CSCE 552 Fall 2012

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

arcs

Vectors and Scalars

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

Vectors describe a magnitude and direction together using multiple values

Examples of vectors

Difference between two points Magnitude is the distance between the points Direction points from one point to the other

Velocity of a projectile Magnitude is the speed of the projectile Direction is the direction in which it’s traveling

A force is applied along a direction

Vectors Add and Subtraction

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

Subtraction reverses the second vector

V

WV + W

V

W

V

V – W–W

Matrix Representation

The entry of a matrix M in the i-th row and j-th column is denoted Mij

For example,

Vectors and Matrices

Example matrix product

Identity Matrix In

For any n n matrix M,the product with theidentity 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

Inverse of 2x2 and 3x3

The inverse of a 2 2 matrix M is

The inverse of a 3 3 matrix M is

Inverse of 4x4

1 1 1 111 12 13

1 1 1 1 1 121 22 23

11 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

Gauss-Jordan Elimination

Gaussian Elimination

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,

Dot Product Usage

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

V

W

Dot Product Properties

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

Self 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)

Illustration

The cross product satisfies the trigonometric relationship

This is the area ofthe parallelogramformed byV and W

V

W

||V|| sin

Area and Cross Product

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

Rules

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

Rules in Figure

Formular

The cross product between V and W is

A helpful tool for remembering this formula is the pseudodeterminant (x, y, z) are unit vectors

Alternative

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

The perpendicularity property means

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

Illustration

R S T

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

2D Example

θ

2D Rotations

Rotation Around Arbitrary VectorRotation about an arbitrary vector Counterclockwise rotation about an arbitrary vector (lx,ly,lz) normalised so that                    by an angle α is given by a matrix

                                                                           

where

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

Translation

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:

Homogeneous coordinates

Four-dimensional space Combines 3 3 matrix and translation

into one 4 4 matrix

Direction Vector and Point

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

Translation

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

Scale

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

Rotation (Z)

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

Rotations (X, Y)

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

Transforming Normal Vectors

Normal vectors are transformed 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

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

Geometry -- Lines

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

Plane

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

Plane Equation

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

Properties of Plane

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

Distance from Point to a Line

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

P

VS

d

Distance from Point to a Line

Use Pythagorean theorem:

Taking square root,

If V is unit length, then V 2 = 1

Line and Plane Intersection

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 SL V

Formular

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

Otherwise, the point of intersection is

t

L SP S VL V