Mathematics for Computer Graphics (Appendix A) Won-Ki Jeong

A-1. Coordinate Reference Frame 2D Cartesian reference frame x y x y

2D Polar Coordinate reference frame

3D Cartesian reference frame Right-handed v.s left-handed Right-handedLeft-handed

3D curvilinear coordinate systems General curvilinear reference frame  Orthogonal coordinate system  Each coordinate surfaces intersects at right angles

Cylindrical-coordinate : vertical cylinder : vertical plane containing z-axis : horizontal plane parallel to xy-plane constant Transform to Cartesian coordinator x axis y axisz axis

Spherical-coordinate x axis y axis z axis : sphere : vertical plane containing z-axis : cone with the apex at the origin constant Transform to Cartesian coordinator

Solid angle 3D Angle defined on a sphere(steradian) Steradian : Total solid angle : steradian

A-2. Points & Vectors Point  Position in some reference frame  Distance from the origin depends on the reference frame P Frame B Frame A x y

Vector  Difference between two point positions  Properties : Magnitude & direction  Same properties within a single coordinate system  Magnitude is independent from coordinate frames Magnitude : Direction :

3D vector Magnitude Directional angle

Vector addition & scalar multiplication Addition Scalar multiplication

Vector multiplication Scalar product(inner product) Commutative : Distributive : Orthogonal :

Vector product(Cross product) Noncommutative : Nonassociative : Distributive : Right-handed rule!

A-3. Basis vectors and the metric tensor Basis of vector space  Linearly independent axis vectors Orthonormal basis  Orthogonal :  Normalized :  Orthonormal = Orthogonal + Normalized  Orthonormal basis of 3D Cartesian reference frame

Metric tensor Tensor  Generalization of a vector with rank & dim. that satisfy certain transformation properties  n-th rank with dim m : m-dimensional space which has n indices  Rank 0: scalar, rank 1: dim m vector rank 2 : vector which has m 2 component Metric tensor  Definition :  The tensor for  Distance metric  Used as transformation equation  Component of differential vector operators (gradient, divergence, and curl)

Example of metric tensor Cartesian coordinate system Polar coordinates If j = k otherwise Pythagorean theorem : For 3D Cartesian coordinate system :

A-4. Matrices Rows & columns Matrix multiplication Column row Properties

Transpose & Determinant Matrix transpose Determinant  Large matrix A

Inverse of a matrix Inverse matrix  Determinant is not 0 : Non-singular matrix  Elements of

A-5. Complex numbers Real + Imaginary part Real axis Imaginary axis

Polar form & Euler ’ s formula Polar form Euler ’ s formula Real axis Imaginary axis

A-6. Quaternions Higher dimension complex number Addition, multiplication, magnitude, & inverse

A-7. Nonparameteric representation Direct description in terms of the reference frame  Surface : or  Curve :  Useful in the given reference frame Example (circle) Implicit form Explicit form

A-8. Parameteric representation Use parameter domain  Curve  Ex. Circle  Surface  Ex. Spherical surface r : radius of the sphere u: latitude v: longitude

A-9. Numerical methods Solving sets of linear equation  Matrix form  Cramer ’ s rule  Adequate for a few variables : matrix A with the kth column replaced with B

 Gauss elimination  Elementary Row Operation  Multiply a row through by a nonzero constant  Interchange two rows  Add a multiple of one row to another row  Make row-echelon form by e.r.o  Row-echelon form  First nonzero number of each row is 1(leading 1)  Entire-zero-rows are grouped together at the bottom of the matrix  In any successive non-entire-zero rows, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row

 Gauss-Seidel method  Start with initial guess and repeatedly calculate successive approximations until their difference is small  Convergence condition  Each diagonal element of a matrix A has a magnitude greater than the sum of the magnitudes of the other elements across that row

Finding roots of nonlinear equation Object  Finding the solution of Newton-Raphson algorithm  Iterative approximation  Fast, but it may be fail to converge Initial guess

Bisection method  Convergence guaranteed

Evaluating integrals Rectangle approximation Polynomial approximation  Simpson ’ s rule

 Monte Carlo method  For high-frequency oscillation function or multiple integrals  Use random positions : uniformly distributed : # of random points between f(x) and x-axis Given two random number r 1 and r 2 :

Fitting curves to data sets  Least-squares algorithm  Fitting a function to a set of data points  Ex. 2D linear case Solve linear equation!