CSCE 552 Fall 2012 Math By Jijun Tang

Applied Trigonometry Trigonometric functions  Defined using right triangle  x y h

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 W V + W V W V V – W –W–W

Matrix Representation The entry of a matrix M in the i-th row and j-th column is denoted M ij For example,

Vectors and Matrices Example matrix product

Identity Matrix I n For any n  n matrix M, the product with the identity matrix is M itself  I n M = M  MI n = 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

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

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 two n-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 of the parallelogram formed by V and W  V W ||V|| sin 

Area and Cross Product The area A of a triangle with vertices P 1, P 2, and P 3 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 Vector Rotation about an arbitrary vector Counterclockwise rotation about an arbitrary vector (l x,l y,l z ) 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

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

Rotation (Z) Rotation matrix Rotates points about the z-axis through the angle 

Rotations (X, Y) Similar matrices for rotations about x, y

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 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

Plane A plane in 3D space can be defined by a normal direction N and a point P Other points in the plane satisfy P Q 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 line S + 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

Formular If L  V = 0, the line is parallel to the plane and no intersection occurs Otherwise, the point of intersection is