1. Efficient Methods for Roots of Univariate Scalar Beziers
Jinesh Machchhar and Gershon Elber
Faculty of Computer Science,
Technion-Israel Institute of Technology
Center for Graphics and Geometric Computing, Technion

2. Center for Graphics and Geometric Computing, Technion
Outline
Motivation
Traditional approach
Our approach
Results
Extension to B-splines
Extension w/o subdivision
Discussion
Center for Graphics and Geometric Computing, Technion

3. Center for Graphics and Geometric Computing, Technion
Zeros of polynomials
Defining manifolds implicitly:
E.g. 𝑥 2 +3 𝑦 2 −1=0
Offsets, blends, Minkowski-sums:
Intersection of manifolds:
Kinematic analysis/synthesis:
Center for Graphics and Geometric Computing, Technion

4. Center for Graphics and Geometric Computing, Technion
Previous methods
Some roots are complex, in this work we look only for real roots.
The analytic method gives exact roots of polynomials with degree up to 5.
Numeric methods such as Newton-Raphson cannot locate all the roots in interval.
Sub-division based methods: Recursively sub-divide the interval until:
Absence of roots is guaranteed in the sub-interval, or
Exactly one root is guaranteed in the sub-interval.
J. Lane and R. Riesenfeld. Bounds on a polynomial, 1981.
T. Sederberg and S. Parry. Comparison of three curve intersection algorithms, 1986.
Center for Graphics and Geometric Computing, Technion

5. The Subdivision approaches
Traditional approach
Our approach
Subdivide at arbitrary location as long as topology not clear (until we verify a single or no solution).
We guess a solution (via NR) and exploit the mathematical representation of Bezier with a root at its end.
Hence the subdivision is converging slower than can be hoped for (like the bisection method).
The high chances of successful guessing and the degree reduction from eliminating the root gives about a x10 boost.
Center for Graphics and Geometric Computing, Technion

6. Our approach: Locate a root
Assume root at 𝑡 2 is found.
𝑡 1
𝑡 2
𝑐(𝑡)
Center for Graphics and Geometric Computing, Technion

7. Our approach: factoring out roots
Lemma: A degree 𝑛 scalar Bernstein polynomial 𝑐 𝑡 = 𝑖=0 𝑛 𝑝 𝑖 𝜃 𝑖,𝑛 such that 𝑐 0 =0 can be written as 𝑐 𝑡 =𝑡 𝑖=0 𝑛−1 𝑞 𝑖 𝜃 𝑖,𝑛−1 =𝑡 𝑟(𝑡), where, 𝑞 𝑖 = 𝑝 𝑖+1 𝑛 𝑖+1 for 𝑖=0, …,𝑛−1. →
𝑐(𝑡)
𝑟 𝑡
Degree reduction: 𝑐 𝑡 of degree 𝑛, 𝑟 𝑡 of degree 𝑛−1.
Similarly, when 𝑐 1 =0, 𝑐 𝑡 = 1−𝑡 𝑟 𝑡 , 𝑟 𝑡 of degree 𝑛−1.
Center for Graphics and Geometric Computing, Technion

8. Our approach: factoring out roots
Proof: Since 𝑐 0 =0, the first coefficient, 𝑝 0 , is zero, and we have,
𝑐 𝑡 = 𝑖=0 𝑛 𝑝 𝑖 𝑛 𝑖 𝑡 𝑖 1−𝑡 𝑛−𝑖 ,
= 𝑖=1 𝑛 𝑝 𝑖 𝑛 𝑖 𝑡 𝑖 1−𝑡 𝑛−𝑖 ,
=𝑡 𝑖=1 𝑛 𝑝 𝑖 𝑛 𝑖 𝑡 𝑖−1 1−𝑡 𝑛−𝑖 ,
cont’d..
Center for Graphics and Geometric Computing, Technion

9. Our approach: factoring out roots
Proof: cont’d.
𝑐 𝑡 =𝑡 𝑖=1 𝑛 𝑝 𝑖 𝑛 𝑖 𝑡 𝑖−1 1−𝑡 𝑛−𝑖 ,
=𝑡 𝑖=0 𝑛−1 𝑝 𝑖+1 𝑛 𝑖+1 𝑡 𝑖 1−𝑡 𝑛−𝑖−1 ,
=𝑡 𝑖=0 𝑛−1 𝑝 𝑖+1 𝑛 𝑖+1 𝑛−1 𝑖 𝑡 𝑖 1−𝑡 𝑛−𝑖−1 ,
=𝑡 𝑖=0 𝑛−1 𝑞 𝑖 𝜃 𝑖,𝑛−1 𝑡 ,
=𝑡𝑟 𝑡 .
Center for Graphics and Geometric Computing, Technion

10. Our approach: Sub-divide at root
𝑐 𝑡 is subdivided at 𝑡 2 .
𝑡 1
𝑡 2
𝑐(𝑡)
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11. Our approach: Sub-divide at root
𝑐 𝑙 𝑡 =−0.2 𝜃 0,4 −0.38 𝜃 1, 𝜃 2, 𝜃 3,4 +𝟎 𝜃 4,4
𝑐 𝑟 𝑡 =𝟎 𝜃 0,4 −0.15 𝜃 1,4 −0.4 𝜃 2,4 −0.6 𝜃 3,4 −0.2 𝜃 4,4
𝑐 𝑡 is subdivided at 𝑡 2 .
𝑡 2
𝑡 1
𝑡 2
𝑐 𝑙 (𝑡)
𝑐 𝑟 (𝑡)
Center for Graphics and Geometric Computing, Technion

12. Our approach: Factor-out the root
𝑐 𝐿 𝑡 =−0.2 𝜃 0,3 −0.5 𝜃 1, 𝜃 2, 𝜃 3,4
𝑐 𝑅 𝑡 =−0.62 𝜃 0,3 −0.81 𝜃 1,3 −0.8 𝜃 2,3 −0.2 𝜃 3,3
The root at 𝑡 2 is factored out from 𝑐 𝑙 𝑡 and 𝑐 𝑟 𝑡 .
𝑐 𝑙 (𝑡)
𝑡 1
𝑐 𝑟 (𝑡)
𝑐 𝐿 (𝑡)
𝑐 𝑅 (𝑡)
Center for Graphics and Geometric Computing, Technion

13. Center for Graphics and Geometric Computing, Technion
The basic algorithm
BZF(𝑐, 𝑡 𝑚𝑖𝑛 , 𝑡 𝑚𝑎𝑥 ,𝜖,𝛿)
if PurgeProblem(𝑐,𝜖) then //Examine if all coefficients are same sign.
return ∅;
end if
𝑡 0 ← NumericStep(𝑐, 𝜖, 0.5); //Newton Raphson, starting from 𝑡=0.5.
if 𝑡 0 ≠∅ then
𝑐 𝑙 , 𝑐 𝑟 ← Subdivide(𝑐, 𝑡 0 );
𝑐 𝐿 ← Factor1MinusT( 𝑐 𝑙 );
𝑐 𝑅 ← FactorT( 𝑐 𝑟 );
return BZF 𝑐 𝐿 , 𝑡 𝑚𝑖𝑛 , 𝑡 0 ,𝜖, 𝛿 ∪ { 𝑡 0 }∪ BZF 𝑐 𝑅 , 𝑡 0 , 𝑡 𝑚𝑎𝑥 ,𝜖, 𝛿 ;
else
cont’d..
𝑐 𝐿 (𝑡)
𝑐 𝑅 (𝑡)
Center for Graphics and Geometric Computing, Technion

14. The basic algorithm ..cont’d.
else
if 𝑡 𝑚𝑎𝑥 − 𝑡 𝑚𝑖𝑛 <𝛿 then
return ∅;
end if
𝑡 0 ← 𝑡 𝑚𝑎𝑥 + 𝑡 𝑚𝑖𝑛 2 ;
𝑐 𝑙 , 𝑐 𝑟 ← Subdivide(𝑐, 0.5);
return BZF 𝑐 𝑙 , 𝑡 𝑚𝑖𝑛 , 𝑡 0 ,𝜖, 𝛿 ∪ BZF( 𝑐 𝑟 , 𝑡 0 , 𝑡 𝑚𝑎𝑥 ,𝜖, 𝛿);
Center for Graphics and Geometric Computing, Technion

15. Counting root multiplicities
Assume root at 𝑡 2 is found.
𝑐(𝑡)
𝑡 1
𝑡 3
𝑡 2
Center for Graphics and Geometric Computing, Technion

16. Counting root multiplicities
𝑐 𝑙 𝑡 =2.12 𝜃 0,4 −1.9 𝜃 1,4 −0.58 𝜃 2,4 +𝟎 𝜃 3,4 +𝟎 𝜃 4,4
𝑐 𝑟 𝑡 =𝟎 𝜃 0,4 +𝟎 𝜃 1,4 −0.58 𝜃 2,4 −1.59 𝜃 3, 𝜃 4,4
𝑐 𝑟 (𝑡)
𝑐 𝑙 (𝑡)
𝑐 𝑡 is subdivided at 𝑡 2 .
𝑡 1
𝑡 3
𝑡 2
𝑡 2
Center for Graphics and Geometric Computing, Technion

17. Counting root multiplicities
𝑐 𝐿 𝑡 =2.12 𝜃 0,3 −2.54 𝜃 1,3 −1.16 𝜃 2,3 +𝟎 𝜃 3,3
𝑐 𝑅 𝑡 =𝟎 𝜃 0,3 −1.16 𝜃 1,3 −2.12 𝜃 2, 𝜃 3,3
𝑐 𝑅 (𝑡)
A root at 𝑡 2 is factored out from 𝑐 𝑙 𝑡 and 𝑐 𝑟 𝑡 .
𝑐 𝐿 (𝑡)
𝑡 1
𝑡 3
𝑡 2
𝑡 2
Center for Graphics and Geometric Computing, Technion

18. Counting root multiplicities
𝑐 𝐿𝐿 𝑡 =2.12 𝜃 0,2 −3.81 𝜃 1,2 −3.5 𝜃 2,2
𝑐 𝑅𝑅 𝑡 =−3.5 𝜃 1,3 −3.18 𝜃 2, 𝜃 3,3
The second root at 𝑡 2 is factored out from 𝑐 𝐿 𝑡 and 𝑐 𝑅 𝑡 .
𝑐 𝑅𝑅 (𝑡)
𝑐 𝐿𝐿 (𝑡)
𝑡 1
𝑡 3
Center for Graphics and Geometric Computing, Technion

19. Counting root multiplicities
BZF(𝑐, 𝑡 𝑚𝑖𝑛 , 𝑡 𝑚𝑎𝑥 ,𝜖,𝛿)
if PurgeProblem(𝑐,𝜖) then
return ∅;
end if
𝑡 0 ← NumericStep(𝑐, 𝜖, 0.5);
if 𝑡 0 ≠∅ then
𝑐 𝑙 , 𝑐 𝑟 ← Subdivide(𝑐, 𝑡 0 );
𝑚 0 ←0; do
𝑐 𝑙 ← Factor1MinusT( 𝑐 𝑙 );
𝑐 𝑟 ← FactorT( 𝑐 𝑟 );
𝑚 0 ← 𝑚 0 +1;
while | 𝒄 𝒓 𝟎 |<𝝐
return BZF 𝑐 𝑙 , 𝑡 𝑚𝑖𝑛 , 𝑡 0 ,𝜖, 𝛿 ∪ { 𝑡 0 }∪ BZF 𝑐 𝑟 , 𝑡 0 , 𝑡 𝑚𝑎𝑥 ,𝜖, 𝛿 ;
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20. Center for Graphics and Geometric Computing, Technion
More extensions
Initialization of Newton-Raphson method
Control polygon of a Bezier curve approximates the curve.
Use zeros of control polygon as seed for NR method.
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21. Center for Graphics and Geometric Computing, Technion
More extensions
Searching for roots outside the domain
Allow the NR method to go outside the domain.
Factor out the roots found outside the domain.
Hence reduce the problem complexity. →
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22. Comparison with previous methods
Analytic: Euler’s method for polynomials of degree up to five.
CCI: Intersect the convex-hull of the polynomial with the 𝑡 – axis, [Sederberg and Parry, 1986].
Convex-hulls Bounding-rectangles
Source: [Sederberg & Parry, 1986]
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23. Comparison with previous methods
BZF: Our algorithm.
NCZ: Use cones bounding the tangent field of the polynomial, [Sederberg and Meyers, 1988].
Non-overlapping wedges Overlapping wedges
Source: [Sederberg & Meyers, 1988]
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24. Results I: Random polynomials
Some of the Bezier polynomials with randomly generated coefficients with degrees between 3 and samples for each degree.
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25. Results I: Random polynomials
#CtrlPt
Avg #Roots
Analytic
CCI
NCZ
BZF
BZF-NRI1
BZF-NRI2 4
0.90
0.31
6.11
12.32
1.17
1.09
1.08 5
1.15
0.48
8.83
16.81
1.53
1.35
1.39 6
1.38 -
10.94
20.45
1.90
1.71
1.78 7
1.50
10.66
23.46
2.16
1.88
1.95 8
1.64
14.23
25.55
2.53
2.25
2.33 10
2.04
19.77
33.58
3.49
2.85
3.03 15
26.84
39.88
4.80
4.20
4.24 20
2.82
42.95
58.17
8.03
6.97
7.01 50
167.54
185.63
37.21
31.24
31.31
100
6.74
342.67
225.62
245.25
All times in microseconds, averaged over runs.
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26. Results II: Degree 𝒏 with 𝒏 real roots
Some of the degree 𝑛 polynomials with 𝑛 real roots with 3≤𝑛≤14. One sample per degree.
Center for Graphics and Geometric Computing, Technion

27. Results II: Degree 𝒏 with 𝒏 real roots
#CtrlPt
#Roots
Analytic
CCI
NCZ
BZF
BZF-NRI1
BZF-NRI2 4 3
0.48
22.59
36.99
3.83
2.83
2.59 5
0.59
43.62
78.57
4.38
4.15
3.74 6 -
43.56
92.04
6.12
4.96
4.72 7
53.10
114.70
6.18
6.40
6.15 8
72.16
136.48
9.10
7.65
7.28 9
93.52
157.09
8.97
8.82
8.80 10
88.21
201.03
12.18
11.06
10.96 11
113.20
195.19
13.80
11.89
12.56 12
107.83
281.53
13.74
14.42
16.35 13
202.44
382.29
15.98
16.59
16.58 14
158.58
306.96
22.65
18.52
19.77 15
166.22
361.96
24.74
22.00
24.31
All times in microseconds, averaged over runs.
Center for Graphics and Geometric Computing, Technion

28. Results III: Repeated roots
Degree 𝑛 polynomials with 𝑛 real roots with one repeated root. 8≤𝑛≤20. One sample per degree.
Center for Graphics and Geometric Computing, Technion

29. Results III: Repeated roots
#CtrlPt
# Unique Roots
Analytic
CCI
NCZ
BZF
BZF-NRI1
BZF-NRI2 9 7 -
102.90
169.35
8.16
9.87
9.02 10 8
107.82
214.43
10.89
9.84
9.66 11
125.18
211.18
11.34
12.27
12.54 12
774.45
240.14
12.62
13.03
14.02 13
351.35
285.84
14.61
15.06
15.27 14
319.09
20.84
19.75
20.36 15
764.16
295.76
20.11
19.50
18.36 16
408.83
22.11
23.50
22.71 17
415.89
24.34
24.79
26.46 18
537.87
422.28
27.79
27.20
29.35 19
365.31
505.46
36.66
41.34
33.74 20
553.07
38.18
41.47
33.76 21
622.01
554.67
36.33
46.53
37.95
All times in microseconds, averaged over runs.
Center for Graphics and Geometric Computing, Technion

30. Results IV: Wilkinson’s polynomials
Degree 13
Degree 20
#CtrlPt
#Roots
Analytic
CCI
NCZ
BZF
BZF-NRI1
BZF-NRI2 14 13 -
187.66
233.68
14.46
15.79
15.32 21 20
347.11
487.15
40.17
32.89
37.00
All times in microseconds, averaged over runs.
Center for Graphics and Geometric Computing, Technion

31. Zeros of B-splines 𝑐(𝑡) Assume the root between consecutive
knots 𝑡 1 and 𝑡 2 is found.
𝑡 1
𝑡 2
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32. Center for Graphics and Geometric Computing, Technion
Zeros of B-splines
𝑐 𝑙 𝑡
𝑐 𝑟 𝑡
𝑐 𝑚 𝑡
Subdivide at 𝑡 1 and 𝑡 2 to obtain two B-splines 𝑐 𝑙 𝑡 , 𝑐 𝑟 𝑡 and a Bezier 𝑐 𝑚 𝑡 .
𝑡 1
𝑡 2
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33. Results: Zeros of B-splines
Order
Length
# Avg. Real Roots
SubdivAllKnot
SelectiveSubdiv 3 4
1.18
0.012
0.011 7
2.76
0.048
0.040 15
6.14
0.148
0.107 25
10.02
0.234
0.186
120
50.99
2.143
1.007
220
93.59
12.304
2.083
0.95
0.010
0.009
2.28
0.035
5.19
0.126
0.101
9.33
0.214
0.181
45.71
1.047
84.57
12.539
2.188 5
2.13
0.033
5.09
0.129
0.106
8.1
0.217
0.185
42.73
2.214
1.114
76.29
13.264
2.279
All times in microseconds, averaged over runs.
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34. Another extension: No subdivision
Guess a root, 𝑡 0
Factor out the root, 𝑟 𝑡 = 𝑐(𝑡) 𝑡− 𝑡 0
Recurse, 𝑟 𝑡
Reverse Bezier polynomial multiplication to factor out root
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35. Bezier polynomial multiplication
𝑟 𝑡 = 𝑖=0 𝑚 𝑟 𝑖 𝜃 𝑖,𝑚 𝑡
𝑔 𝑡 = 𝑖=0 𝑘 𝑔 𝑖 𝜃 𝑖,𝑘 𝑡
𝑐 𝑡 =𝑟 𝑡 𝑔 𝑡 = 𝑖=0 𝑚+𝑘 𝑗= max 0,𝑖−𝑘 min 𝑚,𝑖 𝑚 𝑗 𝑘 𝑖−𝑗 𝑚+𝑘 𝑖 𝑟 𝑗 𝑔 𝑖−𝑗 𝜃 𝑖,𝑚+1 (𝑡)
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36. Center for Graphics and Geometric Computing, Technion
Factoring out root
Assume root at 𝑡 0 is known
Express polynomial 𝑡− 𝑡 0 in Bernstein basis as 𝑔 0 𝜃 0,1 𝑡 + 𝑔 1 𝜃 1,1 𝑡 , hence,
𝑐 𝑡 = 𝑖=0 𝑚+1 𝑗= max 0,𝑖−1 min 𝑚,𝑖 𝑚 𝑗 1 𝑖−𝑗 𝑚+1 𝑖 𝑟 𝑗 𝑔 𝑖−𝑗 𝜃 𝑖,𝑚+1 (𝑡)
𝑐 𝑡 = 𝑟 0 𝑔 0 𝜃 0,𝑚+1 𝑡
+ 𝑖=1 𝑚 𝑚 𝑖−1 𝑚+1 𝑖 𝑟 𝑖−1 𝑔 𝑚 𝑖 𝑚+1 𝑖 𝑟 𝑖 𝑔 0 𝜃 𝑖,𝑚+1 (𝑡)
+ 𝑟 𝑚 𝑔 1 𝜃 𝑚+1,𝑚+1 𝑡
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37. Center for Graphics and Geometric Computing, Technion
Factoring out root
𝑟 𝑖 = 𝑐 0 𝑔 0 , 𝑖=0, 𝑐 𝑖 𝑚+1 𝑖 − 𝑟 𝑖−1 𝑔 1 𝑚 𝑖−1 𝑚 𝑖 𝑔 0 , 1≤𝑖≤𝑚.
𝑟 𝑖 = 𝑐 𝑚+1 𝑔 1 , 𝑖=𝑚, 𝑐 𝑖+1 𝑚+1 𝑖+1 − 𝑟 𝑖+1 𝑔 0 𝑚 𝑖 𝑚 𝑖 𝑔 1 , 𝑚−1≥𝑖≥0.
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38. Another extension: No subdivision
Assume
the root
at 𝑡 1 is known.
𝑡 1
𝑡 2
𝑐(𝑡)
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39. Another extension: No subdivision
After
factoring
out the
root at 𝑡 1 .
𝑡 2
𝑞(𝑡)
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40. Results All times in microseconds, averaged over 10000 runs. #CtrlPt
#Roots
CagdCrvZeroSet
ZerosNoSubdiv 3 2
1.94
1.21 4
3.03
1.72 5
4.01
2.69 6
5.09
3.52 7
6.51
6.33 8
8.27
5.82 9
9.55
8.99 10
12.01
9.05
All times in microseconds, averaged over runs.
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41. Results All times in microseconds, averaged over 10000 runs. #CtrlPt
#Roots
CagdCrvZeroSet
ZerosNoSubdiv 3
0.78
0.99
0.70 4
0.89
1.41
1.03 5
1.14
1.51
1.39 6
1.37
1.81
1.87 7
1.38
1.96
2.17 8
1.62
2.56
2.70 10
1.84
2.98
3.54 15
2.29
4.89
6.81 20
2.83
8.20
12.77
All times in microseconds, averaged over runs.
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42. Center for Graphics and Geometric Computing, Technion
Future directions
Handling numerical instabilities while detecting higher-order multiplicities.
Inspect first 𝑘 coefficients for detecting a multiplicity of order 𝑘.
Extension to finding zeros of multi-variates.
Extension to finding solution of CCI and SSI.
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43. Center for Graphics and Geometric Computing, Technion
Thank You!
Center for Graphics and Geometric Computing, Technion