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Graphics Mathematics for Computer Graphics CSE 581 – Roger Crawfis (slides developed from Korea University slides) CSE 581 – Interactive Computer Graphics Spaces Scalars (Linear) Vector Space Affine Space Scalars, vectors, and points Euclidean Space Scalars and vectors Scalars, vectors, points Concept of distance Projections CSE 581 – Interactive Computer Graphics Scalars (1/2) Scalar Field Ex) Ordinary real numbers and operations on them Two Fundamental Operations Addition and multiplication , S , S , S Commutative Associative Distributive CSE 581 – Interactive Computer Graphics Scalars (2/2) Two Special Scalars Additive identity: 0, multiplicative identity: 1 0 0 1 1 1 Additive inverse and multiplicative inverse of 0 1 1 CSE 581 – Interactive Computer Graphics Vector Spaces (1/4) Two Entities: Scalars and Vectors Vectors Directed line segments n-tuples of numbers Two operations: vector-vector addition, scalarvector multiplication Special Vector: Zero Vector u0u u u 0 Let u denote a vector Directed line segments CSE 581 – Interactive Computer Graphics Vector Spaces (2/4) Scalar-Vector Multiplication u v u v u u u u and v: vectors, α and β: scalars Scalar-vector multi. Vector-Vector Addition Head-to-tail axiom: to visualize easily Head-to-tail axiom CSE 581 – Interactive Computer Graphics Vector Spaces (3/4) Vectors = n-tuples v v1 , v2 ,, vn Vector-vector addition u v u1 , u2 , , un v1 , v2 , , vn u1 v1 , u2 v2 , , un vn Scalar-vector multiplication Vector space: R n v v1 , v2 ,, vn Linear Independence Linear combination: u 1u1 2u2 nun Vectors are if the only set of scalars linear independent 1u1 2u2 nun 0 is 1 2 n 0 CSE 581 – Interactive Computer Graphics Vector Spaces (4/4) Dimension The greatest number of linearly independent vectors Basis n linearly independent vectors (n: dimension) Unique expression in terms of the basis vectors Representation i v 1v1 2v2 n vn Change of Basis: Matrix M Other basis v1 , v2 , , vn v 1v1 2v2 nvn 1 1 2 M 2 n n CSE 581 – Interactive Computer Graphics 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 Basis vectors located at the origin Identical vectors Arbitrary placement of basis vectors CSE 581 – Interactive Computer Graphics Affine Spaces (2/2) Third Type of Entity: Points New Operation: Point-Point Subtraction P and Q : any two points v P Q vector-point addition P v Q P Q Q R P R Head-to-tail axiom for points Frame: a Point P0 and a Set of Vectors v1 , v2 , , vn Representations of the vector and point: n scalars Vector v 1v1 2 v2 n vn Point P P0 1v1 2 v2 n vn CSE 581 – Interactive Computer Graphics 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 u v v u u v w u w v w v v 0 if v 0 00 0 Orthogonal: u v 0 CSE 581 – Interactive Computer Graphics Euclidean Spaces (2/2) Magnitude (length) of a vector v Distance between two points P Q vv P Q P Q Measure of the angle between two vectors u v u v cos cosθ = 0 orthogonal cosθ = 1 parallel CSE 581 – Interactive Computer Graphics 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 w v u w v v v u v v v wv vv wv u w v w v vv Projection of one vector onto another CSE 581 – Interactive Computer Graphics Matrices Definitions Matrix Operations Row and Column Matrices Rank Change of Representation Cross Product CSE 581 – Interactive Computer Graphics What is a Matrix? A matrix is a set of elements, organized into rows and columns rows columns a b c d 6.837 Linear Algebra Review CSE 581 – Interactive Computer Graphics 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 aij , i 1, , n, j 1, , m A aij Transpose: interchanging the rows and columns of a matrix T A a ji b1 Column Matrices and Row Matrices b2 Column matrix (n x 1 matrix): b bi Row matrix (1 x n matrix): b T bn CSE 581 – Interactive Computer Graphics Basic Operations Addition, Subtraction, Multiplication a b e c d g f a e b f h c g d h Just add elements a b e c d g f a e b f h c g d h Just subtract elements a b e c d g f ae bg h ce dg 6.837 Linear Algebra Review af bh cf dh Multiply each row by each column CSE 581 – Interactive Computer Graphics Matrix Operations (1/2) Scalar-Matrix Multiplication A aij Matrix-Matrix Addition C A B aij bij Matrix-Matrix Multiplication A: n x l matrix, B: l x m C: n x m matrix C AB cij l cij aik bkj k 1 CSE 581 – Interactive Computer Graphics Matrix Operations (2/2) Properties of Scalar-Matrix Multiplication Properties of Matrix-Matrix Addition A A A A AB BA Commutative: Associative: A B C A B C Properties of Matrix-Matrix Multiplication Identity Matrix I (Square Matrix) ABC AB C AB BA 1 if i j I aij , aij 0 otherwise AI A IB B CSE 581 – Interactive Computer Graphics Multiplication Is AB = BA? Maybe, but maybe not! a b e c d g f ae bg ... h ... ... e g f a b ea fc ... h c d ... ... Heads up: multiplication is NOT commutative! 6.837 Linear Algebra Review CSE 581 – Interactive Computer Graphics Row and Column Matrices pT: row matrix Concatenations x p y z Column Matrix p Ap p ABCp Associative By Row Matrix AB T BT A T pT pT CT BT AT CSE 581 – Interactive Computer Graphics Vector Operations Vector: 1 x N matrix Interpretation: a line in N dimensional space Dot Product, Cross Product, and Magnitude defined on vectors only a v b c y v x 6.837 Linear Algebra Review CSE 581 – Interactive Computer Graphics 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 x a b x' y c d y ' 6.837 Linear Algebra Review V V’ CSE 581 – Interactive Computer Graphics Vectors: Dot Product Interpretation: the dot product measures to what degree two vectors are aligned A B C B A+B = C (use the head-to-tail method to combine vectors) A 6.837 Linear Algebra Review CSE 581 – Interactive Computer Graphics Vectors: Dot Product d a b abT a b c e ad be cf f a aa T aa bb cc 2 a b a b cos( ) 6.837 Linear Algebra Review 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 CSE 581 – Interactive Computer Graphics 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” a b a b sin( ) 6.837 Linear Algebra Review CSE 581 – Interactive Computer Graphics 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 noncommutativity property 6.837 Linear Algebra Review 1 0 0 I 0 1 0 0 0 1 CSE 581 – Interactive Computer Graphics 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 a b A c d det( A) ad bc 1 d b A ad bc c a 1 CSE 581 – Interactive Computer Graphics Determinant of a Matrix a b d e g h c f aei bfg cdh afh bdi ceg i a b d e g h c a b f d e i g h c a b f d e i g h 6.837 Linear Algebra Review c f i Sum from left to right Subtract from right to lef Note: N! terms CSE 581 – Interactive Computer Graphics Change of Representation (1/2) Matrix Representation of the Change between the Two Bases Ex) two bases u1 , u2 , , un and v1 , v2 ,, vn v 1u1 2u2 nun or v 1v1 2v2 n vn Representations of v a 1 2 n T or b 1 2 n T Expression of u1 , u2 , , un in the basis v1 , v2 ,, vn ui i1v1 i 2v2 invn , i 1,, n CSE 581 – Interactive Computer Graphics Change of Representation (2/2) u1 v1 A : n x n matrix ij u v 2 A 2 un v n u1 a and b i i u v aT 2 or un By direct substitution v1 v1 u1 v1 v v u v aT A 2 b T 2 aT 2 bT 2 vn v n u n vn v1 v v bT 2 vn b T aT A CSE 581 – Interactive Computer Graphics Cross Product In 3D Space, a unit Vector, w, is Orthogonal to Given Two Nonparallel Vectors, u and v wu w v 0 Definition Consistent Orientation 2 3 3 2 w u v 3 1 1 3 1 2 2 1 where u 1 , 2 , 3 , v 1 , 2 , 3 Ex) x-axis x y-axis = z-axis CSE 581 – Interactive Computer Graphics