Download Three basic (data) types in CG: scalars, points, and vectors

Math Primer for CG
Ref: Interactive Computer
Graphics, Chap. 4, E. Angel
Contents
Scalar, Vector, Point
Change of Basis
Frame
Change of Frame
Affine Sum, Convex Combination,
Convex Hull, …
Case Study: shooting game
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…
Scalars: Live in Real Space
Two fundamental
operations are
defined between
pairs:
Addition,
multiplication
 Closure
a,bS,
a+bS, a·bS

a,b,gS
Commutative
a+b=b+a
a · b=b · a
Associative
a ·(b · g)=(a · b) · g
a+(b+g)=(a+b)+g
Distributive
a ·(b+g)=(a · b)+(a · g)
Additive & multiplicative
inverse
a+(-a)=0, a · a-1=1
Real Analysis
The study of real numbers
…
If you are interested, see Analysis
WebNotes:
http://www.math.unl.edu/~webnotes/contents/chapters.htm
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, u+vV
Commutative
u+v=v+u

Associative
 u+(v+w)=(u+v)+w


Distributive
 a(u+v)=au+av
 (a+b)u=au+bu
Vector Space (cont)
Linear combination
u=a1u1+a2u2+…+anun
Linear independent
Only set of scalars such
that
u=a1u1+a2u2+…+anun
is zero
 a1=a2=…=an=0
Basis

a set of linearly
independent vectors that
span the space
Vector Space: change of basis
represent any vector
uniquely in a basis
u1

u2

 
u3  = v1 v2
a11 a 21 a 31 

v3 a12 a 22 a 32 
a13 a 23 a 33 




u = a1u1 + a 2u2 + a 3u3
  
u = u1 u 2




 
u = b1v1 + b 2v2 + b 3v3 = v1 v2
a1 

u3 a 2 
a 3 
change of basis




u1 = a11v1 + a12v2 + a13v3




u2 = a 21v1 + a 22v2 + a 23v3




u3 = a 31v1 + a 32v2 + a 33v3
v1

v2
 b1 

v3  b 2 
 b 3 
a11 a 21 a 31  a1 

 
v3 a12 a 22 a 32  a 2  = v1 v2
a13 a 23 a 33  a 3 
a11 a 21 a 31  a1   b1 
a
 a  =  b 
a
a
12
22
32

 2   2 
a13 a 23 a 33  a 3   b 3 
 b1 

v3  b 2 
 b 3 
Example
1

 
u = 2 = e1 e2
3
1 

e3 2
3
v1

v2
-1
e1

e2
New Basis
1

  
v3 = 1 = e1 + e2 + e3
1

 
e3  = v1 v2
 
= v1 v2
1


v1 = 0 = e1
0
1

 
v2 = 1 = e1 + e2
0

 
v3  = e1 e2
1 1 1

e3 0 1 1
0 0 1
  
u = v1 v2
1 1 1

v3 0 1 1
0 0 1
1 - 1 0 

v3 0 1 - 1
0 0 1 
1 - 1 0  1

 
v3 0 1 - 1 2 = v1 v2
0 0 1  3
- 1

v3 - 1
 3 
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
Affine Space: change of frames



P = b1v1 + b 2v2 + b 3v3 + Q0
Represent point in a
frame



P = a1u1 + a 2u2 + a 3u3 + P0
u1
u2
Change of frame




u1 = a11v1 + a12v2 + a13v3




u2 = a 21v1 + a 22v2 + a 23v3 v1 v2




u3 = a 31v1 + a 32v2 + a 33v3



P0 = a 41v1 + a 42v2 + a 43v3 + Q0
v3
u3
P0  = v1 v2
v3
a11 a 21 a 31 a 41 
a

a
a
a
12
22
32
42

Q0 
a13 a 23 a 33 a 43 


0
0
0
1


a11 a 211 a 31 a 41  a1 
a
 a 
a
a
a
12
221
32
42
  2  = v1 v2
Q0 
a13 a 23 a 33 a 43  a 3 

 
0
0
1  1 
0
a11 a 211 a 31 a 41  a1   b1 
a
 a   b 
a
a
a
12
221
32
42

 2  =  2 
a13 a 23 a 33 a 43  a 3   b 3 

   
0
0
0
1

 1   1 
v3
 b1 
b 
Q0  2 
 b3 
 
1
Euclidean Space
Supplement vector
space with the
notion of distance
New operation

Inner (dot) product
a,bS u,v,wV
u · v=v · u
(au+bv) · w=au · w + bv ·
w
v · v>0 (v0)
0 · 0=0
u · v=0  u and v are
orthogonal
|v|=(v · v)½
u · v=|u||v| cosq
Summary: Mathematical Spaces
Real Space
(Scalar)
Vector Space
(Scalar, Vector)
Euclidean Space
(Scalar, Vector)
[distance]
Affine Space
(Scalar, Vector, Point)
[location]
Affine Sum
In affine space, point addition is only
defined in the following case:

P = Q + av = Q + a ( R - Q)
P = aR + (1 - a )Q
P = a1R + a 2Q where a1 + a 2 = 1
Point addition is allowed
only if their weights add
up to one
Q
R
P(a)
Affine Sum (cont)
More generally,
P = a i Pi where a i = 1
i
i
Convex Combination
 A particular affine sum where weights are nonnegative
P = a i Pi where a i = 1 and a i  0
i
i
Convex Hull
Now, how to apply these math
Besides change of basis/frame…
Ex: Parametric Equation of a Line
R
S (a ) = aQ + (1 - a ) R, a  R
S(a)
Q
R
Q
S(a)
S (a ) = aQ + (1 - a ) R, 0  a  1
Ex: Plane, Triangle, Parallelogram

u  Q - P;

v  R-P


T (a , b ) = P + au + bv
R

v
P
T(a,b)

u
S(a)
= P + a (Q - P) + b ( R - P)
Q
= (1 - a - b ) P + aQ + bR
(infinite) Plane
a, b  R
Parallelogram
0  a  1 and 0  b  1
Triangle
0  a  1, 0  b  1, 0  a + b  1
Ex: Homogeneous Coordinates
A unified representation scheme in
affine space for points and vectors
Change of Frame: subsume change of
basis
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



P = a1u1 + a 2u2 + a 3u3 + P0
 
P = u1 u2

u3
Point: [a1 a2 a3 1]
Frame: [u1 u2 u3 P0]
  
u = u1 u2

u3
Vector: [a1 a2 a3 0]
Frame: [u1 u2 u3 P0]
a1 
a 
P0  2 
a 3 
 
1
a1 
a 
P0  2 
a 3 
 
0
Ex: Shooting (AABB)
p0
p0
Intersection can be
checked using 3D
Sutherland
algorithm
Cohen-Sutherland Line-Clipping Algorithm
Trivially accept
Trivially reject
clipping
Ex: Shooting (OBB)
p0
p0
Change of frame
then apply
Sutherland
algorithm in the
local frame
Ex: Ray-Triangle Intersection
Ray :

w
q0

v

p (t ) = p0 + tu , t  0
q2
Triangle

u
q1
q (a , b ) = (1 - a - b )q0 + aq1 + bq2 ,
p0
0  a  1, 0  b  1, 0  a + b  1
Solve
p (t ) = q (a , b ) for triple t (t , a , b )
Ex: Shooting (discrete version)
p0 = p(t0 ); p0 = p (t0 + t )
Bullet Path :

w
q0

v
p(t ) = (1 - t ) p0 + tp0
p0
Triangle
p0
q (a , b ) = (1 - a - b )q0 + aq1 + bq2 ,
0  a  1,0  b  1,0  a + b  1
Solve
p(t ) = q (a , b ) for triple t (t , a , b )
Ex: Shooting (continuous version)
Before (t0 ) : p0 ; q0 , q1 , q2
After (t0 + t ) : p0 ; q0 , q1, q2
Bullet Path :
p (t ) = (1 - t ) p0 + tp0 , 0  t  1
q2
q0
p0
q2
Triangle (in motion) :
p0
q0
q1
q1
q (a , b ) = (1 - a - b )q0 (t ) + aq1 (t ) + bq2 (t )
q0 (t ) = (1 - t )q0 + tq0
q1 (t ) = (1 - t )q1 + tq1
q2 (t ) = (1 - t )q2 + tq2
0  a  1, 0  b  1, 0  a + b  1
Solve
p (t ) = q (a , b , t ) for triple t (t , a , b )
Exercise
Convex Hull: prove convex combination
yields the shape you imagine