CSCE 590E Spring 2007

Basic Math

By Jijun Tang

Applied Trigonometry

Trigonometric functions Defined using right triangle

x

yh

Applied Trigonometry

Angles measured in radians

Full circle contains 2 radians

Trigonometry

Trigonometric identities

Inverse trigonometric functions

Return angle for which sin, cos, or tan function produces a particular value

If sin = z, then = sin-1 z

If cos = z, then = cos-1 z

If tan = z, then = tan-1 z

arcs

Vectors and Matrices

Scalars represent quantities that can be described fully using one value Mass Time Distance

Vectors describe a magnitude and direction together using multiple values

Vectors and Matrices

Two vectors V and W are added by placing the beginning of W at the end of V

Subtraction reverses the second vector

V

W

V + W

V

W

V

V – W–W

Vectors and Matrices

Vectors add and subtract componentwise

Vectors and Matrices

The magnitude of an n-dimensional vector V is given by

In three dimensions, this is

Vectors and Matrices

A vector having a magnitude of 1 is called a unit vector

Any vector V can be resized to unit length by dividing it by its magnitude:

This process is called normalization

Vectors and Matrices

A matrix is a rectangular array of numbers arranged as rows and columns A matrix having n rows and m columns is

an n m matrix At the right, M is a

2 3 matrix If n = m, the matrix is a square matrix

Vectors and Matrices

The transpose of a matrix M is denoted MT and has its rows and columns exchanged:

Vectors and Matrices

An n-dimensional vector V can be thought of as an n 1 column matrix:

Or a 1 n row matrix:

Vectors and Matrices

Product of two matrices A and B Number of columns of A must equal

number of rows of B Entries of the product are given by

If A is a n m matrix, and B is an m p matrix, then AB is an n p matrix

Vectors and Matrices

Example matrix product

Vectors and Matrices

Matrices are used to transform vectors from one coordinate system to another

In three dimensions, the product of a matrix and a column vector looks like:

Identity Matrix In

For any n n matrix M,

the product with the

identity matrix is M itself InM = M

MIn = M

Invertible

An n n matrix M is invertible if there exists another matrix G such that

The inverse of M is denoted M-1

1 0 0

0 1 0

0 0 1

n

MG GM I

Determinant

The determinant of a square matrix M is denoted det M or |M|

A matrix is invertible if its determinant is not zero

For a 2 2 matrix,

deta b a b

ad bcc d c d

Determinant

The determinant of a 3 3 matrix is

Inverse

Explicit formulas exist for matrix inverses These are good for small matrices, but

other methods are generally used for larger matrices

In computer graphics, we are usually dealing with 2 2, 3 3, and a special form of 4 4 matrices

Vectors and Matrices

A special type of 4 4 matrix used in computer graphics looks like

R is a 3 3 rotation matrix, and T is a translation vector

11 12 13

21 22 23

31 32 33

0 0 0 1

x

y

z

R R R T

R R R T

R R R T

M

Vectors and Matrices

The inverse of this 4 4 matrix is

1 1 1 111 12 13

1 1 1 1 1 121 22 23

1

1 1 1 131 32 33

1 0 0 0 1

x

y

z

R R R

R R R

R R R

R T

R R T R TM

R T

0

The Dot Product

The dot product is a product between two vectors that produces a scalar

The dot product between twon-dimensional vectors V and W is given by

In three dimensions,

The Dot Product

The dot product can be used to project one vector onto another

V

W

The Dot Product

The dot product satisfies the formula

is the angle between the two vectors Dot product is always 0 between

perpendicular vectors If V and W are unit vectors, the dot

product is 1 for parallel vectors pointing in the same direction, -1 for opposite

The Dot Product

The dot product of a vector with itself produces the squared magnitude

Often, the notation V 2 is used as shorthand for V V

The Cross Product

The cross product is a product between two vectors the produces a vector The cross product only applies in three

dimensions The cross product is perpendicular to both

vectors being multiplied together The cross product between two parallel

vectors is the zero vector (0, 0, 0)

The Cross Product

The cross product between V and W is

A helpful tool for remembering this formula is the pseudodeterminant

The Cross Product

The cross product can also be expressed as the matrix-vector product

The perpendicularity property means

The Cross Product

The cross product satisfies the trigonometric relationship

This is the area ofthe parallelogramformed byV and W

V

W

||V|| sin

The Cross Product

The area A of a triangle with vertices P1, P2, and P3 is thus given by

The Cross Product

Cross products obey the right hand rule If first vector points along right thumb, and

second vector points along right fingers, Then cross product points out of right palm

Reversing order of vectors negates the cross product:

Cross product is anticommutative

Transformations

Calculations are often carried out in many different coordinate systems

We must be able to transform information from one coordinate system to another easily

Matrix multiplication allows us to do this

Transformations

Suppose that the coordinate axes in one coordinate system correspond to the directions R, S, and T in another

Then we transform a vector V to the RST system as follows

ILLustration

Transformation matrix

We transform back to the original system by inverting the matrix:

Often, the matrix’s inverse is equal to its transpose—such a matrix is called orthogonal

Transformations

A 3 3 matrix can reorient the coordinate axes in any way, but it leaves the origin fixed

We must add a translation component D to move the origin:

Transformations

Homogeneous coordinates Four-dimensional space Combines 3 3 matrix and translation

into one 4 4 matrix

Transformations

V is now a four-dimensional vector The w-coordinate of V determines whether

V is a point or a direction vector If w = 0, then V is a direction vector and

the fourth column of the transformation matrix has no effect

If w 0, then V is a point and the fourth column of the matrix translates the origin

Normally, w = 1 for points

Transformations

The three-dimensional counterpart of a four-dimensional homogeneous vector V is given by

Scaling a homogeneous vector thus has no effect on its actual 3D value

Transformations

Transformation matrices are often the result of combining several simple transformations Translations Scales Rotations

Transformations are combined by multiplying their matrices together

Transformation Steps

Orderings

Orderings

Orderings of different type is important A rotation followed by a translation is different from a translation followed by a rotation

Orderings of the same type does not matter

Transformations

Translation matrix

Translates the origin by the vector T

translate

1 0 0

0 1 0

0 0 1

0 0 0 1

x

y

z

T

T

T

M

Transformations

Scale matrix

Scales coordinate axes by a, b, and c If a = b = c, the scale is uniform

scale

0 0 0

0 0 0

0 0 0

0 0 0 1

a

b

c

M

Transformations

Rotation matrix

Rotates points about the z-axis through the angle

-rotate

cos sin 0 0

sin cos 0 0

0 0 1 0

0 0 0 1

z

M

Transformations

Similar matrices for rotations about x, y

-rotate

1 0 0 0

0 cos sin 0

0 sin cos 0

0 0 0 1

x

M

-rotate

cos 0 sin 0

0 1 0 0

sin 0 cos 0

0 0 0 1

y

M

Transformations

Normal vectors transform differently than do ordinary points and directions A normal vector represents the direction

pointing out of a surface A normal vector is perpendicular to the

tangent plane If a matrix M transforms points from one

coordinate system to another, then normal vectors must be transformed by (M-1)T

Geometry

A line in 3D space is represented by

S is a point on the line, and V is the direction along which the line runs

Any point P on the line corresponds to a value of the parameter t

Two lines are parallel if their direction vectors are parallel

t t P S V

Geometry

A plane in 3D space can be defined by a normal direction N and a point P

Other points in the plane satisfy

PQ

N

Geometry

A plane equation is commonly written

A, B, and C are the components of the normal direction N, and D is given by

for any point P in the plane

Geometry

A plane is often represented by the 4D vector (A, B, C, D)

If a 4D homogeneous point P lies in the plane, then (A, B, C, D) P = 0

If a point does not lie in the plane, then the dot product tells us which side of the plane the point lies on

Geometry

Distance d from a point P to a lineS + t V

P

VS

d

Geometry

Use Pythagorean theorem:

Taking square root,

If V is unit length, then V 2 = 1

Geometry

Intersection of a line and a plane Let P(t) = S + t V be the line Let L = (N, D) be the plane We want to find t such that L P(t) = 0

Careful, S has w-coordinate of 1, and V has w-coordinate of 0

x x y y z z w

x x y y z z

L S L S L S Lt

L V L V L V

L S

L V

Geometry

If L V = 0, the line is parallel to the plane and no intersection occurs

Otherwise, the point of intersection is

t

L S

P S VL V