[PPT] - Algebraic and Geometric Computations with Rational Curves Elias PowerPoint Presentation

SLIDE 1

Algebraic and Geometric Computations with Rational Curves

Elias TSIGARIDAS

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 2

Short intro to rational curves

Outline

1 Short intro to rational curves

2 Univariate real solving

3 Special points

4 Curves in ■

❘♥

5 Bezier curves/surfaces

6 Random Bernstein polynomials

7 ToDo List

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 3

Short intro to rational curves

Parametric Curves

✥ ✿ ■ ❘ ✦■ ❘✷ t ✼✦

✏❢✶❀◆✭t✮

❢❉✭t✮ ❀ ❢✷❀◆✭t✮ ❢◆✭t✮

✑

❈ ❂ ■♠✭✥✮ ❬ ■ ❘✷

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 4

Short intro to rational curves

Parametric Curves

✥ ✿ ■ ❘ ✦■ ❘✷ t ✼✦

✏❢✶❀◆✭t✮

❢❉✭t✮ ❀ ❢✷❀◆✭t✮ ❢◆✭t✮

✑

❈ ❂ ■♠✭✥✮ ❬ ■ ❘✷ ✥ ✿ ■ ❘ ✦■ ❘✷ t ✼✦

✒ ✶ ✸t

✭✶ ✰ t✷✮✷ ❀ t✭✶ ✸t✮ ✭✶ ✰ t✷✮✷

✓

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 5

Short intro to rational curves

Parametric Curves

✥ ✿ ■ ❘ ✦■ ❘✷ t ✼✦

✏❢✶❀◆✭t✮

❢❉✭t✮ ❀ ❢✷❀◆✭t✮ ❢◆✭t✮

✑

❈ ❂ ■♠✭✥✮ ❬ ■ ❘✷

✥ ✿ ■ ❘ ✦■ ❘✷ t ✼✦

✒

✼t✹ ✰ ✷✽✽t✷ ✰ ✷✺✻ t✹ ✰ ✸✷t✷ ✰ ✷✺✻ ❀ ✽✵t✸ ✰ ✷✺✻t t✹ ✰ ✸✷t✷ ✰ ✷✺✻

✓

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 6

Short intro to rational curves

Proper Curve

Definition A (parametric) curve ❈ is proper if almost all the points in ■♠✭✥✮ are reached by one value of the parameter t. Example (Proper parametrization)

t ✼✦

✒

t ✶ ✰ t✷ ❀ ✶ ✶ ✰ t✷

✓ Example (non-Proper parametrization)

t ✼✦

✥

t✷ ✶ ✰ t✷ ❀ ✶ ✶ ✰ t✷

✦

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 7

Short intro to rational curves

Questions

Is it proper? Special points: extreme, singular, inflexion, cusps Topology computations Drawing Arrangement computation

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 8

Short intro to rational curves

Extreme points

❴ ① ❂ ✵ ✮ ❞ ❞t ❢✶❀◆✭t✮ ❢❉✭t✮ ❂ ✵ ❴ ② ❂ ✵ ✮ ❞ ❞t ❢✷❀◆✭t✮ ❢❉✭t✮ ❂ ✵ ✥ ✿ ■ ❘ ✦■ ❘✷ t ✼✦

✒ ✶ ✸t

✭✶ ✰ t✷✮✷ ❀ t✭✶ ✸t✮ ✭✶ ✰ t✷✮✷

✓

❴ ① ❂ ✵ ✮ ✾ t✷ ✹ t ✸ ❂ ✵ ❴ ② ❂ ✵ ✮ ✻ t✸ ✸ t✷ ✻ t ✰ ✶ ❂ ✵

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 9

Short intro to rational curves

Extreme points

❴ ① ❂ ✵ ✮ ❞ ❞t ❢✶❀◆✭t✮ ❢❉✭t✮ ❂ ✵ ❴ ② ❂ ✵ ✮ ❞ ❞t ❢✷❀◆✭t✮ ❢❉✭t✮ ❂ ✵ ✥ ✿ ■ ❘ ✦■ ❘✷ t ✼✦

✒ ✶ ✸t

✭✶ ✰ t✷✮✷ ❀ t✭✶ ✸t✮ ✭✶ ✰ t✷✮✷

✓

❴ ① ❂ ✵ ✮ ✾ t✷ ✹ t ✸ ❂ ✵ ❴ ② ❂ ✵ ✮ ✻ t✸ ✸ t✷ ✻ t ✰ ✶ ❂ ✵

We need to solve (isolate) univariate polynomials!

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 10

Univariate real solving

Outline

1 Short intro to rational curves

2 Univariate real solving

3 Special points

4 Curves in ■

❘♥

5 Bezier curves/surfaces

6 Random Bernstein polynomials

7 ToDo List

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 11

Univariate real solving

Univariate Real Solving

Problem Given ❆ ✷ ❩❬❳❪ such that

❆ ❂ ❛❞❳❞ ✰ ✁ ✁ ✁ ✰ ❛✶❳ ✰ ❛✵ ✷ ❩❬❳❪

where

d ❂ ❞❡❣✭❆✮

and

▲ ✭❆✮ ❂ ♠❛①

✵✔✐✔❞ ❢❧❣ ❥❛✐❥❣ ❂ ✜

Compute isolating intervals for the real roots.

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 12

Univariate real solving

Example Let

❢ ❂ ①✺ ✼①✹ ✰ ✷✷①✸ ✹①✷ ✹✽① ✰ ✸✻ ❂ ✭① ✶✮ ✁ ✭①✷ ✻① ✰ ✶✽✮ ✁ ✭①✷ ✷✮

real roots

♣

✷ ✶ ✰ ♣ ✷

utput

✭✹✾❀ ✵✮ ✭ ✹✾

✻✹❀ ✶✹✼ ✶✷✽✮

✭ ✶✹✼

✶✷✽❀ ✹✾✮

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 13

Univariate real solving

The history of (expected) complexity bounds

cf STURM descartes bernstein

❁ 1980

❡ ❖❇✭✷✜✮ ❡ ❖❇✭❞✼✜ ✸✮ ❡ ❖❇✭❞✻✜ ✷✮

[Uspensky;1948] [Heidel;1971] [Collins,Akritas;1976] [L,R;81]

❁ 2005

❖❇✭❞✺✜ ✸✮ ❡ ❖❇✭❞✻✜ ✸✮ ❡ ❖❇✭❞✺✜ ✷✮ ❡ ❖❇✭❞✻✜ ✸✮

[Akritas;1980] [Davenport;1988] [Krandick;95,Johnson;98] [MVY;2004]

✔2006

❡ ❖❇✭❞✹✜ ✷✮ ❡ ❖❇✭❞✹✜ ✷✮ ❡ ❖❇✭❞✹✜ ✷✮ ❡ ❖❇✭❞✹✜ ✷✮

[Du,Sharma,Yap;2005] [Eigenwillig,Sharma,Yap;06] [ESY;2006] [Emiris,T.;2006] [Emiris,Mourrain,T.;2006] [EMT;2006]

2006+

❡ ❖❇✭❞✸✜✮ [E,T.; 09] ❡ ❖❇✭❞✺✜ ✷✮ [S;08] ❡ ❖❇✭r ❞✷✜✮ ❡ ❖❇✭❞✹✜ ✷✮ [M,R;09]

[Emiris,Galligo,T.;10]

❡ ❖❇✭❞✹✜ ✷✮ [T;11] ❡

❖❇✭❞✸✜✮ [Schoenhage:1982-] [Sagraloff: 2012]

Numerical bound ❡

❖❇✭❞✷✜✮ [Pan; 2001]

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 14

Univariate real solving

How hard is the problem?

Definition (Separation bound)

✁ ❂ s❡♣✭❆✮ ❂ ♠✐♥

✐✻❂❥ ❥✌✐ ✌❥❥ ✘ ✷❞ ✜ ❂ ✷s

Example Consider the Wilkinson polynomial

❆ ❂ ✭① ✶✮✭① ✷✮ ✁ ✁ ✁ ✭① ✷✵✮ ✁ ✘ ✶✵✸✹✹

actual

s❡♣✭❆✮ ❂ ✶

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 15

Univariate real solving

Experimental results

300 400 500 600 700 800 900 1000 L cf 9.14 25.27 55.86 110.13 214.99 407.09 774.22 1376.34 #roots 300 400 500 600 700 800 900 1000 C1 cf 3.16 8.61 19.67 38.23 77.75 139.18 247.11 414.51 #roots 300 400 500 600 700 800 900 1000 W cf 2.54 6.09 12.07 21.43 34.52 53.35 81.88 120.21 #roots 300 400 500 600 700 800 900 1000 R1 cf 0.07 0.33 0.06 0.37 0.66 0.76 1.03 1.77 #roots 2 6 2 4 4 2 4 4

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 16

Special points

Outline

1 Short intro to rational curves

2 Univariate real solving

3 Special points

4 Curves in ■

❘♥

5 Bezier curves/surfaces

6 Random Bernstein polynomials

7 ToDo List

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 17

Special points

Singular points (Self-intersections)

For different values of t we get the same point (different tangents!)

❢✶❀◆✭t✶✮ ❢❉✭t✶✮ ❂ ❢✶❀◆✭t✷✮ ❢❉✭t✷✮ ❢✷❀◆✭t✶✮ ❢❉✭t✶✮ ❂ ❢✷❀◆✭t✷✮ ❢❉✭t✷✮

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 18

Special points

Singular points (Self-intersections)

For different values of t we get the same point (different tangents!)

❢✶❀◆✭t✮ ❢❉✭t✮ ❂ ❢✶❀◆✭s✮ ❢❉✭s✮ ✮ ❢✶❀◆✭t✮❢❉✭s✮ ❢✶❀◆✭s✮❢❉✭t✮ ❢❉✭t✮❢❉✭s✮ ❂ ✵ ❢✷❀◆✭t✮ ❢❉✭t✮ ❂ ❢✷❀◆✭s✮ ❢❉✭s✮ ✮ ❢✷❀◆✭t✮❢❉✭s✮ ❢✷❀◆✭s✮❢❉✭t✮ ❢❉✭t✮❢❉✭s✮ ❂ ✵

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 19

Special points

❣✶✭s❀ t✮ ❂ ❢✶◆✭s✮❢❉✭t✮ ❢✶◆✭t✮❢❉✭s✮ s t ❣✷✭s❀ t✮ ❂ ❢✷◆✭s✮❢❉✭t✮ ❢✷◆✭t✮❢❉✭s✮ s t ❘t ❂ Res✭❣✶✭s❀ t✮❀ ❣✷✭s❀ t✮❀ s✮ ✷ ❩❬t❪

Theorem

✥ is proper iff ❘t ✻✑ ✵

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 20

Special points

Example

✥ ✿ ■ ❘ ✦■ ❘✷ t ✼✦✭ ✼t✹ ✰ ✷✽✽t✷ ✰ ✷✺✻ t✹ ✰ ✸✷t✷ ✰ ✷✺✻ ❀ ✽✵t✸ ✰ ✷✺✻t t✹ ✰ ✸✷t✷ ✰ ✷✺✻ ✮ ❣✶✭s❀ t✮ ❂ ✺✶✷ s✸t✷ ✷✵✹✽ s✸ ✺✶✷ s✷t✸ ✷✵✹✽ s✷t ✰ ✻✺✺✸✻ s ✷✵✹✽ st✷ ✰ ✻✺✺✸✻ t ✷✵✹✽ t✸ ❣✷✭s❀ t✮ ❂ ✷✺✻ s✸t ✰ ✽✵ s✸t✸ ✷✽✶✻ s✷t✷ ✷✵✹✽✵ s✷ ✷✽✻✼✷ st ✷✺✻ st✸ ✰ ✻✺✺✸✻ ✷✵✹✽✵ t✷ ❘t ❂ Res✭❣✶❀ ❣✷❀ s✮ ❂ ❝ ✺ t✷ ✶✻✁ t✷ ✰ ✶✻✁✽

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 21

Special points

Complexity of singular points

Theorem We can identify the singular points in ❡

❖❇✭❞✺✜✮

r

❡

❖❇✭◆✻✮

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 22

Special points

Inflection points

Definition (Curvature)

✔✭t✮ ❂ ❦❈✵✭t✮ ✂ ❈

✵✵✭t✮❦

❦❈✵✭t✮❦✸ ❑✭t✮ ❂ ❈✵✭t✮ ✂ ❈

✵✵✭t✮ ❂

✔❤✶✭t✮

❤✭t✮ ❀ ❤✷✭t✮ ❤✭t✮ ❀ ❤✸✭t✮ ❤✭t✮

✕

❣✭t✮ ❂ ❣❝❞✭❤✶✭t✮❀ ❤✷✭t✮❀ ❤✸✭t✮✮

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 23

Special points

Strange (hidden) real points

Definition (Hidden points) Points in ■

❘✷ reached only by complex values of t.

(isolated points)

t ❂ ✉ ✰ ✐ ✐ ✐✈ ❢✶◆✭✉ ✰ ✐ ✐ ✐✈✮ ✁ ❢❉✭✉ ✰ ✐ ✐ ✐✈✮ ❥❢❉❥✷ ❂ ✶ ❥❢❉❥✷ ✁ ✭❆✭✉❀ ✈✮ ✰ ✐ ✐ ✐❇✭✉❀ ✈✮✮ ❢✷◆✭✉ ✰ ✐ ✐ ✐✈✮ ✁ ❢❉✭✉ ✰ ✐ ✐ ✐✈✮ ❥❢❉❥✷ ❂ ✶ ❥❢❉❥✷ ✁ ✭❈✭✉❀ ✈✮ ✰ ✐ ✐ ✐❉✭✉❀ ✈✮✮

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 24

Special points

Strange (hidden) real points

Definition (Hidden points) Points in ■

❘✷ reached only by complex values of t.

(isolated points) Helpful construction Let t ❂ ✉ ✰ ✐

✐ ✐✈ ❢✶◆✭✉ ✰ ✐ ✐ ✐✈✮ ✁ ❢❉✭✉ ✰ ✐ ✐ ✐✈✮ ❥❢❉❥✷ ❂ ✶ ❥❢❉❥✷ ✁ ✭❆✭✉❀ ✈✮ ✰ ✐ ✐ ✐❇✭✉❀ ✈✮✮ ❢✷◆✭✉ ✰ ✐ ✐ ✐✈✮ ✁ ❢❉✭✉ ✰ ✐ ✐ ✐✈✮ ❥❢❉❥✷ ❂ ✶ ❥❢❉❥✷ ✁ ✭❈✭✉❀ ✈✮ ✰ ✐ ✐ ✐❉✭✉❀ ✈✮✮

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 25

Special points

Hidden points

Theorem

✭❙✶✮

✽ ❃ ❃ ❁ ❃ ❃ ✿

❇✭✉❀ ✈✮ ❂ ✵ ❉✭✉❀ ✈✮ ❂ ✵ ✈ ✁ ❥❢❉❥✷ ✁ ✇ ✶ ❂ ✵ ✭❙✷✮

✽ ❃ ❃ ❃ ❃ ❃ ❃ ❁ ❃ ❃ ❃ ❃ ❃ ❃ ✿

①✭t✮ ❂ ❆✭✉❀ ✈✮ ❥❢❉✭✉❀ ✈✮❥✷ ②✭t✮ ❂ ❇✭✉❀ ✈✮ ❥❢❉✭✉❀ ✈✮❥✷ ✈ ✁ ❢❉ ✁ ✇ ✶ ❂ ✵

The hidden points are the real solutions of ✭❙✶✮ that are not solutions of ✭❙✷✮. ❡

❖❇✭❞✽ ✰ ❞✼✜✮

❡

❖❇✭◆✽✮

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 26

Special points

Hidden points

Theorem

✭❙✶✮

✽ ❃ ❃ ❁ ❃ ❃ ✿

❇✭✉❀ ✈✮ ❂ ✵ ❉✭✉❀ ✈✮ ❂ ✵ ✈ ✁ ❥❢❉❥✷ ✁ ✇ ✶ ❂ ✵ ✭❙✷✮

✽ ❃ ❃ ❃ ❃ ❃ ❃ ❁ ❃ ❃ ❃ ❃ ❃ ❃ ✿

①✭t✮ ❂ ❆✭✉❀ ✈✮ ❥❢❉✭✉❀ ✈✮❥✷ ②✭t✮ ❂ ❇✭✉❀ ✈✮ ❥❢❉✭✉❀ ✈✮❥✷ ✈ ✁ ❢❉ ✁ ✇ ✶ ❂ ✵

The hidden points are the real solutions of ✭❙✶✮ that are not solutions of ✭❙✷✮. Theorem We can identify the hidden points in ❡

❖❇✭❞✽ ✰ ❞✼✜✮ or ❡ ❖❇✭◆✽✮.

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 27

Curves in ■

❘♥

Outline

1 Short intro to rational curves

2 Univariate real solving

3 Special points

4 Curves in ■

❘♥

5 Bezier curves/surfaces

6 Random Bernstein polynomials

7 ToDo List

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 28

Curves in ■

❘♥

Rational Curves in ■

❘♥

❣✐✭s❀ t✮ ❂ ❢✐❀◆✭s✮❢❉✭t✮ ❢✐◆✭t✮❢❉✭s✮ s t ❘t ❂ Res✭❣✶✭s❀ t✮❀ ✉✷ ❣✷✭s❀ t✮ ✰ ✁ ✁ ✁ ✰ ✉♥ ❣♥✭s❀ t✮❀ s✮ ❂

❳

☛

❤☛✭t✮✉☛ ❘t ✷ ❩❬t❪❬✉✷❀ ✿ ✿ ✿ ❀ ✉♥❪ ❙ ❂ ❩ ✭❤☛✭t✮✮ ✚ ■ ❘

Theorem

✥ is proper iff ❘t ✻✑ ✵

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 29

Curves in ■

❘♥

Rational Curves in ■

❘♥

❘t ❂ Res✭❣✶✭s❀ t✮❀ ✉✷ ❣✷✭s❀ t✮ ✰ ✁ ✁ ✁ ✰ ✉♥ ❣♥✭s❀ t✮❀ s✮ ❂

❳

☛

❤☛✭t✮✉☛

Theorem Let ❈ be a proper rational curve. Then there exist ❬❧✐✰✶❀ ✿ ✿ ✿ ❀ ❧♥❪ such that

❈✐ ❂

✵ ❅❢✶✭t✮

❢✭t✮ ❀ ✿ ✿ ✿ ❀ ❢✐✶✭t✮ ❢✭t✮ ❀ ❢✐✭t✮ ❢✭t✮ ✰

♥

❳

❥❂✐✰✶

❧❥ ❢❥✭t✮ ❢✭t✮

✶ ❆ ❀

t ✷ ❬❛❀ ❜❪

is proper for ✷ ✔ ✐ ❁ ♥ Lemma Any singularity of ❈❥ is a singularity of ❈✐ with ✷ ✔ ✐ ❁ ❥ ✔ ♥.

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 30

Bezier curves/surfaces

Outline

1 Short intro to rational curves

2 Univariate real solving

3 Special points

4 Curves in ■

❘♥

5 Bezier curves/surfaces

6 Random Bernstein polynomials

7 ToDo List

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 31

Bezier curves/surfaces

Bezier Curves/Patches

Exact (?), Complete, Efficient algorithms Arrangement of Bezier curves Univariate real solving Complexity analysis?

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 32

Bezier curves/surfaces

Bezier Curves/Patches

Exact (?), Complete, Efficient algorithms Arrangement of Bezier curves Arrangement of Rational Bezier curves

❈✭✉✮ ❂

♥

❳

✐❂✵

❇♥❀✐✭✉✮✇✐

P♥

❥❂✵ ❇♥❀❥✭✉✮✇❥

P✐

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 33

Bezier curves/surfaces

Bezier Curves/Patches

Exact (?), Complete, Efficient algorithms Arrangement of Bezier curves Arrangement of Rational Bezier curves Computations with NURBS

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 34

Random Bernstein polynomials

Outline

1 Short intro to rational curves

2 Univariate real solving

3 Special points

4 Curves in ■

❘♥

5 Bezier curves/surfaces

6 Random Bernstein polynomials

7 ToDo List

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 35

Random Bernstein polynomials

Random polynomials

1 1−ρ

1 1 − ρ −

1 1−ρ

−1 + ρ

① ✼✦

① ①✰✶

1 r 1 2 1 2−ρ 1−ρ 2−ρ

− 1−ρ

ρ

1

(equi-)distribution of the roots [Erdos,Tur´

an;1950], [Hughes,Nikeghbali;2004]

E❬★real roots❪

monomial

✷ ✙ ❧♦❣✭❞✮ ✰ ♦✭✶✮

◆✭✵❀ ✶✮

[Kac;1943] [Edelman,Kostlan;1995]

Bernstein

♣ ❞ ◆✭✵❀ ❞❞✮

[Dedieu,Armentano;2009]

Bernstein

♣ ✷ ❞ ✝ ❖✭✶✮ ◆✭✵❀ ❞✮

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 36

Random Bernstein polynomials

Integral Geometry

a ✁ x ❂ ✭❛✷❀ ❛✶❀ ❛✵✮ ✁ ✭①✷❀ ①❀ ✶✮ ❂ P❞❂✷

✐❂✵ ❛✐ ①✐ ❂ ✵

#(real roots) ✘ area ✘ length of the curve

[Edelman,Kostlan;1995]

Images and video by [George Koulieris; 2009]

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 37

Random Bernstein polynomials

Theorem (Expected number of real roots [Kac] [Edelman-Kostlan])

✈✭t✮ ❂ ✭✶❀ t❀ ✿ ✿ ✿ ❀ t♥✮❃

is a vector of differentiable functions and ❝✵❀ ✿ ✿ ✿ ❀ ❝♥ elements of a multivariate normal distribution with zero mean and covariance matrix ❈. The expected number of real zeros on an interval (or a measurable set) ■ of the equation

❝✵ ✁ ✶ ✰ ❝✶ ✁ t ✰ ✁ ✁ ✁ ✰ ❝♥ ✁ t♥ ❂ ✵

is ❩

■

✶ ✙❦w✵✭t✮❦❞t❀ w ❂ ✇✭t✮❂❦✇✭t✮❦✿

where ✇✭t✮ ❂ ❈✶❂✷✈✭t✮. In logarithmic derivative notation, this is

✶ ✙

❩

■

s

❅✷ ❅①❅② ❧♦❣ ✭✈✭①✮❃❈✈✭②✮✮❥①❂②❂t❞t✿

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 38

Random Bernstein polynomials

Change the polynomial

The coefficients ❜❦ are standard normal r.v. ❜

P ✿❂ P❦❂❞

❦❂✵ ❜❦

❞

❦

✁③❦✭✶ ③✮❞❦

P ❂ P❦❂❞

❦❂✵ ❜❦

❞

❦

✁②❦

P ❂ P❦❂❞

❦❂✵ ❜❦

❞

❦

✁①✷❦ ❞

❦

✁ ✘ rq

❞ ✙

q

✶ ❦✭❞❦✮

q✷❞

✷❦

✁ ❂

♣ ❙

q✷❞

✷❦

✁

P ❂ P❦❂❞

❦❂✵ ❜❦

q✷❞

✷❦

✁①✷❦

❙ ✔ ♣ ❞

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 39

Random Bernstein polynomials

Change the polynomial

The coefficients ❜❦ are standard normal r.v. ❜

P ✿❂ P❦❂❞

❦❂✵ ❜❦

❞

❦

✁③❦✭✶ ③✮❞❦

P ❂ P❦❂❞

❦❂✵ ❜❦

❞

❦

✁②❦

P ❂ P❦❂❞

❦❂✵ ❜❦

❞

❦

✁①✷❦ ❞

❦

✁ ✘ rq

❞ ✙

q

✶ ❦✭❞❦✮

q✷❞

✷❦

✁ ❂

♣ ❙

q✷❞

✷❦

✁

P ❂ P❦❂❞

❦❂✵ ❜❦

q✷❞

✷❦

✁①✷❦ Put ❙ ✔

♣ ❞ in the variance

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 40

Random Bernstein polynomials

The curve and some tricks

P ❂

❦❂❞

❳

❦❂✵

❛❦

✈ ✉ ✉ t ✥

✷❞ ✷❦

✦

①✷❦ ✶ ✙

❩

■

s

❅✷ ❅①❅② ❧♦❣ ✭✈✭①✮❃♣ ❈ ♣ ❈✈✭②✮✮❥①❂②❂t❞t ✈✭①✮❃❈✈✭②✮ ❂

❞

❳

❦❂✵

✥

✷❞ ✷❦

✦

✭①②✮✷❦

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 41

Random Bernstein polynomials

Theorem The expected number of real roots of a random polynomial (coefficients i.i.d. Gaussians, with 0 mean and moderate variance) in the Bernstein basis, is

♣ ✷❞ ✝ ❖✭✶✮

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 42

Random Bernstein polynomials

Random polynomials in the Bernstein basis

❞ ♣ ✷❞ ✭✶❀ ✶✮ ✭✶❀ ✶✮ ✭✶❀ ✵✮ ✭✵❀ ✶✮ ✭✶❀ ✶✮

100 14.142 13.640 0.760 2.740 6.530 3.610 150 17.321 16.540 0.890 3.260 8.090 4.300 200 20.000 19.740 1.100 3.780 9.740 5.120 250 22.361 21.400 1.350 3.970 10.610 5.470 300 24.495 24.320 1.270 4.760 12.300 5.990 350 26.458 26.540 1.620 5.100 13.400 6.420 400 28.284 27.980 1.490 5.430 14.080 6.980 450 30.000 29.460 1.620 5.890 14.970 6.980 500 31.623 31.200 1.830 5.960 15.620 7.790 550 33.166 32.740 1.770 6.360 16.290 8.320 600 34.641 34.300 1.850 6.570 17.270 8.610 650 36.056 35.480 2.050 6.840 17.240 9.350 700 37.417 37.200 2.160 7.510 18.650 8.880 750 38.730 38.180 2.190 7.300 19.360 9.330 800 40.000 39.160 2.220 7.830 19.490 9.620 850 41.231 40.420 2.130 8.010 20.320 9.960 900 42.426 41.780 2.390 8.070 20.530 10.790 950 43.589 42.680 2.200 8.330 21.570 10.580 1000 44.721 43.540 2.400 8.610 21.770 10.760

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 43

Random Bernstein polynomials

Theorem Consider a ♥ ✂ ♥ polynomial system, with polynomials of degree ❞ in the Bernstein basis. The coefficients are Gaussian random variables, with mean zero and moderate variance. The expected number of real roots is (asymptotically)

♣ ✷♥ ❞♥

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 44

ToDo List

Outline

1 Short intro to rational curves

2 Univariate real solving

3 Special points

4 Curves in ■

❘♥

5 Bezier curves/surfaces

6 Random Bernstein polynomials

7 ToDo List

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 45

ToDo List

Example

t ✼✦

t ✶✰t✷ ❀ t✰✶ ✶✰t✷

✁

❣✶ ❂

t✁✭✶✰s✷✮s✁✭✶✰t✷✮ st

❂ ✶ ts ❣✷ ❂

✭t✰✶✮✭✶✰s✷✮✭s✰✶✮✁✭✶✰t✷✮ st

❂ t s s t ✰ ✶ ❘t ❂ Res✭❣✶❀ ❣✷✮ ❂ ✶ ✰ t✷ t ✼✦

✏

✶✰t✷ t

❀ ✶✰t✷

t✰✶

✑

❣✶ ❂

✭✶✰t✷✮✁s✭✶✰s✷✮✁t st

❂ ts ✶ ❣✷ ❂

✭✶✰t✷✮✭s✰✶✮✭✶✰s✷✮✭t✰✶✮ st

❂ t s ✰ s ✰ t ✶ ❘t ❂ ✭❣✶❀ ❣✷✮ ❂ ✶ ✰ t✷

■ ❘♥ ✷♥

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 46

ToDo List

Example

t ✼✦

t ✶✰t✷ ❀ t✰✶ ✶✰t✷

✁

❣✶ ❂

t✁✭✶✰s✷✮s✁✭✶✰t✷✮ st

❂ ✶ ts ❣✷ ❂

✭t✰✶✮✭✶✰s✷✮✭s✰✶✮✁✭✶✰t✷✮ st

❂ t s s t ✰ ✶ ❘t ❂ Res✭❣✶❀ ❣✷✮ ❂ ✶ ✰ t✷ t ✼✦

✏

✶✰t✷ t

❀ ✶✰t✷

t✰✶

✑

❣✶ ❂

✭✶✰t✷✮✁s✭✶✰s✷✮✁t st

❂ ts ✶ ❣✷ ❂

✭✶✰t✷✮✭s✰✶✮✭✶✰s✷✮✭t✰✶✮ st

❂ t s ✰ s ✰ t ✶ ❘t ❂ Res✭❣✶❀ ❣✷✮ ❂ ✶ ✰ t✷

■ ❘♥ ✷♥

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 47

ToDo List

Example

t ✼✦

t ✶✰t✷ ❀ t✰✶ ✶✰t✷

✁

❣✶ ❂

t✁✭✶✰s✷✮s✁✭✶✰t✷✮ st

❂ ✶ ts ❣✷ ❂

✭t✰✶✮✭✶✰s✷✮✭s✰✶✮✁✭✶✰t✷✮ st

❂ t s s t ✰ ✶ ❘t ❂ Res✭❣✶❀ ❣✷✮ ❂ ✶ ✰ t✷ t ✼✦

✏

✶✰t✷ t

❀ ✶✰t✷

t✰✶

✑

❣✶ ❂

✭✶✰t✷✮✁s✭✶✰s✷✮✁t st

❂ ts ✶ ❣✷ ❂

✭✶✰t✷✮✭s✰✶✮✭✶✰s✷✮✭t✰✶✮ st

❂ t s ✰ s ✰ t ✶ ❘t ❂ Res✭❣✶❀ ❣✷✮ ❂ ✶ ✰ t✷

In ■

❘♥ you have ✷♥ ‘‘redundant’’ information.

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 48

ToDo List

Example

t ✼✦

t ✶✰t✷ ❀ t✰✶ ✶✰t✷

✁

❣✶ ❂

t✁✭✶✰s✷✮s✁✭✶✰t✷✮ st

❂ ✶ ts ❣✷ ❂

✭t✰✶✮✭✶✰s✷✮✭s✰✶✮✁✭✶✰t✷✮ st

❂ t s s t ✰ ✶ ❘t ❂ Res✭❣✶❀ ❣✷✮ ❂ ✶ ✰ t✷ t ✼✦

✏

✶✰t✷ t

❀ ✶✰t✷

t✰✶

✑

❣✶ ❂

✭✶✰t✷✮✁s✭✶✰s✷✮✁t st

❂ ts ✶ ❣✷ ❂

✭✶✰t✷✮✭s✰✶✮✭✶✰s✷✮✭t✰✶✮ st

❂ t s ✰ s ✰ t ✶ ❘t ❂ Res✭❣✶❀ ❣✷✮ ❂ ✶ ✰ t✷

In ■

❘♥ you have ✷♥ ‘‘redundant’’ information.

ToDo Better algorithm!

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

SLIDE 49

ToDo List

Future directions

Better algorithm for curves in ■

❘♥.

Complete average case analysis. Extension to surfaces.

E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC