Bspline Notes

1
Bspline Notes

• Jordan Smith
• UC Berkeley
• CS184

2
Outline

• Bézier Basis Polynomials
• Linear
• Quadratic
• Cubic
• Uniform Bspline Basis Polynomials
• Linear
• Quadratic
• Cubic
• Uniform Bsplines from Convolution

3
Review of Bézier CurvesDeCastlejau Algorithm
V001
V011
V111
V000
Insert at t ¾
4
Review of Bézier CurvesDeCastlejau Algorithm
001
011
111
Insert at t ¾
000
5
Review of Bézier CurvesDeCastlejau Algorithm
001
011
0¾1
00¾
¾11
111
Insert at t ¾
000
6
Review of Bézier CurvesDeCastlejau Algorithm
001
011
0¾1
0¾¾
00¾
¾¾1
¾11
111
Insert at t ¾
000
7
Review of Bézier CurvesDeCastlejau Algorithm
001
011
0¾1
0¾¾
00¾
¾¾¾
¾¾1
¾11
111
Insert at t ¾
000
8
Review of Bézier CurvesDeCastlejau Algorithm
001
011
0¾1
0¾¾
00¾
¾¾¾
¾¾1
¾11
111
Insert at t ¾
000
9
Review of Bézier CurvesDeCastlejau Algorithm
001
011
0¾1
0¾¾
00¾
¾¾¾
¾¾1
¾11
111
Insert at t ¾
000
10
Review of Bézier CurvesDeCastlejau Algorithm
001
011
0¾1
0¾¾
00¾
¾¾¾
¾¾1
¾11
111
Insert at t ¾
000
11
Bézier Curves Summary

• DeCastlejau algorithm
• Evaluate Position(t) and Tangent(t)
• Subdivides the curve into 2 subcurves with
  independent control polygons
• Subdivision of Bézier curves and convex hull
  property allows for
• Adaptive rendering based on a flatness criterion
• Adaptive collision detection using line segment
  tests

12
Linear Bézier Basis Polys
V0
Vt
V1
Vt
Bez1(t)
(1-t) V0 t V1
1-t
t
V0
V1
Knots
13
Quadratic Bézier Basis Polys
V01
Vtt
Vt1
V0t
V00
V11
14
Quadratic Bézier Basis Polys
Bez2(t) (1-t)2 V00 2(1-t)t V01 t2 V11
Knots
15
Cubic Bézier Basis Polys
0t1
001
011
ttt
tt1
0tt
00t
t11
111
000
16
Cubic Bézier Basis Polys
Bez3(t) (1-t)3 V000 3(1-t)2t V001 3(1-t)t2
V011 t3 V111
Knots
17
Blossoming of Bsplines
234
345
456
123
Knots
0
1
7
6
5
4
3
2
18
Blossoming of Bsplines
234
345
33.54
233.5
3.545
456
123
Knots
0
1
7
6
5
4
3
2
3.5
19
Blossoming of Bsplines
234
345
33.54
33.53.5
3.53.54
233.5
3.545
456
123
Knots
0
1
7
6
5
4
3
2
3.5
20
Blossoming of Bsplines
234
345
33.54
33.53.5
3.53.54
3.53.53.5
233.5
3.545
456
123
Knots
0
1
7
6
5
4
3
2
3.5
21
Bspline Blossoming Summary

• Blossoming of Bsplines is a generalization of the
  DeCastlejau algorithm
• Control point index triples on the same control
  line share 2 indices with each other
• Inserting a knot (t value)
• Adds a new control point and curve segment
• Adjusts other control points to form a control
  polygon
• Inserting the same t value reduces the parametric
  continuity of the curve
• A control point triple with all 3 indices equal
  is a point on the Bspline curve

22
Uniform Linear Bspline Basis Polys
V0
Vt
V1
Vt
B1(t)
(1-t) V0 t V1
1-t
t
V0
V1
Knots
23
Uniform Quadratic Bspline Basis Polys
V01
Vt1
V0t
Vtt
V-10
V12
B2(t)
24
Uniform Quadratic Bspline Basis Polys
V-10
V01
V12
Knots
-2
-1
0
1
3
2
25
Uniform Cubic Bspline Basis Polys
B3(t)
26
Uniform Cubic Bspline Basis Polys
V-2-10
V-101
V012
V123
27
Uniform Bsplines from Convolution

?

?

?