Blossoms

1
Blossoms

• CS 419
• Advanced Topics in Computer Graphics
• John C. Hart
• Borrowed somewhat from Tom Sederbergs notes

2
de Casteljau
p1
p12
p2
p012
1-t

• de Casteljau algorithmevaluates a point on
  aBezier curve byscaffolding lerps
• Blossoming renames thecontrol and
  intermediatepoints, like p12, using apolar
  form, like p(0,t,1)

p123
p0123
p01
p23
t
p0
p3
3
de Casteljau
p(1,0,0)
p12
p(1,1,0)
p012
1-t

• de Casteljau algorithmevaluates a point on
  aBezier curve byscaffolding lerps
• Blossoming renames thecontrol and
  intermediatepoints, like p12, using apolar
  form, like p(0,t,1)

p123
p0123
p01
t
p23
p(0,0,0)
p(1,1,1)
4
de Casteljau
p(1,0,0)
p(1,t,0)
p(1,1,0)
p012
1-t

• de Casteljau algorithmevaluates a point on
  aBezier curve byscaffolding lerps
• Blossoming renames thecontrol and
  intermediatepoints, like p12, using apolar
  form, like p(0,t,1)

p123
p0123
p(t,0,0)
t
p(1,1,t)
p(0,0,0)
p(1,1,1)
5
de Casteljau
p(1,0,0)
p(1,t,0)
p(1,1,0)
p(t,t,0)
1-t

• de Casteljau algorithmevaluates a point on
  aBezier curve byscaffolding lerps
• Blossoming renames thecontrol and
  intermediatepoints, like p12, using apolar
  form, like p(0,t,1)

p(1,t,t)
p0123
p(t,0,0)
t
p(1,1,t)
p(0,0,0)
p(1,1,1)
6
de Casteljau
p(1,0,0)
p(1,t,0)
p(1,1,0)
p(t,t,0)
1-t

• de Casteljau algorithmevaluates a point on
  aBezier curve byscaffolding lerps
• Blossoming renames thecontrol and
  intermediatepoints, like p12, using apolar
  form, like p(0,t,1)

p(1,t,t)
p(t,t,t)
p(t,0,0)
t
p(1,1,t)
p(0,0,0)
p(1,1,1)
7
Blossoming Rules
p(1,0,0)
p(1,t,0)
p(1,1,0)
p(t,t,0)
1-t

• of parameters degree
• Order doesnt matter
• p(a,b,c) p(b,a,c)
• Linear in any parameter
• ap(a,b,c) p(aa,b,c) p(a,ab,c)
• p(a,bc,d) p(a,b,d) p(a,c,d)
• Lerping
• (1-t) p(0,0,0) t p(1,0,0) p(0 (1-t),0,0)
  p(1 t,0,0)
• p(0 (1-t) 1 t,0,0) p(t,0,0)

p(1,t,t)
p(t,t,t)
p(t,0,0)
t
p(1,1,t)
p(0,0,0)
p(1,1,1)
8
Evaluation
p(1,0,0)
p(1,t,0)
p(1,1,0)
p(t,t,0)
1-t

• Goal is to find p(t) bydiagonalization,
  bymanipulating blossomsinto p(t,t,t)
• de Casteljau algorithm blossomsinto Bernstein
  polynomials

p(1,t,t)
p(t,t,t)
p(t,0,0)
t
p(1,1,t)
p(0,0,0)
p(1,1,1)
p(t) p(t,t,t) (1-t) p(t,t,0) t p(t,t,1)
(1-t)(1-t) p(t,0,0) t p(t,0,1) t (1-t)
p(t,0,1) t p(t,1,1) (1-t)2 p(t,0,0)
2 (1-t) t p(t,0,1) t2
p(t,1,1) (1-t)2(1-t)p(0,0,0)tp(1,0,0)2
(1-t)t (1-t)p(0,0,1)tp(1,0,1)t2(1-t)p(0,1,1)
tp(1,1,1) (1-t)3 p(0,0,0) 3 (1-t)2 t
p(0,0,1) 3 (1-t) t2 p(0,1,1) t3 p(1,1,1)
9
B-Spline Blossoms

• Three segments 2,3,3,4,4,5
• Points within each segment influenced by four
  surrounding control points
• Knots influenced by three surrounding control
  points
• Need two extra knots ateach end of knot
  vector(in general, d-1 extraknots at each end)

p(2,3,4)
p(3,4,5)
t3
t4
p(1,2,3)
p(4,5,6)
t2
t5
p(0,1,2)
p(5,6,7)
knot vector 0 1 2 3 4 5 6 7
10
Bohm Blossoms

• Trick Think of each segmentas a Bezier curve

p(2,3,4)
p(3,4,5)
t3
t4
p(1,2,3)
p(4,5,6)
t2
t5
p(0,1,2)
p(5,6,7)
knot vector 0 1 2 3 4 5 6 7
11
Bohm Blossoms

• Trick Think of each segmentas a Bezier curve

p(2,3,4)
p(3,4,5)
p(3,3,3)
p(4,4,4)
p(1,2,3)
p(4,5,6)
p(2,2,2)
p(5,5,5)
p(0,1,2)
p(5,6,7)
knot vector 0 1 2 3 4 5 6 7
12
Bohm Blossoms

• Trick Think of each segmentas a Bezier curve
• Where should the othertwo control points gofor
  the 2,3 segment?

p(2,3,4)
p(3,4,5)
p(3,3,3)
p(1,2,3)
p(4,5,6)
p(2,2,2)
p(0,1,2)
p(5,6,7)
knot vector 0 1 2 3 4 5 6 7
13
Bohm Blossoms

• Trick Think of each segmentas a Bezier curve
• Where should the othertwo control points gofor
  the 2,3 segment?
• Need to find
• p(2,2,3) 2/3 p(1,2,3) 1/3 p(4,2,3)
• p(3,2,3) 1/3 p(1,2,3) 2/3 p(4,2,3)

p(4,2,3)
p(3,4,5)
p(3,2,3)
p(3,3,3)
p(2,2,3)
p(1,2,3)
p(4,5,6)
p(2,2,2)
p(0,1,2)
p(5,6,7)
knot vector 0 1 2 3 4 5 6 7
14
Bohm Blossoms

• Where are the endpoints located?
• Need to find
• p(2,1,2) 1/3 p(0,1,2) 2/3 p(3,1,2)
• p(3,4,3) 2/3 p(3,4,2) 1/3 p(3,4,5)
• p(2,2,2) 1/2 p(2,1,2) 1/2 p(2,3,2)
• p(3,3,3) 1/2 p(3,2,3) 1/2 p(3,4,3)
• Reveals how to turn a B-splineinto a Bezier
  curve!

p(3,4,2)
p(3,4,5)
p(3,4,3)
p(3,2,3)
p(3,3,3)
p(2,3,2)
p(3,1,2)
p(2,2,2)
p(2,1,2)
p(0,1,2)
knot vector 0 1 2 3 4 5 6 7
15
Knot Insertion

• Suppose we want to add aknot at t 3.5

p(2,3,4)
p(3,4,5)
t3
t4
p(1,2,3)
p(4,5,6)
t2
t5
p(0,1,2)
p(5,6,7)
knot vector 0 1 2 3 4 5 6 7
16
Knot Insertion

• Suppose we want to add aknot at t 3.5
• Then we need new cpsp(2,3,3.5), p(3,3.5,4)and
  p(3.5,4,5)and can get rid ofp(2,3,4) and
  p(3,4,5)

p(3,3.5,4)
p(2,3,4)
p(3,4,5)
p(3.5,4,5)
p(2,3,3.5)
t3
t4
p(1,2,3)
p(4,5,6)
t2
t5
p(0,1,2)
p(5,6,7)
knot vector 0 1 2 3 4 5 6 7
17
Knot Insertion

• Suppose we want to add aknot at t 3.5
• Then we need new cpsp(2,3,3.5), p(3,3.5,4)and
  p(3.5,4,5)and can get rid ofp(2,3,4) and
  p(3,4,5)

p(3,3.5,4)
p(3.5,4,5)
p(2,3,3.5)
t3.5
p(1,2,3)
p(4,5,6)
t2
t5
p(0,1,2)
p(5,6,7)
knot vector 0 1 2 3 3.5 4 5 6 7
18
de Boor Algorithm

• What if we want toevaluate p(3.5)?
• Then create a triple knotat t 3.5 and figure
  outwhere to put the controlpoint p(3.5,3.5,3.5)

p(3,3.5,4)
p(3.5,4,5)
p(2,3,3.5)
t3.5
p(1,2,3)
p(4,5,6)
t2
t5
p(0,1,2)
p(5,6,7)
knot vector 0 1 2 3 3.5 4 5 6 7
19
de Boor Algorithm

• What if we want toevaluate p(3.5)?
• Then create a triple knotat t 3.5 and figure
  outwhere to put the controlpoint p(3.5,3.5,3.5)
• Need p(3,3.5,3.5)and p(3.5,3.5,4)
• Also subdivides B-spline into0,1,2,3,3.5,3.5,3.5
  and3.5,3.5,3.5,4,5,6,7

p(3,3.5,4)
p(3.5,3.5,4)
p(3,3.5,3.5)
p(3.5,4,5)
p(2,3,3.5)
p(3.5,3.5,3.5)
p(1,2,3)
p(4,5,6)
p(0,1,2)
p(5,6,7)
knot vector 0 1 2 3 3.5 3.5 3.5 4 5 6 7