Math Primer for CG

1
Math Primer for CG

• Ref Interactive Computer Graphics, Chap. 4, E.
  Angel

2
Contents

• Scalar, Vector, Point
• Change of Basis
• Frame
• Change of Frame
• Affine Sum, Convex Combination, Convex Hull,
• Case Study shooting game

3
Introduction

• Three basic data types in CG scalars, points,
  and vectors
• scalar not a geometric type per se used in
  measurement
• point a location in space exist regardless of
  any coordinate system
• vector any quantity with direction and
  magnitude does not have fixed location
• Examine these concepts in a mathematically more
  rigorous way

4
Scalars Live in Real Space

• ?a,b,g?S
• Commutative
• abba
• a bb a
• Associative
• a (b g)(a b) g
• a(bg)(ab)g
• Distributive
• a (bg)(a b)(a g)
• Additive multiplicative inverse
• a(-a)0, a a-11

• Two fundamental operations are defined between
  pairs
• Addition, multiplication
• Closure
• ?a,b?S,
• ab?S, ab?S

5
Real Analysis

• The study of real numbers
• If you are interested, see Analysis WebNotes
• http//www.math.unl.edu/webnotes/contents/chapter
  s.htm

6
Vectors Live in Vector Space

• Two kinds of entities
• Scalar, vector
• Two operations and corresponding geometric
  interpretations
• v-v addition
• (head-tail)
• scalar-v multiplication
• (scaling of vector)

• Properties
• Closure
• ?u,v?V, uv?V
• Commutative
• uvvu
• Associative
• u(vw)(uv)w
• Distributive
• a(uv)auav
• (ab)uaubu

7
Vector Space (cont)

• Linear combination
• ua1u1a2u2anun

• Linear independent
• Only set of scalars such that
• ua1u1a2u2anun
• is zero
• a1a2an0
• Basis
• a set of linearly independent vectors that span
  the space

8
Vector Space change of basis

• represent any vector uniquely in a basis
• change of basis

9
Example
New Basis
10
Points Live in Affine Space

• Vector space lacks
• Location, distance,
• Concept of coordinate system (frame)
• Reference point origin
• Frame origin basis defines position in space
• Add another entity to vector spaces
• Point

• New operations
• Point Point ? Vector
• Point Vector ? Point (translation)
• Note that the following are not defined
• point addition
• multiplication of scalar point

11
Affine Space change of frames

• Represent point in a frame
• Change of frame

12
Euclidean Space

• Supplement vector space with the notion of
  distance
• New operation
• Inner (dot) product

• ?a,b?S ?u,v,w?V
• u vv u
• (aubv) wau w bv w
• v vgt0 (v?0)
• 0 00
• u v0 ? u and v are orthogonal
• v(v v)½
• u vuv cosq

13
Summary Mathematical Spaces
Real Space (Scalar)
Vector Space (Scalar, Vector)
Euclidean Space (Scalar, Vector) distance
Affine Space (Scalar, Vector, Point) location
14
Affine Sum

• In affine space, point addition is only defined
  in the following case

Point addition is allowed only if their weights
add up to one
15
Affine Sum (cont)

• More generally,
• Convex Combination
• A particular affine sum where weights are
  non-negative

16
Convex Hull
17
Now, how to apply these math

• Besides change of basis/frame

18
Ex Parametric Equation of a Line
19
Ex Plane, Triangle, Parallelogram
R
T(a,b)
P
Q
S(a)
(infinite) Plane
Parallelogram
Triangle
20
Ex Homogeneous Coordinates

• A unified representation scheme in affine space
  for points and vectors
• Change of Frame subsume change of basis

21
Affine Space coordinate system

• fixing the origin of the vectors of coordinate
  system at P0
• 4x1 vector
• Unified vector point representation
• homogeneous coordinate
• distinguish point vector
• 4x4 matrix algebra for change in frames

Point a1 a2 a3 1 Frame u1 u2 u3 P0
Vector a1 a2 a3 0 Frame u1 u2 u3 P0
22
Ex Shooting (AABB)

• Intersection can be checked using 3D Sutherland
  algorithm

23
Cohen-Sutherland Line-Clipping Algorithm
Trivially accept Trivially reject clipping
24
Ex Shooting (OBB)

• Change of frame then apply Sutherland algorithm
  in the local frame

25
Ex Ray-Triangle Intersection
26
Ex Shooting (discrete version)
27
Ex Shooting (continuous version)
28
Exercise

• Convex Hull prove convex combination yields the
  shape you imagine