Transcript Bspline Notes Jordan Smith UC Berkeley CS184 Outline • Bézier Basis Polynomials – Linear – Quadratic – Cubic • Uniform Bspline Basis Polynomials – Linear – Quadratic – Cubic • Uniform Bsplines.

Bspline Notes
Jordan Smith
UC Berkeley
CS184
Outline
• Bézier Basis Polynomials
– Linear
– Quadratic
– Cubic
• Uniform Bspline Basis Polynomials
– Linear
– Quadratic
– Cubic
• Uniform Bsplines from Convolution
V001
Review of Bézier Curves
DeCastlejau Algorithm
V011
V111
V000
Insert at t = ¾
001
000
Review of Bézier Curves
DeCastlejau Algorithm
011
Insert at t = ¾
111
001
Review of Bézier Curves
DeCastlejau Algorithm
0¾1
011
00¾
¾11
000
Insert at t = ¾
111
001
Review of Bézier Curves
DeCastlejau Algorithm
0¾1
011
0¾¾
00¾
¾¾1
¾11
000
Insert at t = ¾
111
001
Review of Bézier Curves
DeCastlejau Algorithm
0¾1
011
0¾¾
00¾
¾¾¾
¾¾1
¾11
000
Insert at t = ¾
111
001
Review of Bézier Curves
DeCastlejau Algorithm
0¾1
011
0¾¾
00¾
¾¾¾
¾¾1
¾11
000
Insert at t = ¾
111
001
Review of Bézier Curves
DeCastlejau Algorithm
0¾1
011
0¾¾
00¾
¾¾¾
¾¾1
¾11
000
Insert at t = ¾
111
001
Review of Bézier Curves
DeCastlejau Algorithm
0¾1
011
0¾¾
00¾
¾¾¾
¾¾1
¾11
000
Insert at t = ¾
111
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
Linear Bézier Basis Poly’s
V0
Vt
Vt
Bez1(t) =
1-t
V0
V1
= (1-t) V0 + t V1
t
V1
Knots:
-1
0
1
2
Quadratic Bézier Basis Poly’s
V01
Vtt
V0t
Vt1
V00
V11
Vtt
Bez2(t) =
1-t
=
t
V0t
1-t
V00
Vt1
t 1-t
V01
t
V11
(1-t)2 V00
+ 2(1-t)t V01
+
t2 V11
Quadratic Bézier Basis Poly’s
Bez2(t) = (1-t)2 V00 + 2(1-t)t V01 + t2 V11
Knots:
-1
0
1
2
Cubic Bézier Basis Poly’s
0t1
001
011
ttt
0tt
tt1
00t
t11
000
111
Vttt
1-t
Bez3(t) =
t
V0tt
Vtt1
1-t
t 1-t
V00t
V0t1
=
t
Vt11
1-t
t 1-t
t 1-t
V000
V001
V011
t
V111
(1-t)3 V000
+ 3(1-t)2t V001
+ 3(1-t)t2 V011
+
t3 V111
Cubic Bézier Basis Poly’s
Bez3(t) = (1-t)3 V000 + 3(1-t)2t V001 + 3(1-t)t2 V011 + t3 V111
Knots:
-1
0
1
2
Blossoming of Bsplines
234
123
Knots:
345
456
0
1
2
3
4
5
6
7
Blossoming of Bsplines
234
33.54
345
233.5
123
Knots:
3.545
456
0
1
2
3 3.5 4
5
6
7
Blossoming of Bsplines
234
33.53.5 33.54 3.53.54
345
233.5
123
Knots:
3.545
456
0
1
2
3 3.5 4
5
6
7
Blossoming of Bsplines
234
33.53.5 33.54 3.53.54
3.53.53.5
233.5
123
Knots:
345
3.545
456
0
1
2
3 3.5 4
5
6
7
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
Uniform Linear Bspline Basis Poly’s
V0
Vt
Vt
B1(t) =
1-t
V0
V1
= (1-t) V0 + t V1
t
V1
Knots:
-1
0
1
2
Uniform Quadratic Bspline Basis Poly’s
V01
V0t
Vtt
Vt1
V-10
V12
Vtt
1 t
1  t 2
t
V0t
B2(t) =
1
V-10
Vt1
t 1
t 1
2
2
1
V01
 
t
t
2
2
V12

2
2
1  2t  2t
2
2
t
2
V -10
V 01
V12
Uniform Quadratic Bspline Basis Poly’s
B 2 (t ) 
1  t 2
2
V -10 
1  2t  2t
-2
-1
V 01 
2
V-10
Knots:
2
V01
0
t
2
2
V12
V12
1
2
3
Uniform Cubic Bspline Basis Poly’s
0t1
-101
-10t
0tt
012
t12
tt1
ttt
Vttt
-2-10
1 t
123
t
V0tt
1
B3(t) =
1
t 1
2
2
1
t2
t
2
2

3
3
1
V-101

Vt12
t 1
3
3
1
V012
4  6 t  3t
2
t
t 1
V - 2 -10
6
V0t1
t2
V-2-10
Vtt1
t 1
V-10t
1  t 3

t
t
3
3
V123

3
6
2
3
1  3t  3t  3t
6
3
t
6
V -101
V 012
V123
Uniform Cubic Bspline Basis Poly’s
B 3 (t ) 
1  t 3
6
4  6 t  3t
2
V - 2 -10 
3
6
1  3t  3t  3t
2
V -101 
6
V-2-10 V-101
Knots:
-3
-2
-1
3
V 012 
t
3
6
V123
V012 V123
0
1
2
3
4
Uniform Bsplines from Convolution

f (t )  g (t ) 

f ( t  s ) g ( s ) ds


0
=
1
0
1
0

0
1
1 0
2
=
0
1
0
2

0
1
1
2
3
=
1
2
30
1
2
3
4