Chun-Yuan Lin Mathematics for Computer Graphics 2015/12/15 1 CG

Coordinate Reference Frames 2015/12/15 CG 2 See the powerpoint: Coordinate Reference Frames.ppt

Points and Vectors (1) 2015/12/15 CG 3 There is a fundamental difference between the concept of a geometric point and that of a vector. A point is a position specified with coordinate values in some reference frame. (depend on the choice for the frame of refernece) A vector has properties that are independent of any particular coordinate system. Point Properties Frame A Frame B x y P

Points and Vectors (2) 2015/12/15 CG 4 Vector Properties We can define a vector as the difference between two point positions. V x and V y are the projection V onto the x and the y axes. We can obtain these same vector components using two other point positions in the same coordinate reference frames. A vector has no fixed position within a coordinate system. We can describe a vector as a directed line segment that has two fundamental properties: magnitude and direction. P2P2 P1P1 V

Points and Vectors (3) 2015/12/15 CG 5 Magnitude: We can specify the vector direction in various ways, such as A vector has the same magnitude and direction within a single coordinate system. If we transform the vector to another reference frame, the value for its components and direction within that reference frame may change. For a three-dimensional Cartesian vector representation

Points and Vectors (4) 2015/12/15 CG 6 We can give the vector direction in terms of the direction angles, α, β, γ. The values cosα, cos β, cos γ are called the direction cosines of the vector. Vectors are used to represent any quantities that have the properties of magnitude and direction. (force and velocity) V γ β α z y x

Points and Vectors (5) 2015/12/15 CG 7 Vector Addition and Scalar Multiplication V2V2 V1V1 V2V2 V1V1 V 1 +V 2

Points and Vectors (6) 2015/12/15 CG 8 Scalar Product of two Vectors This multiplication scheme is called the scalar product or dot product. (inner product) is the projection of vector V 2 in the direction of V 1. In addition to the coordinate-independent form of the scalar product. V2V2 V1V1 θ

Points and Vectors (7) 2015/12/15 CG 9 The scalar product of two vectors is zero if and only if the two vectors are perpendicular (orthogonal)

Points and Vectors (8) 2015/12/15 CG 10 Vector Product of Two Vectors V2V2 V1V1 V 1 × V 2 u Cross product

Points and Vectors (9) 2015/12/15 CG 11

Matrices (1) 2015/12/15 CG 12 A matrix is a rectangular array of quantities, called the elements of the matrix. We identify matrices according to the number of rows and number of columns. When the number of rows is the same as the number of columns, this matrix is called a square matrix. An r by c matrix Row vector Column vector

Matrices (2) 2015/12/15 CG 13 The matrix representation for a three-dimensional vector in Cartesian coordinates as We use this standard matrix representation for both points and vectors.

Scalar Multiplication and Matrix Addition 2015/12/15 CG 14

Matrix Multiplication(1) 2015/12/15 CG 15 The product of two matrices is defined as a generalization of the vector dot product.

Matrix Multiplication(2) 2015/12/15 CG 16 AB≠BA A(B+C)=AB+AC

Matrix Transpose The transpose M T of a matrix is obtained by interchanging rows and columns. 2015/12/15 CG 17 (M1M2) T =M 2 T M 1 T

Determinant of a Matrix If we have a square matrix, we can combine the matrix elements to produce a single number called the determinant of the matrix. 2015/12/15 CG 18

Matrix Inverse With square matrices, we can obtain an inverse matrix if and only of the determinant of the matrix is nonzero. 2015/12/15 CG 19 Identity matrix