Outline

1
Outline

• What is polar form?
• B-Spline
• Labeling, why (0,1,2), (1,2,3) ?
• Knot Vector, why (-2,-1,0,1,2,) ?
• Derivation of B-Spline points
• Maybe(!?)

2
What is Polar Form Labeling?

• The sequences of parameters applied at each level
  of pyramid for de Casteljau evaluation.

P0
P1
P2
1-u0
1-u0
u0
u0
u0
u1
1-u1
u1
Pnew (u0, u1)
3
Connecting Bezier Curves
u0 1, u1 0
u0 0
u1 1
4
Subdivision of Curve
u 0.5
u 0
u 1
5
Ideal Uniform ParameterizationBut Hard to
Control !!!
u 1
u 0
u 2
6
Remember Curve-Subdivision
u 1
u 1
u 1
u 0
u 0
u 0
u 2
u 2
7
Controllability
u 1
u 0
u 2
8
Adding the 3rd Segment
u 1
u 0
u 2
u 3
9
Reverse Subdivision
u 1
u 0
u 2
u 3
10
Adding New Control
u 1
u 0
u 2
u 3
11
Prepend Segments u-10
u 1
u 0
u 2
u -1
u 3
12
Add the New Control
u 1
u 0
u 2
u -1
u 3
13
Prepend new Segment u -2-1
u 1
u -2
u 0
u 2
u -1
u 3
14
Add new control
u 1
u -2
u 0
u 2
u -1
u 3
15
u 1
u -2
u 0
u 2
u -1
u 3
16
(No Transcript)
17
(0,1,1)
(0,0,1)
(1,1,1)
(0,0,0)
18
(0,1,1)
(1,1,2)
(0,0,1)
(1,2,2)
(1,1,1)
(2,2,2)
(0,0,0)
19
(0,1,2)
Extrapolate
(0,1,1)
(1,1,2)
(0,0,1)
(1,2,2)
(1,1,1)
(2,2,2)
(0,0,0)
20
(0,1,2)
(0,1,2)
(0,1,2)
(0,1,2)
(0,1,2)
(0,1,2)
(0,1,1)
(0,1,1)
(0,1,1)
(0,1,1)
(0,1,1)
(0,1,1)
(0,1,1)
(1,1,2)
(1,1,2)
(1,1,2)
(1,1,2)
(0,0,1)
(0,0,1)
(0,0,1)
(0,0,1)
(0,0,1)
(0,0,1)
(0,0,1)
(0,0,1)
(1,2,2)
(1,2,2)
(1,2,2)
(1,1,1)
(1,1,1)
(1,1,1)
(1,1,1)
(1,1,1)
(2,2,2)
(2,2,2)
(0,0,0)
(0,0,0)
(0,0,0)
(0,0,0)
(0,0,0)
(0,0,0)
(0,0,0)
(0,0,0)
21
(0,1,2)
(0,1,1)
(1,1,2)
(0,0,1)
(1,2,2)
(1,1,1)
(1,2,3)
(2,2,2)
(0,0,0)
(2,2,3)
(2,3,3)
(3,3,3)
22
(0,1,2)
(0,1,1)
(1,1,2)
(0,0,1)
(1,2,2)
(1,1,1)
(1,2,3)
(1,0,-1)
(2,2,2)
(0,0,0)
(2,2,3)
(-1,-1,-1)
(0,0,-1)
(0,-1,-1)
(2,3,3)
(3,3,3)
23
(0,1,2)
(0,1,1)
(1,1,2)
(0,0,1)
(1,2,2)
(1,1,1)
(1,2,3)
(-1,0,1)
(-2,-2,-2)
(-1,-2,-2)
(2,2,2)
(0,0,0)
(2,2,3)
(-1,-1,-1)
(-1,-1,-2)
(0,0,-1)
(0,-1,-1)
(2,3,3)
(3,3,3)
(-2,-1,0)
24
(0,1,2)
u 1
(1,2,3)
(-1,0,1)
u 0
(-2,-1,0)