pair of equations

0)z,y,x(g0)z,y,x(f

, yielding 0)z,x(i , which is the

XZ-projecting cylinder.

for the XY-projecting cylinder, eliminate z from the pair of equations

0)z,y,x(g0)z,y,x(f

, yielding 0)y,x(j , which is the

XY-projecting cylinder.

14.9 Tracing Space Curves by its Projecting Cylinders

a. To trace a space curve using its YZ, XZ, and XY projecting cylinders,

Find the YZ, XZ, and XY projecting cylinders of the space curve.

Draw the space curve at the intersection of any two projecting cylinders.

14.10 Sketching Solids Bounded by Surfaces

a. To sketch a solid bounded by surfaces,

Use the method given in Sec. 14.4 to inspect and visualize the bounding surfaces, and find the curves of intersection of these bounding surfaces by inspection if possible.

Use the method given in Sec. 14.9 using projecting cylinders if the curves of intersection are difficult to find by inspection.

Draw the relevant portions of the surfaces bounding the solid.

14.11 Equation of a Given Surface

a. To find the equation of a given surface:

Consider a general point P(x, y, z) on the surface.


Express the geometric properties or the relation among the coordinates (x, y, z) and other quantities involved for every general point P on the surface, by means of an equation involving one, two, or all of the three variables x, y, and z.

Converting the equation of a surface in rectangular coordinates to another space coordinate system is possible using the relations given in Sec. 13.3.

99

(15.4)0DCzByAx

(15.2)ky

(15.3)kz

Chapter 15The Plane

15.1 A Plane Parallel to a Coordinate Plane

a. An equation of the first degree in one variable represents a plane parallel to the plane of the other two missing variables. That is, the equation of a plane,

parallel to the YZ-coordinate plane, and at a distance k from it is,

parallel to the XZ-coordinate plane, and at a distance k from it is,

parallel to the XY-coordinate plane, and at a distance k from it is,

15.2 General Equation of a Plane

a. Every plane may be represented by an equation of the first degree in three variables, having the form,

If 0,BA the line is parallel the XY-plane, 0,CA the line is parallel the XZ-plane, 0,CB the line is parallel the YZ-plane.

(15.1)kx


(15.7)212121 kCCkBBkAA and , ,

(15.6)0DzCyBxA

0DzCyBxA

2222

1111

and

(15.8)

0CBCB

0CACA

0BABA

22

11

22

11

22

11

and

,

,

(15.5)0

1zyx1zyx1zyx1zyx

333

222

111

15.3 Three-Point Form

a. The equation of a plane having three points P1(x1, y1, z1), P2(x2, y2, z2), and P3(x3, y3, z3), not in a straight line, has the form,

15.4 Parallel and Perpendicular Planes

a. The two planes, represented by the equations,

are parallel if and only if their coefficients are related such that,

where k is any constant. In matrix form, the above conditions may be stated as,

THE PLANE 101

(15.9)0CCBBAA 212121

(15.10)1

cz

by

ax

(15.11) nzmylx

b. The two planes given by equations 15.6, 0DzCyBxA 1111 and 0DzCyBxA 2222 , are

perpendicular if and only if their coefficients are related such that,

15.5 Intercept Form

a. The equation of a plane with x-intercept a, y-intercept b, and z-intercept c, Fig. 15.1, is given by,

15.6 Normal Form

a. The equation of a plane located at a perpendicular distance ρfrom the origin to the point N on the plane, Fig. 15.2, has the form,

Fig. 15.1

Y

X

Z

O

(0, 0, c)

(0, b, 0)

(a, 0, 0)

the plane

1cz

by

ax

ρ

Fig. 15.2

Y

X

Z

ON

the plane nzmylx

P(x, y, z)


(15.12)222

111

CBA

DCzByAxd

where l, m, and n, are the direction cosines of the line of length ρ perpendicular to the plane from the origin. This line is called the normal of the plane.

b. To reduce the general equation of a plane to the normal form,

Divide each term of the general equation,

0DCzByAx , by 222 CBA ,depending on the sign of C. If 0C , the radical takes on the sign of B. If 0CB , like the sign of A.

Transfer the constant term to the other side of the equation.

15.7 Directed Perpendicular Distance of a Plane to a Point

a. The directed perpendicular distance d of a plane, whose equation in the general form is 0DCzByAx , to the point P1(x1, y1, z1), is given by,

where the radical in the denominator takes on the sign of C. If 0C , it follows the sign of B. If 0CB , like the sign of A. A positive value for the distance d results if the point lies above the plane (or in the positive direction of thecoordinate axis perpendicular to the coordinate plane, in the case of planes perpendicular to this coordinate plane), and negative if the point lies below the plane (or in the negative direction of the coordinate axis perpendicular to thecoordinate plane, in the case of planes perpendicular to this coordinate plane).

15.8 Three Conditions Determine a Plane

a. A set of three independent conditions is required to determine the equation of a plane since its equation contains three essential constants that must be evaluated by three consistent equations.

103

(16.2) 0DzCyBxAcDzCyBxA 22221111

Chapter 16The Straight Line in Space

16.1 General Equation of a Line in Space

a. A straight line in space is the curve of intersection of two planes (see Sec. 14.7 for the general case). So that, the general equation of every straight line in space may bewritten as a set of two simultaneous linear equations in three variables as,

16.2 A Family of Planes Through a Given Space Line

a. A family of planes through a straight line in space whose equation is given by equation 16.1,

0DzCyBxA

0DzCyBxA

2222

1111 taken simultaneously, has the

form,

where c is an arbitrary constant. See Sec. 14.7.

16.3 Parametric Equations of a Line in Space

a. Every line in space having a point P1(x1, y1, z1) and one set of direction numbers L, M, and N, (see Sec. 12.4d), may be represented by the parametric equations,

with t as the parameter.

(16.1)

0DzCyBxA

0DzCyBxA

2222

1111

(16.3)

Ntzz

Mtyy

Ltxx

1

1

1

and

,

,


(16.4)N

zzM

yyLxx 111

b. Elimination of the parameter t from equation 16.3 yields the general set of equations of a line in space given by equation 16.1.

16.4 Symmetric Equations of a Line

a. The symmetric equation of a line in space is also obtained by eliminating the parameter t from equation 16.3, and expressed in the form,

b. To reduce the general form,

0DzCyBxA

0DzCyBxA

2222

1111 , to

the symmetric form, Nzz

Myy

Lxx 111

, of the

equation of a space line,

Find any two of the projecting cylinders (in the case of space lines, it is called projecting planes) of the space line from the general equation as presented in Sec. 14.8b. For example, the projecting planes

0 y)j(x, and 0z)i(x, .

Solve for the common variable in the equations of the chosen projecting planes

0 y)j(x, and 0z)i(x, and equate values to obtain f(z)f(y)x .

Reduce the coefficient of each variable to unity by dividing the equation through by a number.

c. The general form on the other hand is obtained from any two of the resulting three equations from the symmetric form, given by,

THE STRAIGHT LINE IN SPACE 105

(16.5)

Nzz

Myy

Nzz

Lxx

Myy

Lxx

11

11

11

and

,

,

(16.6)0NCMBLA

Any one of these three equations follows from the other two. This means that only two are independent equations that areenough to describe the line. These three equations are also the projecting planes of the space line to the XY, XZ, and YZ coordinate planes respectively.

16.5 Lines Parallel and Perpendicular to a Plane

a. A space lineNzz

Myy

Lxx 111

is parallel to the plane

0DCzByAx if and only if,

b. A space line Nzz

Myy

Lxx 111

is perpendicular to the

plane 0DCzByAx if and only if, the quantities L, M, and N, are proportional to the quantities A, B, and C, respectively.

107

Chapter 17Quadric Surfaces

17.1 Quadric Surfaces

a. A quadric surface (or quadric) is the locus of a second degree equation in three variables. A quadric surface corresponding to a general quadratic equation is not easy to study geometrically. This chapter therefore would only be dealing with quadrics whose equations are in the simplest form. That is, quadratic equations in three variables with second degree terms that are just squares of the variables and not products of any two variables.

b. Except for degenerate cases (two parallel planes, two coincident planes, two intersecting planes, a single straight line, or a point), the quadric surfaces can be classified into nine types or species. These are the ellipsoid (the sphere is a special case), the hyperboloids (one sheet and two sheets), the paraboloids (elliptic and hyperbolic), the quadric cylinders (elliptic, parabolic, and hyperbolic), and the elliptic cone.

17.2 Ellipsoids. Spheres

a. An ellipsoid, Fig. 17.1, is a surface represented by the equation,

where the segments of length 2a, 2b, 2c, cut off on the coordinate axes, are the axes of the ellipsoid. Its point of symmetry, the origin O, is the center of the ellipsoid.

Fig. 17.1

Y

X

Z

O

(0, 0, c)

(0, b, 0)

(a, 0, 0)

the ellipsoid

1c

z

b

y

a

x2

2

2

2

2

2

traces on eachcoordinate plane

(17.1)1c

z

b

y

a

x2

2

2

2

2

2


b. The ellipsoid is symmetric is symmetric with respect to each of the three coordinate planes, so it is also symmetric with respect to the three coordinate axes. The ellipsoid is a closed surface.

c. The x, y, and z intercepts are c and b, ,a respectively.

d. The XY, XZ, and YZ traces are all ellipses, having the respective forms,

e. All sections made by planes parallel to a coordinate plane are ellipses that decrease in size as the intersecting plane moves farther from the coordinate plane, becoming a “point-ellipse”, a degenerate ellipse (Sec. 6.1d), at the planes whose equations are,

f. An ellipsoid of revolution is formed when two of the semi-axes a, b, c, are equal since all sections made by planes perpendicular to the third axis are circles. That is, an ellipsoid formed by revolving an elliptic trace about its major axis, generating a prolate spheroid, or about its minor axis, generating an oblate spheroid.

g. A sphere is a special case of an ellipsoid where the semi-axes a, b, c, are all equal.

(17.2)

1c

z

b

y

1c

z

a

x

1b

y

a

x

2

2

2

2

2

2

2

2

2

2

2

2

and

,

,

(17.3) and ,cz,by,ax

QUADRIC SURFACES. TRANSFORMATION OF COORDINATES IN SPACE 10917.3 Hyperboloids of One Sheet

a. A hyperboloid of one sheet, Fig. 17.2, is a surface represented by the equation,

where the Z-axis is the axis of the hyperboloid of one sheet. Its center is at the origin O, also its point of symmetry.

b. The hyperboloid of one sheet is symmetric with respect to each of the three coordinate planes. It is an open surface extending indefinitely in the direction of the positive and the negative Z-axis. In other words, it encloses the Z-axis.

c. The x and y intercepts are b and a respectively. The z-intercept is imaginary (meaning it does not intersect the Z-axis of the space rectangular coordinate system).

d. The XY-trace is an ellipse, having the form,

The XZ and YZ traces are hyperbolas whose respective equations are,

Fig. 17.2

YX

Z

O

the hyperboloidof one sheet

1c

z

b

y

a

x2

2

2

2

2

2

(17.6)

1c

z

b

y

1c

z

a

x

2

2

2

2

2

2

2

2

and

(17.4)1c

z

b

y

a

x2

2

2

2

2

2

(17.5)1b

y

a

x2

2

2

2


e. The XY-sections (or the sections on the planes kz ) are ellipses which increase in size as the intersecting plane recedes from the XY-plane (or as |k| increases in the equation of the plane kz ). The XZ and the YZ sections (or the sections on the planes k xand ky respectively) are hyperbolas.

f. A hyperboloid of revolution of one sheet is formed when ba in equation 17.4, forming circular XY-sections instead

of ellipses. That is, by revolving one of the hyperbolic traces (equation 17.6) around the Z-axis. For cbor ca , but

ba , the surface is not a surface of revolution, but it just means that the XZ and the YZ sections are equilateral hyperbolas.

g. The following equations also represent hyperboloids of one sheet,

where the first encloses the Y-axis, and the second the X-axis.

17.4 Hyperboloids of Two Sheets

a. A hyperboloid of two sheets, Fig. 17.3, is a surface represented by the equation, Fig. 17.3

YX

Z

O

the hyperboloidof two sheets

1c

z

b

y

a

x2

2

2

2

2

2

(17.7)

1c

z

b

y

a

x

1c

z

b

y

a

x

2

2

2

2

2

2

2

2

2

2

2

2

and

(17.8)1c

z

b

y

a

x2

2

2

2

2

2

QUADRIC SURFACES. TRANSFORMATION OF COORDINATES IN SPACE 111

where the X-axis is the axis of the hyperboloid of two sheets, and the origin O is the center, also a point of symmetry of the surface.

b. The hyperboloid of two sheets possesses symmetry with respect to each coordinate plane. It is an open surface consisting of two disconnected sheets, one in the region

ax , the other in the region -ax .

c. The x-intercepts are a . The y and z intercepts are imaginary.

d. There is no YZ-trace. The XY and XZ traces are hyperbolas having the respective equations,

e. The YZ-sections are ellipses, starting with a “point-ellipse” when ak for the cutting plane kx , and increasing in

size as |k| increases numerically. No section exists for ak .

The XZ and the XY sections are hyperbolas.

f. A hyperboloid of revolution of two sheets is formed when cb in equation 17.8 since the YZ-sections become circular. This surface is generated by revolving one of the hyperbolic traces (equation 17.9) around the X-axis. No surface of revolution is formed whether

c.b if b,aor ca This just means that the XZ and the XY sections are equilateral hyperbolas.

(17.9)

1c

z

a

x

1b

y

a

x

2

2

2

2

2

2

2

2

and


g. The following equations also represent hyperboloids of two sheets,

where the Y-axis is the axis of the first hyperboloid of two sheets, and the Z-axis is the axis of the second.

17.5 Elliptic Paraboloids

a. An elliptic paraboloid, Fig. 17.4, is a surface whose equation is,

where the Z-axis (the only axis of symmetry of the elliptic paraboloid) is the axis of the surface. The origin O (or the intersection of the axis with the surface) is the vertex.

b. The elliptic paraboloid is symmetric with respect to the XZ and the YZ coordinate planes. It is an open surface lying entirely on one side of the XY-coordinate plane, starting from the origin and extending indefinitely in the direction of the positive Z-axis when c is positive (equation 17.11), and in the direction of the negative Z-axis when c is negative.

c. The x, y, and z intercepts of the elliptic paraboloid are all equal to zero.

Fig. 17.4

YX

Z

O

the ellipticparaboloid

czb

y

a

x2

2

2

2

(17.10)

1c

z

b

y

a

x

1c

z

b

y

a

x

2

2

2

2

2

2

2

2

2

2

2

2

and

(17.11)czb

y

a

x2

2

2

2


d. The XY-trace is a “point-ellipse” at the origin O (the vertex of the surface). The XZ and the YZ traces are parabolas having the respective forms,

e. The XY-sections are ellipses that increase in size as the cutting plane recedes from the XY-coordinate plane. The XZ and the YZ sections are parabolas, either both opening upward, or both opening downward, depending on whether c is positive or negative respectively, in the equation of the elliptic paraboloid (equation 17.11) .

f. A paraboloid of revolution is formed when ba in equation 17.11, forming circular XY-sections instead of ellipses. This surface could be generated by revolving either the XZ or the YZ parabolic trace (equation 17.12) about the Z-axis.

g. The following equations also represent elliptic paraboloids,

where the Y-axis is the axis of the first elliptic paraboloid, and the X-axis is the axis of the second.

(17.12)

zcby

zcax

22

22

and

(17.13)

axc

z

b

y

byc

z

a

x

2

2

2

2

2

2

2

2

and


17.6 Hyperbolic Paraboloids

a. A hyperbolic paraboloid, Fig. 17.5 is a surface represented by the equation,

where the Z-axis is the axis of the surface, and the origin O is the vertex.

b. The hyperbolic paraboloid is symmetric with respect to the XZ and the YZ coordinate planes. It is a saddle-shaped surface extending indefinitely in all directions.

c. The x, y, and z intercepts of the hyperbolic paraboloid are all zero.

d. The XY-trace is a pair of intersecting lines (a degenerate hyperbola, see Sec. 6.1d) having the form,

or by an equivalent pair of simultaneous linear equations,

(17.14)czb

y

a

x2

2

2

2

(17.15)0b

y

a

x2

2

2

2

(17.16)

0by

ax

0by

ax

Fig. 17.5

Y

X

Z

O

the hyperbolicparaboloid

czb

y

a

x2

2

2

2


The XZ and the YZ traces are parabolas whose respective equations are,

e. The XY-sections on the positive side of the Z-axis are hyperbolas whose transverse axes are parallel to the X-axis, and the XY-sections on the negative side of the Z-axis are hyperbolas whose transverse axes are parallel to the Y-axis. The XZ-sections are parabolas, opening upward when c is positive in the equation of the hyperbolic paraboloid (equation 17.14), or opening downward when c is negative. The YZ-sections are also parabolas, opening in the direction opposite to that of the XZ-sections.

f. The hyperbolic paraboloid cannot in any way be a surface of revolution. Only equilateral hyperbolic XY-sections would be formed for the case ba in equation 17.14.

g. Hyperbolic paraboloids may also be represented by the equations,

where the Y-axis is the axis of the first equation of the hyperbolic paraboloid, and the X-axis is the axis of the second.

17.7 Quadric Cylinders

a. A quadric cylinder is a cylinder (see Sec. 14.6) whose right section is a conic parallel to one of the coordinate planes. It

(17.17)

zcby

zcax

22

22

and

(17.18)

axc

z

b

y

byc

z

a

x

2

2

2

2

2

2

2

2

and


may be called elliptic, parabolic, or hyperbolic, depending on the nature of its trace on that coordinate plane.

b. A quadric cylinder whose generator is perpendicular to the XY-plane for example, would have an equation of the form

0 y)f(x, (see equation 14.6), provided that this equation represents a conic section. The following equations represent an elliptic, a parabolic, and a hyperbolic cylinder respectively, with generator perpendicular to the XY-plane,

The graphs of these quadric cylinders are shown in Figs. 17.6 to 17.8.

Fig. 17.7

YX

Z

O(a, b, 0)

(-a, b, 0)

the parabolic cylinder

by

a

x2

2

Fig. 17.6

YX

Z

O

(0, b, 0)(a, 0, 0)

the elliptic cylinder

1b

y

a

x2

2

2

2

(17.19)

1b

y

a

x

by

a

x

1b

y

a

x

2

2

2

2

2

2

2

2

2

2

and

,

,


17.8 Elliptic Cones

a. An elliptic cone, Fig. 17.9, is a surface represented by the equation,

where the Z-axis is the axis of the elliptic cone, and the origin O is the vertex and at the same time the point of symmetry of the surface.

b. The elliptic cone is symmetric with respect to the three coordinate planes. It is an open surface extending to infinity in both directions along the Z-axis.

c. The x, y, and z intercepts of the elliptic cone are all zero.

Fig. 17.9

YX

Z

O

the elliptic cone

2

2

2

2

2

2

c

z

b

y

a

x

Fig. 17.8

Y

X

Z

O

the hyperbolic cylinder

1b

y

a

x2

2

2

2

2b

2a

asymptotes

(17.20)2

2

2

2

2

2

c

z

b

y

a

x


d. The XY-trace is a “point-ellipse” at the origin O. The XZ and the YZ traces are the respective pairs of intersecting lines,

e. The XY-sections (or right sections, see Sec. 14.6c) are ellipses that increase in size as the cutting plane moves farther from the XY-plane. The XZ and the YZ sections are hyperbolas.

f. A right circular cone (or cone of revolution) is formed when ba in equation 17.20, so that the right sections are all circles. This surface is generated by revolving one of the lines from either the XZ or the YZ-trace around the Z-axis.

g. Elliptic cones may also be represented by the equations,

where the Y-axis is the axis of the first equation, and the X-axis is the axis of the second.

(17.21)

0cz

by

0cz

by

0cz

ax

0cz

ax

and

(17.22)

2

2

2

2

2

2

2

2

2

2

2

2

a

x

c

z

b

y

b

y

c

z

a

x

and


17.9 Ruled Surfaces

a. A ruled surface is a surface generated by a moving straight line called a ruling or generator. A cylinder is a ruled surface, all of whose rulings parallel, while a cone is a ruled surface, all of whose rulings are concurrent (see Sec. 14.6).

b. A double ruled surface is a surface generated by two distinct sets of generators. The hyperboloid of one sheet and the hyperbolic paraboloid are double ruled surfaces.

17.10 Translation of Axes in Space

a. The set of transformation equations for translation in space is analogous to the planar case (Sec. 7.1), and is written in the form,

where the coordinates (x, y, z) represents a general point P in the original coordinate system with coordinate axes X, Y, Z, while (x’, y’, z’) are the coordinates of the same point P referred to the new coordinate system with origin at the point O’(h, k, l) and coordinate axes X’, Y’, Z’.

17.11 Rotation of Axes in Space

a. The set of transformation equations for rotation in space (see also Sec. 7.2) has the form,

(17.23)

lz'zky'yhx'x

and, ,

(17.24)

znymxl'z

znymxl'y

znymxl'x

333

222

111

and

,

,


or in matrix form, it is,

where l1, m1, n1, are the direction cosines of the rotated X’-axis relative to the original X, Y, Z, axes respectively; l2, m2, n2, are the direction cosines of the rotated Y’-axis relative to the original X, Y, Z, axes respectively; and l3, m3, n3, are the direction cosines of the rotated Z’-axis relative to the original X, Y, Z, axes respectively.

b. From equation 17.25, we see at once that the coordinate transformation matrix for rotation in space was already modified as,

which is still orthogonal. That is,

(17.25)

zyx

nmlnmlnml

'z'y'x

333

222

111

(17.26)

333

222

111

nmlnmlnml

T

(17.27)

321

321

321T1

nnnmmmlll

TT

121

INTRODUCTORY VECTOR ANALYSIS

Chapter 18Vector Operations

18.1 Vectors

a. Vectors are quantities possessing magnitude and direction (for example force, velocity, acceleration, displacement, etc.). Scalars, on the other hand, possess only magnitude (for example volume, mass, time, any real number, etc.).

b. The symbol or notation for a vector in most textbooks is set in boldface (for example the vectors A and b). If written by hand, an arrow is placed above the symbol (for example the

vectors A

and b

, or without the arrows as the vectors OP and OR if two characters are used, with the first character representing the origin of the vector and the second its terminus). This handwritten notation for a vector would be used throughout this text to avoid confusion. A scalar is represented by letters of ordinary type (for example the scalars A and b, or the scalars |AB| and |AD| in case two characters are used).

c. A vector is represented graphically by a directed line segment (see Sec. 1.2e), whose length (which is equal to some convenient scale) represent the magnitude (also called the absolute value) of the vector, and whose direction is the same as the direction of the

vector, Fig. 18.1. The vector A

may also be written asOP (without the arrow, the same with a directed line

Fig. 18.1

Y

X

Z

O

P

O

OPor A

122 INTRODUCTORY VECTOR ANALYSIS

segment represented by two characters). The magnitude may be denoted by |OP|or |A| . Scalars on the other hand are just represented graphically by marks on a fixed scale.

d. A vector whose absolute value is unity, regardless of direction is called a unit vector. A null or zero vector (represented by the symbol O

) has zero magnitude and no specific

direction. A vector which is not null is a proper vector.

18.2 Equality of Vectors. Negative of a Vector

a. Two vectors A and B ofFig. 18.2, are equalvectors (written as

BA ), if and only if their absolute values are equal,

|B||A| , and their directions are the same regardless of the points in space from which they may be drawn (that is, their corresponding direction cosines should be equal, see Sec. 12.4),

BABABA nn and ,mm ,ll .

b. The negative of a vector

A (denoted by the symbol

A ) is the vector whose direction is opposite to that

of vector A but having the same magnitude.

18.3 Sum of Vectors. Difference

a. The sum or resultant of

vectors A and B is the vector C , written as,

YX

Z

O

Fig. 18.3

A

B

BAC

BAD

(18.1)CBA

Fig. 18.2

Y

X

Z

O

A

B

VECTOR OPERATIONS 123

YX

Z

O

Fig. 18.4

A

B

BAC

BAD

where C is represented by the diagonal of the parallelogram

constructed with A and B as sides, Fig. 18.3. This is known as the parallelogram law for vector addition.

Alternatively, the sum may be thought of to be the

vector C formed by joining

the initial point of A to the

terminal point of B , if the

initial point of B is placed first on the terminal point

of A , Fig. 18.4.

b. The difference of vectors A and B is the vector D , written as,

where D is defined to be the sum of A and B . That is,

)B(AD , Figs. 18.3 and 18.4.

c. Properties of vector addition:

CommutativeABBA

AssociativeC)BA()CB(A

18.4 Product of a Scalar and a Vector. The Unit Vector

a. The product of a scalar a, and a vector A

, is the vector B

, written as,

(18.2)DBA

(18.3)BAa


where B

has a magnitude equal to the product of a and |A|(or |A|a|B| ), and direction the same with the direction of

A

if a is positive, and opposite to the direction of A

if a is negative.

b. A unit vector a

in the direction of a given vector A

could be determined by,

c. Properties of scalar and vector multiplication:

CommutativeaAAa

AssociativeA(ab))Aa(b

Distributive

BaAa)BAa(

or ,AbAaA)ba(

18.5 Dot Product of Two Vectors

a. The dot product (or scalar product) of two vectors

B and A

is the scalar C defined to be the product of the

magnitudes of B and A

and the cosine of the angle θ between them, having the form,

b. Two vectors B and A

are perpendicular vectors if and only if,

(18.4)AA

a

)0A(

(18.5))0(cosBABAC

(18.6))OBA(0BA

,

VECTOR OPERATIONS 125

YX

Z

O

Fig. 18.5

A

B

O

R

Pvector projection of

OP ,A on B

vector projection of

OR ,B on A

θ

c. The vector projection (or component, see Fig. 18.5) of by,givenOP vector theis,A on B

The component of B on A

on the other hand, is the vector OR given by,

d. The scalar projection (or the magnitude of the vector projection) of A on B

, is the scalar |OP| given by,

(18.7)AA

BAOP

2

(18.8)BB

BAOR

2

(18.9)A

BAcosBOP


The scalar projection of B on A

, is the scalar |OR| given by,

e. Properties of dot multiplication:

CommutativeABBA

DistributiveCABA)CB(A

18.6 Cross Product of Two Vectors

a. The cross product (or vector product) of two vectors B and A

is the vector C

defined to be the product of the

magnitudes of B and A

, the sine of the smaller angle θ between them, and a unit vector u

defined to have a

direction perpendicular to the plane determined by B and A

(and so sensed that a right-handed screw turned from A

towards B

through the angle θ would advance in the direction of this unit vector u

). That is,

The magnitude |C|, of the vector product C

is given by,

We will notice that |C| is the scalar to be multiplied by the unit vector u

in equation 18.11 to yield C

. Geometrically,

the magnitude of the vector product of any two vectors is the

(18.10)B

BAcosAOR

(18.11) )0(usinBABAC

(18.12) sinBAC

VECTOR OPERATIONS 127area of the parallelogram constructed with the magnitude of the two vectors as sides, Fig. 18.6.

The direction of the vector product naturally follows the direction of the unit vector u

defined in this

section.

b. Two vectors B and A

are parallel vectors if and only if,

c. Properties of cross multiplication:

Anti-commutativeA BB A

DistributiveCABA)CB(A

YX

Z

O

Fig. 18.6

A

B

θ

sinBAArea

u

usinBA

BAC

(18.13))OBA()BA(0BA

,,

129

Chapter 19Vectors in Cartesian Coordinates

19.1 Cartesian Unit Vectors

a. The rectangular (or Cartesian) unit vectors

,k and ,j ,i

is an important set of unit vectors in the direction of the positive X, Y, and Z coordinate axes respectively, Fig. 19.1.

b. Products of the rectangular unit vectors:

Dot Products

Cross Products

(19.1)0ikkjji

(19.2)1kkjjii

(19.3)0kkjjii

(19.4)kijji

(19.5)ijkkj

(19.6)jkiik

Fig. 19.1

Y

X

Z

Oi

k

j


19.2 Cartesian Representation of a Vector

a. A vector R

joining a point P1(x1, y1, z1) to another point P2(x2, y2, z2), Fig. 19.2, may be written as,

where ,k)zz( ,j)yy( ,i)xx( 121212

are the

components of the vector R

along the X, Y, Z, coordinate axes respectively.

b. The magnitude of R

is given by,

19.3 Operations on Two Vectors with Cartesian Representations

a. The sum of two vectors B and A

, represented by the equations,

(19.7)k)zz(j)yy(i)xx(R 121212

(19.8)212

212

212 )zz()yy()xx(R

(19.9)kbjbibB

kajaiaA

321

321

and

Fig. 19.2

Y

X

Z

O

P1(x1, y1, z1)

P2(x2, y2, z2)

k)zz(j)yy(i)xx(PPR 12121221

VECTORS IN CARTESIAN COORDINATES 131

is the vector C

whose value is,

b. The difference of the two vectors B and A

, is the vector D

having the form,

c. The dot product of the two vectors B and A

, is the scalar C having the form,

d. The cross product of the two vectors B and A

, is the

vector C

given by,

In matrix form, the above equation is written as,

19.4 Products Involving Three Vectors. The Lagrange’s Identity

a. The product C)BA(

of the three vectors ,C and ,B ,A

represented by the equations,

(19.10)k)ba(j)ba(i)ba(BAC 332211

(19.11)k)ba(j)ba(i)ba(BAD 332211

(19.12)332211 bababaBAC

(19.13)k)abba(j)abba(i)abba(BAC 212131313232

(19.14)

321

321

bbbaaakji

BAC


is a vector C)BA(D

having the following properties:

The magnitude of C)BA(D

is,

The direction of C)BA(D

is the same as C

if the dot

product )BA( is positive, and opposite to C

if )BA(

is

negative.

Note that the product,

b. The product )CB(A

(also called the scalar triple product or the box product) of the three vectors

,C and ,B ,A

given by equations 19.15 is a scalar S having the form,

(19.15)

kcjcicC

kbjbibB

kajaiaA

321

321

321

and

(19.16)C)BA(D

(19.17))CB(AC)BA(

(19.18)321

321

321

cccbbbaaa

)CB(AS


The vectors ,C and ,B ,A

are vectors parallel to one and the same plane if and only if,

The scalar triple product has the following properties:

It may be written without the parenthesis as,

since C)BA(

is meaningless.

The dot and the cross may be interchanged without changing the product (since interchanging any two rows in a determinant only changes the sign of the determinant, and interchanging the dot and the cross is the same as interchanging rows twice, therefore changing back the sign of the determinant to its original, see equation 19.18). That is,

The preceding properties allow us to device a new notation for the scalar triple product as,

A cyclic permutation of the factors does not change the product. That is,

(19.19))OCBA()CBA(0)CB(A

,, ,

(19.20)CBA)CB(A

(19.21) CBACBA

(19.22) ABC)CB(A

(19.23) CBABACACBBCACABABC


Geometrically, the scalar triple product

ABCS is the volume of the parallelepiped having the vectors

,C and ,B ,A

as concurrent edges, Fig. 19.3.

c. The product )CB(A

(also called the vector triple

product) of the vectors ,C and ,B ,A

given by equations

19.15 is a vector V

having the form,

The vector triple product is not associative, or

since the product,

d. The products presented in the previous sections can be used to obtain the following relations;

The product ),DC()BA(

or

A

YX

Z

O

Fig. 19.3

B

Cθ

CB

(19.24)C)BA(B)CA()CB(AV

(19.25)C)BA()CB(A

(19.26)B)AC(A)BC()BA(CC)BA(

(19.27)D]ABC[C]ABD[)DC()BA(

(19.28)A]CDB[B]CDA[)DC()BA(


The Lagrange’s identity,

(19.29))CB)(DA()DB)(CA()DC()BA(

137

Chapter 20Vector Analysis of Planes

and Lines

20.1 The Equation of a Plane

a. The equation of a plane having a point P1(x1, y1, z1), and

perpendicular to the vector knjninN 321

is obtained

from the dot product,

where PP1 is the vector from P1(x1, y1, z1) to a general point P(x, y, z) on the plane. Equation 20.1 can be reduced to the form,

20.2 The Parametric Equation of a Line

a. The parametric equation of a line through the point P1(x1, y1,

z1), and parallel to the vector knjninN 321

is obtained

from,

where t is a scalar (called the parameter), and PP1 is the vector from P1(x1, y1, z1) to a general point P(x, y, z) on the line. The equation above reduces to the form,

(20.1)0PPN 1

(20.2)0)zz(n)yy(n)xx(n 131211

(20.3)NtPP1

(20.4)ktnjtnitnk)zz(j)yy(i)xx( 321111


Equating corresponding coefficients of ,k and ,j ,i

gives the parametric equations of the line in the form,

(20.5)

tnzz

tnyy

tnxx

31

21

11

and

Analytic Geometry With Introduction to Vector Analysis

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