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

CGVR Graphics Korea University cgvr.korea.ac.kr Contents Complex Numbers Pure imaginary number Complex conjugate Representation with polar coordinates Quaternions Definition Addition, scalar multiplication, multiplication, division, and inverse 3D Rotations Nonparametric Representations Parametric Representations

CGVR Graphics Korea University cgvr.korea.ac.kr Complex Numbers Real Part + Imaginary Part: Addition and Subtraction Scalar Multiplication Multiplication y x z imaginary axis real axis

CGVR Graphics Korea University cgvr.korea.ac.kr Pure Imaginary Number & Complex Conjugate Imaginary Unit: Complex Conjugate Modulus or absolute value Division

CGVR Graphics Korea University cgvr.korea.ac.kr Representation with Polar Coordinates Euler’s Formula Complex Multiplication and Division n th Roots r θ z=(x, y) imaginary axis real axis

CGVR Graphics Korea University cgvr.korea.ac.kr Quaternions One Real Part + Three Imaginary Part Properties: Addition and Scalar Multiplication

CGVR Graphics Korea University cgvr.korea.ac.kr Ordered-Pair Notation Scalar ‘ s ’ + Vector “ v = (a, b, c) ” Addition: Multiplication Magnitude Inverse

CGVR Graphics Korea University cgvr.korea.ac.kr 3D Rotation For a 3D Point ( α, β, γ ) A unit quaternion its conjugate  Rotating ( α, β, γ ) by angle 2θ about the axis parallel to ( a, b, c ) For R q is a 3D Rotation about ( a, b, c ) by 2θ

CGVR Graphics Korea University cgvr.korea.ac.kr Nonparametric Representations Definition Object descriptions directly in terms of the coordinates of the reference frame Ex. implicit expression: Ex. explicit Expression: Advantages Useful in describing objects Disadvantages Change the independent variable as derivatives Implicit vs. Explicit Equations

CGVR Graphics Korea University cgvr.korea.ac.kr Parametric Representations 3D Curves with Parameter u Ex. 3D Surfaces with Parameter u, v Ex.