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

Algebraic and Geometric Computations with Rational Curves Elias TSIGARIDAS E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

Short intro to rational curves Outline Short intro to rational curves 1 Univariate real solving 2 Special points 3 ❘ ♥ Curves in ■ 4 Bezier curves/surfaces 5 Random Bernstein polynomials 6 ToDo List 7 E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

Short intro to rational curves Parametric Curves ❘ ✷ ✥ ✿ ■ ❘ ✦ ■ ✏ ❢ ✶ ❀◆ ✭ t ✮ ✑ ❢ ❉ ✭ t ✮ ❀ ❢ ✷ ❀◆ ✭ t ✮ t ✼✦ ❢ ◆ ✭ t ✮ ❘ ✷ ❈ ❂ ■♠ ✭ ✥ ✮ ❬ ■ E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

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

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

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

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

❘ ✷ ✥ ✿ ■ ❘ ✦ ■ ✒ ✶ � ✸ t ✓ ✭✶ ✰ t ✷ ✮ ✷ ❀ t ✭✶ � ✸ t ✮ t ✼✦ ✭✶ ✰ t ✷ ✮ ✷ ① ❂ ✵ ✮ ✾ t ✷ � ✹ t � ✸ ❂ ✵ ❴ ② ❂ ✵ ✮ ✻ t ✸ � ✸ t ✷ � ✻ t ✰ ✶ ❂ ✵ ❴ Short intro to rational curves Extreme points ① ❂ ✵ ✮ ❞ ❢ ✶ ❀◆ ✭ t ✮ ❴ ❢ ❉ ✭ t ✮ ❂ ✵ ❞t ② ❂ ✵ ✮ ❞ ❢ ✷ ❀◆ ✭ t ✮ ❴ ❢ ❉ ✭ t ✮ ❂ ✵ ❞t E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

Short intro to rational curves Extreme points ❘ ✷ ✥ ✿ ■ ❘ ✦ ■ ✒ ✶ � ✸ t ✓ ① ❂ ✵ ✮ ❞ ❢ ✶ ❀◆ ✭ t ✮ ✭✶ ✰ t ✷ ✮ ✷ ❀ t ✭✶ � ✸ t ✮ ❴ ❢ ❉ ✭ t ✮ ❂ ✵ t ✼✦ ❞t ✭✶ ✰ t ✷ ✮ ✷ ② ❂ ✵ ✮ ❞ ❢ ✷ ❀◆ ✭ 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

Univariate real solving Outline Short intro to rational curves 1 Univariate real solving 2 Special points 3 ❘ ♥ Curves in ■ 4 Bezier curves/surfaces 5 Random Bernstein polynomials 6 ToDo List 7 E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

Univariate real solving Univariate Real Solving Problem Given ❆ ✷ ❩ ❬ ❳ ❪ such that ❆ ❂ ❛ ❞ ❳ ❞ ✰ ✁ ✁ ✁ ✰ ❛ ✶ ❳ ✰ ❛ ✵ ✷ ❩ ❬ ❳ ❪ where d ❂ ❞❡❣✭ ❆ ✮ ▲ ✭ ❆ ✮ ❂ ♠❛① ✵ ✔ ✐ ✔ ❞ ❢ ❧❣ ❥ ❛ ✐ ❥❣ ❂ ✜ and Compute isolating intervals for the real roots. E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

Univariate real solving Example Let ❢ ❂ ① ✺ � ✼ ① ✹ ✰ ✷✷ ① ✸ � ✹ ① ✷ � ✹✽ ① ✰ ✸✻ ❂ ✭ ① � ✶✮ ✁ ✭ ① ✷ � ✻ ① ✰ ✶✽✮ ✁ ✭ ① ✷ � ✷✮ ♣ ♣ real roots � ✷ ✶ ✰ ✷ ✭ ✹✾ ✻✹ ❀ ✶✹✼ ✭ ✶✹✼ output ✭ � ✹✾ ❀ ✵✮ ✶✷✽ ✮ ✶✷✽ ❀ ✹✾✮ E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

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] ❡ ❖ ❇ ✭ ❞ ✸ ✜ ✮ [E,T.; 09] 2006+ ❡ ❡ ❖ ❇ ✭ ❞ ✺ ✜ ✷ ✮ [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

Univariate real solving How hard is the problem? Definition (Separation bound) ✐ ✻ ❂ ❥ ❥ ✌ ✐ � ✌ ❥ ❥ ✘ ✷ � ❞ ✜ ❂ ✷ � s ✁ ❂ s❡♣ ✭ ❆ ✮ ❂ ♠✐♥ Example Consider the Wilkinson polynomial ❆ ❂ ✭ ① � ✶✮✭ ① � ✷✮ ✁ ✁ ✁ ✭ ① � ✷✵✮ ✁ ✘ ✶✵ � ✸✹✹ s❡♣ ✭ ❆ ✮ ❂ ✶ actual E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

Univariate real solving Experimental results 300 400 500 600 700 800 900 1000 9.14 25.27 55.86 110.13 214.99 407.09 774.22 1376.34 cf L #roots 300 400 500 600 700 800 900 1000 cf 3.16 8.61 19.67 38.23 77.75 139.18 247.11 414.51 C1 #roots 300 400 500 600 700 800 900 1000 cf 2.54 6.09 12.07 21.43 34.52 53.35 81.88 120.21 W #roots 300 400 500 600 700 800 900 1000 0.07 0.33 0.06 0.37 0.66 0.76 1.03 1.77 cf R1 #roots 2 6 2 4 4 2 4 4 E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

Special points Outline Short intro to rational curves 1 Univariate real solving 2 Special points 3 ❘ ♥ Curves in ■ 4 Bezier curves/surfaces 5 Random Bernstein polynomials 6 ToDo List 7 E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC

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

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

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

Special points Example ❘ ✷ ✥ ✿ ■ ❘ ✦ ■ ✼✦ ✭ � ✼ t ✹ ✰ ✷✽✽ t ✷ ✰ ✷✺✻ t ❀ t ✹ ✰ ✸✷ t ✷ ✰ ✷✺✻ ❣ ✶ ✭ s❀ t ✮ ❂ � ✺✶✷ s ✸ t ✷ � ✷✵✹✽ s ✸ � ✺✶✷ s ✷ t ✸ � ✷✵✹✽ s ✷ t � ✽✵ t ✸ ✰ ✷✺✻ t t ✹ ✰ ✸✷ 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

Special points Complexity of singular points Theorem We can identify the singular points in ❖ ❇ ✭ ❞ ✺ ✜ ✮ ❡ ❖ ❇ ✭ ◆ ✻ ✮ ❡ or E.T. ( PolSys @ INRIA ) Rational Curves GC 2013 @ ACMAC