Download Computer Graphics (CS 543) Lecture 4(Part 2): Linear Algebra for

Computer Graphics (CS 543)
Lecture 4(Part 2): Linear Algebra for Graphics (Points, Scalars, Vectors)
Prof Emmanuel Agu
Computer Science Dept.
Worcester Polytechnic Institute (WPI)
Points, Scalars and Vectors
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Points, vectors defined relative to a coordinate system
Example: Point (5,4)
y
(5,4)
(0,0)
x
Vectors

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Magnitude
Direction
NO position
Can be added, scaled, rotated
CG vectors: 2, 3 or 4 dimensions
Length
Angle
Points
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Location in coordinate system
Cannot add or scale
Subtract 2 points = vector
Point
Vector‐Point Relationship
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Diff. b/w 2 points = vector
v = Q – P
point + vector = point
v + P = Q
P
v
Q
Vector Operations
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Define vectors
a  (a1, a2 , a3 )
b  (b1,b2 , b3 )
Then vector addition:
a  b  (a1  b1, a2  b2 , a3  b3 )
a
a+b
b
Vector Operations
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Define scalar, s
Scaling vector by a scalar
as  (a1s, a2 s, a3 s )
Note vector subtraction:
ab
 (a1  (b1 ), a2  (b2 ), a3  (b3 ))
a
a-b
a
2.5a
b
Vector Operations: Examples
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Scaling vector by a scalar
as  (a1s, a2 s, a3 s )
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•Vector addition:
a  b  (a1  b1, a2  b2 , a3  b3 )
For example, if a=(2,5,6) and b=(‐2,7,1)
and s=6, then
a  b  (a1  b1, a2  b2 , a3  b3 )  (0,12,7)
as  (a1s, a2 s, a3 s )  (12,30,36)
Affine Combination
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Given a vector a  (a1, a2 , a3 ,..., an )
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a1  a2  .........an  1
Affine combination: Sum of all components = 1
Convex affine = affine + no negative component
i.e
a1 , a2 ,.........an  non  negative
Magnitude of a Vector
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Magnitude of a
2
2
| a | a1  a2 ..........  an

2
Normalizing a vector (unit vector)
a
vector
â  
a magnitude

Note magnitude of normalized vector = 1. i.e
2
2
2
a1  a2 ..........  an  1
Magnitude of a Vector
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Example: if a = (2, 5, 6)
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Magnitude of a
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Normalizing a
| a | 2 2  52  6 2  65
5
6 
 2
â  
,
,

 65 65 65 
Convex Hull
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Smallest convex object containing P1P2,…..Pn
Formed by “shrink wrapping” points
12
Dot Product (Scalar product)
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Dot product,
d  a  b  a1  b1  a2  b2 ........  a3  b3
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For example, if a=(2,3,1) and b=(0,4,‐1)
then
a b  (2  0)  (3  4)  (1 1)
 0  12  1  11
Properties of Dot Products
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Symmetry (or commutative):
a b  b a
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Linearity:
(a  c)  b  a  b  c  b
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Homogeneity:
( sa)  b  s (a  b)
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And
b2  b  b
Angle Between Two Vectors
y
b   b cos b , b sin b 
c
c

b
c   c cos c , c sin c 
b  c  b c cos
b
x
b
b
b
Sign of b.c:
c
b.c > 0
c
b.c = 0
c
b.c < 0
Angle Between Two Vectors
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Find the angle b/w the vectors b = (3,4) and c = (5,2)
Angle Between Two Vectors
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Find the angle b/w the vectors b = (3,4) and c = (5,2)
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|b|= 5, |c|= 5.385
3 4
b̂   , 
5 5
  31.326
bˆ  cˆ  0.85422  cos
Standard Unit Vectors
y
Define
i  1,0,0
j  0,1,0 
k  0,0,1
So that any vector,
i
k
0
z
v  a, b, c   ai  bj  ck
j
x
Cross Product (Vector product)
If
a  a x , a y , a z 
b  bx , by , bz 
Then
a  b  (a y bz  a z by )i  (a x bz  a z bx ) j  (a x by  a y bx )k
Remember using determinant
i
ax
bx
Note: a x b is perpendicular to a and b
j
ay
by
k
az
bz
Cross Product
Note: a x b is perpendicular to both a and b
axb
a
0
b
Cross Product
Calculate a x b if a = (3,0,2) and b = (4,1,8)
Cross Product
Calculate a x b if a = (3,0,2) and b = (4,1,8)
a x b = -2i – 16j + 3k
Normal for Triangle using Cross Product Method
n
plane
p2
n·(p - p0 ) = 0
n = (p2 - p0 ) ×(p1 - p0 )
normalize n  n/ |n|
p
p1
p0
Note that right‐hand rule determines outward face
Newell Method for Normal Vectors
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Problems with cross product method:
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calculation difficult by hand, tedious
If 2 vectors almost parallel, cross product is small
Numerical inaccuracy may result
p1
p0

p2
Proposed by Martin Newell at Utah (teapot guy)
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Uses formulae, suitable for computer
Compute during mesh generation
Robust!
Newell Method Example
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Example: Find normal of polygon with vertices P0 = (6,1,4), P1=(7,0,9) and P2 = (1,1,2)
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Using simple cross product:
((7,0,9)‐(6,1,4)) X ((1,1,2)‐(6,1,4)) = (2,‐23,‐5)
P1 - P0
P2 - P0
P2
(1,1,2)
PO
(6,1,4)
P1 (7,0,9)
Newell Method for Normal Vectors
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Formulae: Normal N = (mx, my, mz)
N 1
mx    yi  ynext ( i ) zi  z next (i ) 
i 0
N 1
m y   zi  z next (i ) xi  xnext (i ) 
i 0
N 1
mz   xi  xnext (i )  yi  ynext (i ) 
i 0
Using Newell method, for previous example plug in values result is same:
Normal is (2, -23, -5)
Finding Vector Reflected From a Surface
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a = original vector
n = normal vector
r = reflected vector
m = projection of a along n
e = projection of a orthogonal to n
n
Note: Θ1 = Θ2
a
r
e  am
r  em
 r  a  2m
m
Θ1
e
-m
Θ2
e
Lines
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Consider all points of the form
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P()=P0 +  d
Line: Set of all points that pass through P0 in direction of vector d
Parametric Form
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Two‐dimensional forms of a line
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Explicit: y = mx +h
Implicit: ax + by +c =0
Parametric: x() = x0 + (1-)x1
y() = y0 + (1-)y1
Parametric form of line
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
α
P1
1-α
Po
More robust and general than other forms
Extends to curves and surfaces
Pα
Convexity
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An object is convex iff for any two points in the object all points on the line segment between these points are also in the object
P
P
Q
convex
Q
not convex
Curves and Surfaces
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Curves: 1‐parameter non‐linear functions of the form P()
Surfaces: two‐parameter functions P(, )
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Linear functions give planes and polygons
P()
P(, )
References
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Angel and Shreiner, Interactive Computer Graphics, 6th edition, Chapter 3
Hill and Kelley, Computer Graphics using OpenGL, 3rd
edition, Sections 4.2 ‐ 4.4