Graphics Graphics Korea University kucg.korea.ac.kr Mathematics for Computer Graphics 고려대학교 컴퓨터 그래픽스 연구실

KUCG Graphics Korea University kucg.korea.ac.kr Spaces Scalars (Linear) Vector Space Scalars and vectors Affine Space Scalars, vectors, and points Euclidean Space Scalars, vectors, points Concept of distance Projections Exercise

KUCG Graphics Korea University kucg.korea.ac.kr Scalars (1/2) Scalar Field Scalar Field Ex) Ordinary real numbers and operations on them Tow Fundamental Operations Addition and multiplication Associative Commutative Distributive

KUCG Graphics Korea University kucg.korea.ac.kr Scalars (2/2) Two Special Scalars Additive identity: 0, multiplicative identity: 1 Additive inverse and multiplicative inverse of

KUCG Graphics Korea University kucg.korea.ac.kr Vector Spaces (1/4) ScalarsVectors Two Entities: Scalars and Vectors Vectors Directed line segments n -tuples of numbers Two operations: vector-vector addition, scalar- vector multiplication Zero Vector Special Vector: Zero Vector Let u denote a vector Directed line segments

KUCG Graphics Korea University kucg.korea.ac.kr 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 Head-to-tail axiom Scalar-vector multi.

KUCG Graphics Korea University kucg.korea.ac.kr Vector Spaces (3/4) Vectors = n -tuples Vector-vector addition Scalar-vector multiplication Vector space: Linear Independence Linear Independence Linear combination: Vectors are linear independent if the only set of scalars is

KUCG Graphics Korea University kucg.korea.ac.kr Vector Spaces (4/4) Dimension Dimension The greatest number of linearly independent vectors Basis Basis n linearly independent vectors ( n : dimension) Representation Unique expression in terms of the basis vectors Change of Basis: Matrix M Other basis 

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

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

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

KUCG Graphics Korea University kucg.korea.ac.kr 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

KUCG Graphics Korea University kucg.korea.ac.kr 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

KUCG Graphics Korea University kucg.korea.ac.kr Matrices Definitions Matrix Operations Row and Column Matrices Rank Change of Representation Cross Product Eigenvalues and Eigenvectors Exercise

KUCG Graphics Korea University kucg.korea.ac.kr 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):

KUCG Graphics Korea University kucg.korea.ac.kr Matrix Operations (1/2) Scalar-Matrix Multiplication Scalar-Matrix Multiplication Matrix-Matrix Addition Matrix-Matrix Addition Matrix-Matrix Multiplication Matrix-Matrix Multiplication A : n x l matrix, B : l x m  C : n x m matrix

KUCG Graphics Korea University kucg.korea.ac.kr Matrix Operations (2/2) Properties of Scalar-Matrix Multiplication Properties of Matrix-Matrix Addition Commutative: Associative: Properties of Matrix-Matrix Multiplication Identity Matrix Identity Matrix I (Square Matrix)

KUCG Graphics Korea University kucg.korea.ac.kr Row and Column Matrices Column Matrix Cf) p T : row matrix Concatenations Concatenations Associative By Row Matrix

KUCG Graphics Korea University kucg.korea.ac.kr Rank (1/2) Inverse Matrix Inverse Matrix B: inverse of A A: nonsingular Cf) singular: noninvertible matrix Determinant Determinant of A : |A| 2 x 2 matrix n x n matrix ( n >2) The inverse of a square matrix exists, if and only if the determinant of the matrix is non zero.

KUCG Graphics Korea University kucg.korea.ac.kr Rank (2/2) For General Nonsquare Matrix Ex) n x m matrix  rows: elements of Euclidean space R m  columns: elements of R n  determine how many rows (or columns) are linearly independent Rank Row (Column) Rank Maximum number of linearly independent rows (columns) A matrix is invertible, if and only if its rows (and columns) are linearly independent.

KUCG Graphics Korea University kucg.korea.ac.kr How to Compute the Rank? Example) Find the Rank of the Matrix Solution) The Reduced Row-Echelon Form of A

KUCG Graphics Korea University kucg.korea.ac.kr Reduced Row-Echelon Form

KUCG Graphics Korea University kucg.korea.ac.kr A Theorem about Rank Example) Find the Column Rank of A Solution) If A is any matrix, then the row space and column space of A have the same dimension.

KUCG Graphics Korea University kucg.korea.ac.kr 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

KUCG Graphics Korea University kucg.korea.ac.kr Change of Representation (2/2) : n x n matrix and By direct substitution or

KUCG Graphics Korea University kucg.korea.ac.kr Cross Product In 3D Space, a Vector, w, is Orthogonal to Given Two Nonparallel Vectors, u and v Definition Consistent Orientation Ex) x -axis x y -axis = z -axis

KUCG Graphics Korea University kucg.korea.ac.kr Eigenvalues and Eigenvectors (1/2) When Does a Transformation Leave a Point of Vector Unchanged? u : eigenvectors (not a matrix of zeros) of M λ : eigenvalues of M Find the Eigenvalues Polynomial of degree n in λ Square matrix (transformation) Column matrix (point or vector)  Nontrivial!!!

KUCG Graphics Korea University kucg.korea.ac.kr Eigenvalues and Eigenvectors (2/2) Similarity Transformation that Convert M to a Diagonal Matrix Q  Diagonal Elements of Q are the Eigenvalues of Q and M Eigenvalues of Q are same as those of M Geometric Interpretation Consider an ellipsoid ( T : nonsingular matrix)

KUCG Graphics Korea University kucg.korea.ac.kr Exercises (1/2) C.1 C.1 In R 3, consider the two bases {(1,0,0), (1,1,0), (1,1,1)} and {(1,0,0), (0,1,0), (0,0,1)}. Find the two matrices that convert representations between the two bases. Show that they are inverses of each other. C.3 C.3 Suppose that i, j, and k represent the unit vectors in the x, y, and z directions, respectively, in R 3. Show that the cross product u x v is given by the matrix

KUCG Graphics Korea University kucg.korea.ac.kr Exercises (2/2) C.5 C.5 Find the eigenvalues and eigenvectors of the two-dimensional rotation matrix C.6 C.6 Find the eigenvalues and eigenvectors of the three-dimensional rotation matrix