Conics Presentation-R.SURYA KIRAN ,ASSISTANT PROFESSOR ,GITAM UNIVERSITY

8/12/2019 Conics Presentation-R.SURYA KIRAN ,ASSISTANT PROFESSOR ,GITAM UNIVERSITY

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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM

Contents

Introduction

Ellipse

Parabola Hyperbola

Cycloids

Involutes


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Definition

CONIC is defined as

the locus of the point

moving in a plane in

such a way that theratio of its distance

from a point to the

fixed straight line is

always constant.

This constant is known

as ECCENTRICITY.

APEX

AXIS

GENERATOR

BASE


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Section Plane

Parallel to Axis.Hyperbola

Section Plane Parallel

to one - end generator.

Section Plane

Through Generators

Ellipse

Conic Sections

Ellipse, Parabola and Hyperbola are called conic

sections

These curves appear on the surface of a cone

when it is cut by some typical cutting planes.


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Engineering Applications

Ellipse:

Arches, bridges, dam, gears, man-holes, flanges

of the pipes, stuffing boxes.

Parabola:

Headlights, solar connectors, reflectors.

Hyperbola:

Cooling towers, hydraulic channels, transmitters,

receivers.


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Ellipse

The locus of a point moving in a plane in such a

way that the ratio of its distance from a point to

the fixed straight line is always constant.

Distance of the moving point from the

directrix


focusE =

Eccentricity (E) less than 1 gives

ELLIPSE


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Concentric circles method

1

2

3

4

5

6

7

8

9

10

BA

D

C

1

23

4

5

6

78

9

10

Q1. Construct an ellipse

when its major axis

is equal to 100 mm

and the minor axis isequal 65mm.

65

100


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Rectangular method

1

2

3

1

2

3

A B

Q2. A plot of ground is in the shape of a rectangle

110mm*50mm. Inscribe an elliptical lawn in it.

C

D

O

50

110


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Parallelogram or Oblong

methodQ2. A plate is in the shape of 100mm and 70mm long sides

with included angle of 75o. Draw an elliptical mark on the plate.

1

2

3

A B

100

1

2

3


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Rectangular + Concentric circles

method

Q3. The major and minor axis of an elliptical

fish pond is 100mm and 65mm

respectively. Draw one half of the ellipse by

rectangular method and the other half by

the concentric circles method.


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Arcs of circles method

Q4. Construct an ellipse when a pair of conjugate

axes is equal to 110mm and 50mm

respectively. The angle between the conjugate

axes is 700

. Determine the lengths of the majorand the minor axes.

[conjugate axes-these are the two lines

intersecting each other at the centre of an

ellipse when they are parallel to the tangents

drawn at their extremities.]


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A Bf1 f2

C

D

1 2 3 4 5

110

50


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Eccentricity method

Q5. Construct an ellipse when the distance

between the focus and the directrix is 30mm

and the eccentricity is . Draw the tangent

and the normal at any point on the curve

using the directrix.


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30

D

D

O


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Parabola





directrix


focusE =

Eccentricity (E) equal to1 gives

PARABOLA


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Eccentricity method

Q. Construct a parabola when the distance

between the focus and the directrix is

50mm. Draw a tangent and the normal at

any point P on your curve.


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A

B

V

(VERTEX)

F( focus)O 1 2 3 4

50

OV =

VF


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Rectangular method

Q2. A cricket ball is thrown from the ground

level reaches the wicket keepers gloves.Maximum height reached by the ball

is 5 cm. The ball travels a horizontal

distance of 10 cm from the point of projection.

Trace the path of the ball.


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1

2

3

4

5

6

1 2 3 4 5 6

1

2

3

4

5

6

5 4 3 2 1


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Parallelogram method

Q3. Construct a parabolic arch within a

parallelogram of sides 110mm*50mm.One

of the included angles between the sides is700.


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Tangent method

Q4. A fountain jet discharges water from the

ground level at an inclination of 500to the

ground. The jet travels a horizontaldistance of 90 mm from the point of

discharge and falls on the ground. Trace

the path of the jet.


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Hyperbola





directrix


focusE =

Eccentricity (E) greater than1 gives

HYPERBOLA


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Methods(Hyperbola)

Eccentricity method

Ordinate, Abscissa &Transverse

method

Asymptotes method.


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Eccentricity method

Q. Construct a hyperbola when the distance

between the focus and the directrix is 40mm. The

eccentricity is 4/3. Draw a tangent and normalat any point on the hyperbola.


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F ( focus)(vertex)

A

B

V

O di t Ab i & T


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Ordinate ,Abscissa & Transverse

axis

Q. Draw a hyperbola when its double ordinate is

90mm, abscissa is 35mm and half the

transverse axis is 45mm.


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O


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Asymptotes method

Q. Construct a rectangular hyperbola when a

point P on it is at a distance of 15mm and

20mm respectively from the two asymptotes.

[when the angle between the asymptotes is

900,the curve is called as rectangular

hyperbola.]


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P

O

20 mm

15 mm

1

2

3

12 1 2 3

1

2


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Definition

A curve is a line traced out by a moving point by constantlychanging its directions

Plane curves:

A plane curve is a curve traced by a point which movesin a constantly changing directions on a plane

surface.Eg: Cycloidal curves ,involutes ,spirals.

Space curves:

A space curve is a curve traced by a point which movesin a constantly changing directions in space.

Eg. Helixes.

Cycloidal curves: -cycloids, epicycloids, hypocycloids,

Helixes:- cylindrical, conical.


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CYCLOIDS

A Cycloid is a curve traced by a point on the

circumference of a circle which rolls along a

straight line without slipping.

Eg.watches,dial gauges.As the rolling circle is assumed to roll

clockwise numbering of the division points on

it should be in counter clockwise direction.


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1 2 3 4 5 6 7 80

7

6

5

4

3

2

1

8

C0 C1 C2 C3 C4 C5 C6 C7 C8

(pD)


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Epicycloids

Epicycloids is a curve generated by a point on

the circumference of a circle which rolls

without slipping on the outside of another

circle. The fixed circle on the outside of which the

generating circle rolls is called as the Base

circle/directing circle.

Eg. Cycloidal teeth gears ,the faces are of

epicycloidal profile.


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Q. Draw an epicycloids of rolling circle

50mm,which rolls outside another circle of

150mm dia for one revolution. Draw a tangent

and normal at any point on the curve. [(r/R)*360]


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rR

3600=

O

C2


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HYPOCYCLOIDS

It is a curve traced by a point on the

circumference of a circle which rolls without

slipping on the inside of another circle.

[(r/R)*360]


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INVOlUTES

It is a single curve line traced out by an end of

a string when unwound itself from a straight

line or a circle or a polygon , the string being

always kept taut.Eg. Involute of a circle is the profile of the gear

teeth wheel.


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Q. A coir is unwound from a drum of dia 30mm.

Draw the locus of the free end of the coir for

unwinding through an angle of 3600.


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(pD)


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INVOLUTE OF A SQUARE

Q. Draw the involute of square of side 20mm.

Draw a tangent and a normal at any point M.


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Conics Presentation-R.SURYA KIRAN ,ASSISTANT PROFESSOR ,GITAM UNIVERSITY

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