Mathematics for Computer Graphics

1. Mathematics for Computer Graphics CSE 581 – Roger Crawfis (slides developed from Korea University slides)

2. Spaces • Scalars • (Linear) Vector Space • Scalars and vectors • Affine Space • Scalars, vectors, and points • Euclidean Space • Scalars, vectors, points • Concept of distance • Projections

3. Scalars (1/2) • Scalar Field • Ex) Ordinary real numbers and operations on them • Two Fundamental Operations • Addition and multiplication Commutative Associative Distributive

4. Scalars (2/2) • Two Special Scalars • Additive identity: 0, multiplicative identity: 1 • Additive inverse and multiplicative inverse of

5. Vector Spaces (1/4) • Two Entities: Scalars and Vectors • Vectors • Directed line segments • n-tuples of numbers • Two operations: vector-vector addition, scalar-vector multiplication • Special Vector: Zero Vector • Let u denote a vector Directed line segments

6. Vector Spaces (2/4) • Scalar-Vector Multiplication • u and v: vectors, α and β: scalars • Vector-Vector Addition • Head-to-tail axiom: to visualize easily Scalar-vector multi. Head-to-tail axiom

7. Vectors are linear independent if the only set of scalars is Vector Spaces (3/4) • Vectors = n-tuples • Vector-vector addition • Scalar-vector multiplication • Vector space: • Linear Independence • Linear combination:

8. Vector Spaces (4/4) • Dimension • The greatest number of linearly independent vectors • Basis • n linearly independent vectors (n: dimension) • Representation • Unique expression in terms of the basis vectors • Change of Basis: Matrix M • Other basis 

9. Affine Spaces (1/2) • No Geometric Concept in Vector Space!! • Ex) location and distance • Vectors: magnitude and direction, no position • Coordinate System • Origin: a particular reference point Identical vectors Basis vectors located at the origin Arbitrary placement of basis vectors

10. Affine Spaces (2/2) • Third Type of Entity: Points • New Operation: Point-Point Subtraction • P and Q : any two points  vector-point addition • Frame: a Point and a Set of Vectors • Representations of the vector and point: n scalars Head-to-tail axiom for points Vector Point

11. Euclidean Spaces (1/2) • No Concept of How Far Apart Two Points in Affine Spaces!! • New Operation: Inner (dot) Product • Combine two vectors to form a real • α, β, γ, …: scalars, u, v, w, …:vectors • Orthogonal:

12. Euclidean Spaces (2/2) • Magnitude (length) of a vector • Distance between two points • Measure of the angle between two vectors • cosθ = 0  orthogonal • cosθ = 1  parallel

13. Projections • Problem of Finding the Shortest Distance from a Point to a Line of Plane • Given Two Vectors, • Divide one into two parts: one parallel and one orthogonal to the other Projection of one vector onto another

14. Matrices • Definitions • Matrix Operations • Row and Column Matrices • Rank • Change of Representation • Cross Product

15. 6.837 Linear Algebra Review What is a Matrix? • A matrix is a set of elements, organized into rows and columns rows columns

16. Definitions • n x m Array of Scalars (n Rows and m Columns) • n: row dimension of a matrix, m: column dimension • m = n square matrix of dimension n • Element • Transpose: interchanging the rows and columns of a matrix • Column Matrices and Row Matrices • Column matrix (n x 1 matrix): • Row matrix (1 x n matrix):

17. Basic Operations • Addition, Subtraction, Multiplication Just add elements Just subtract elements Multiply each row by each column 6.837 Linear Algebra Review

18. Matrix Operations (1/2) • Scalar-Matrix Multiplication • Matrix-Matrix Addition • Matrix-Matrix Multiplication • A: n x l matrix, B: l x m C: n x m matrix

19. Matrix Operations (2/2) • Properties of Scalar-Matrix Multiplication • Properties of Matrix-Matrix Addition • Commutative: • Associative: • Properties of Matrix-Matrix Multiplication • Identity MatrixI (Square Matrix)

20. Multiplication • Is AB = BA? Maybe, but maybe not! • Heads up: multiplication is NOT commutative! 6.837 Linear Algebra Review

21. Row and Column Matrices • Column Matrix • pT: row matrix • Concatenations • Associative • By Row Matrix

22. Vector Operations • Vector: 1 x N matrix • Interpretation: a line in N dimensional space • Dot Product, Cross Product, and Magnitude defined on vectors only y v x 6.837 Linear Algebra Review

23. Vector Interpretation • Think of a vector as a line in 2D or 3D • Think of a matrix as a transformation on a line or set of lines V V’ 6.837 Linear Algebra Review

24. 6.837 Linear Algebra Review Vectors: Dot Product • Interpretation: the dot product measures to what degree two vectors are aligned A A+B = C (use the head-to-tail method to combine vectors) B C B A

25. Vectors: Dot Product Think of the dot product as a matrix multiplication The magnitude is the dot product of a vector with itself The dot product is also related to the angle between the two vectors – but it doesn’t tell us the angle 6.837 Linear Algebra Review

26. Vectors: Cross Product • The cross product of vectors A and B is a vector C which is perpendicular to A and B • The magnitude of C is proportional to the cosine of the angle between A and B • The direction of C follows the right hand rule – this why we call it a “right-handed coordinate system” 6.837 Linear Algebra Review

27. Inverse of a Matrix • Identity matrix: AI = A • Some matrices have an inverse, such that:AA-1 = I • Inversion is tricky:(ABC)-1 = C-1B-1A-1 Derived from non-commutativity property 6.837 Linear Algebra Review

28. Determinant of a Matrix • Used for inversion • If det(A) = 0, then A has no inverse • Can be found using factorials, pivots, and cofactors! • Lots of interpretations – for more info, take 18.06 6.837 Linear Algebra Review

29. 6.837 Linear Algebra Review Determinant of a Matrix Sum from left to right Subtract from right to left Note: N! terms

30. Change of Representation (1/2) • Matrix Representation of the Change between the Two Bases • Ex) two bases and • Representations of v • Expression of in the basis or or

31. Change of Representation (2/2) • : n x n matrix • and • By direct substitution or

32. Cross Product • In 3D Space, a unit Vector, w, is Orthogonal to Given Two Nonparallel Vectors, u and v • Definition • Consistent Orientation • Ex) x-axis x y-axis = z-axis