Computer Graphics 1 Last Updated 13 Jan12 Linear

Computer Graphics 1 Last Updated: 13 -Jan-12 Linear Algebra (Mathematics for CG) Reading: HB Appendix A

Notation: Scalars, Vectors, Matrices 2 Scalar Ø (lower case, italic) Vector Ø (lower case, bold) Matrix Ø (upper case, bold)

Vectors 3 Arrow: Length and Direction Oriented segment in 2 D or 3 D space

Column vs. Row Vectors 4 Row vectors Column vectors Switch back and forth with transpose

Vector-Vector Addition 5 Add: vector + vector = vector Parallelogram rule tail to head, complete the triangle geometric examples: algebraic

Vector-Vector Subtraction Subtract: vector - vector = vector 6

Vector-Vector Subtraction Subtract: vector - vector = vector argument reversal 7

Scalar-Vector Multiplication Multiply: scalar * vector = vector is scaled 8

Vector-Vector Multiplication Multiply: vector * vector = scalar Dot product or Inner product 9

Vector-Vector Multiplication Multiply: vector * vector = scalar dot product, aka inner product 10

Vector-Vector Multiplication Multiply: vector * vector = scalar dot product or inner product • geometric interpretation • lengths, angles • can find angle between two vectors 11

Dot Product Geometry 12 Can find length of projection of u onto v as lines become perpendicular,

Dot Product Example 13

Vector-Vector Multiplication 14 Multiply: vector * vector = vector cross product

Vector-Vector Multiplication multiply: vector * vector = vector cross product algebraic geometric parallelogram area perpendicular to parallelogram 15

RHS vs. LHS Coordinate Systems 16 Right-handed coordinate system right hand rule: index finger x, second finger y; right thumb points up Left-handed coordinate system left hand rule: index finger x, second finger y; left thumb points down

Basis Vectors Take any two vectors that are linearly independent (nonzero and nonparallel) can use linear combination of these to define any other vector: 17

Orthonormal Basis Vectors 18 If basis vectors are orthonormal (orthogonal (mutually perpendicular) and unit length) o we have Cartesian coordinate system o familiar Pythagorean definition of distance Orthonormal algebraic properties

Basis Vectors and Origins 19 Coordinate system: just basis vectors can only specify offset: vectors Coordinate frame (or Frame of Reference): basis vectors and origin can specify location as well as offset: points It helps in analyzing the vectors and solving the kinematics.

Working with Frames 20 F 1

Working with Frames 21 F 1 p = (3, -1)

Working with Frames 22 F 1 F 2 p = (3, -1)

Working with Frames 23 F 1 F 2 F 1 p = (3, -1) F 2 p = (-1. 5, 2)

Working with Frames 24 F 1 F 2 F 3 F 1 p = (3, -1) F 2 p = (-1. 5, 2) F 3

Working with Frames 25 F 1 F 2 F 3 F 1 p = (3, -1) F 2 p = (-1. 5, 2) F 3 p = (1, 2)

Named Coordinate Frames 26 Origin and basis vectors Pick canonical frame of reference Ø then don’t have to store origin, basis vectors Ø just Ø convention: Cartesian orthonormal one on previous slide Handy to specify others as needed Ø airplane nose, looking over your shoulder, . . . Ø really common ones given names in CG vobject, world, camera, screen, . . .

Lines 27 Slope-intercept form y = mx + b Implicit form y – mx – b = 0 Ax + By + C = 0 f(x, y) = 0

Implicit Functions 28 An implicit function f(x, y) = 0 can be thought of as a height field where f is the height (top). A path where the height is zero is the implicit curve (bottom). Find where function is 0 plug in (x, y), check if =0 <0 >0 on line inside outside

Implicit Circles 29 circle is points (x, y) where f(x, y) = 0 points p on circle have property that vector from c to p dotted with itself has value r 2 points p on the circle have property that squared distance from c to p is r 2 points p on circle are those a distance r from center point c

Parametric Curves 30 Parameter: index that changes continuously (x, y): point on curve t: parameter Vector form

2 D Parametric Lines 31 start at point p 0, go towards p 1, according to parameter t p(0) = p 0, p(1) = p 1

Linear Interpolation 32 Parametric line is example of general concept Interpolation p goes through a at t = 0 p goes through b at t = 1 Linear weights t, (1 -t) are linear polynomials in t

Matrix-Matrix Addition 33 Add: matrix + matrix = matrix Example

Scalar-Matrix Multiplication 34 Multiply: scalar * matrix = matrix Example

Matrix-Matrix Multiplication 35 Can only multiply (n, k) by (k, m): number of left cols = number of right rows legal undefined

Matrix-Matrix Multiplication 36 row by column

Matrix-Matrix Multiplication 37 row by column

Matrix-Matrix Multiplication 38 row by column

Matrix-Matrix Multiplication 39 row by column

Matrix-Matrix Multiplication 40 row by column Non commutative: AB != BA

Matrix-Vector Multiplication points as column vectors: post-multiply 41

Matrices 42 Transpose Identity Inverse

Matrices and Linear Systems 43 Linear system of n equations, n unknowns Matrix form Ax=b

Exercise 1 44 Q 1: If the vectors, u and v in figure are coplanar with the sheet of paper, does the cross product v x u extend towards the reader or away assuming right-handed coordinate system? For u = [1, 0, 0] & v = [2, 2, 1]. Find Q 2: The length of v Q 3: length of projection of u onto v Q 4: v x u Q 5: Cos t, if t is the angle between u & v

Exercise 2 45 F 1 F 3 F 2 F 1 p = (? , ? ) F 2 p = (? , ? ) F 3 p = (? , ? )

Exercise 3 46 Given P 1(3, 6, 9) and P 2(5, 5, 5). Find a point on the line with end points P 1 & P 2 at t = 2/3 using parametric equation of line, where t is the parameter.

Things to do 47 Reading for Open. GL practice: HB 2. 9 Ready for the Quiz

References 48 1. 2. http: //faculty. cs. tamu. edu/schaefer/teaching/441_Spring 2012/index. html http: //www. ugrad. cs. ubc. ca/~cs 314/Vjan 2007/