Du calcul de courbes dextrme de courbure sur une surface au calcul - - PowerPoint PPT Presentation

Differential Geometry Problem General Algebraic Problem

Du calcul de courbes d’extrême de courbure sur une surface au calcul de la topologie de courbes algébriques en général.

Marc Pouget1

1LORIA, INRIA Nancy - Grand Est, VEGAS

JNCF’08 Luminy, CIRM, 20-24, Oct. 2008

Differential Geometry Problem General Algebraic Problem

Outline

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Differential Geometry Problem

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General Algebraic Problem

Differential Geometry Problem General Algebraic Problem

Local Differential Formulation

Locally in a surface can be parameterized by a height function : z = 1 2(k1x2 + k2y2) + 1 6(b0x3 + 3b1x2y + 3b2xy2 + b3y3) + . . . k1 is the maximal principal curvature : blue curvature. k2 is the minimal principal curvature : red curvature. Umbilics are characterized by k1 = k2 Taylor expansion of the blue curvature along the blue curvature line going through the origin : k1(x) = k1 + b0x + . . . Rk : switching the orientation of the principal directions reverts the sign of odd degree coefficients.

Differential Geometry Problem General Algebraic Problem

Blue (red) ridges

Expansion of k1 along the blue line d1: k1(x) = k1+b0x+ P1 2(k1 − k2)x2+ . . . P1 = 3b2

1+(k1−k2)(c0−3k3 1 ).

A blue ridge point is characterized by b0 =< ▽k1, d1 >= 0. elliptic if P1 < 0 then the blue curvature is maximal along its line; hyperbolic if P1 > 0 then the blue curvature is minimal along its line. Remark : Two types of Red ridges Red curves (minimum). Yellow curves (maximum).

Differential Geometry Problem General Algebraic Problem

Special points of the ridge curve

3-ridge umbilic 1-ridge umbilic Purple point

Differential Geometry Problem General Algebraic Problem

Illustrations: ridges and crest lines

Computed using approximation of local differentail quantities on meshes.

Differential Geometry Problem General Algebraic Problem

Global Algebraic Formulation

The surface is parameterized: Φ : R2 − → R3 Define an implicit curve in the parametric domain P : R2 − → R such that P = 0 is the ridge curve in the parametric domain and characterize its singularities. (P is a function of the derivatives up to the the third order of Φ)

Differential Geometry Problem General Algebraic Problem

Systems for Singularities

3-ridge umbilic 1-ridge umbilic Purple point

3-ridge umbilics S3R = {p2 = P = Pu = Pv = 0, δ(P3) > 0} 1-ridge umbilics S1R = {p2 = P = Pu = Pv = 0, δ(P3) < 0} Purple points Sp = {a = b = a′ = b′ = 0, δ(P2) > 0, p2 = 0} δ(P2) (δ(P3)) is the discriminant of the quadratic (cubic) form of the 2nd (3rd) derivatives of P.

Differential Geometry Problem General Algebraic Problem

Example

For the degree 4 Bézier surface Φ(u, v) = (u, v, h(u, v)) with

h(u, v) =116u4v4 − 200u4v3 + 108u4v2 − 24u4v − 312u3v4 + 592u3v3 − 360u3v2 + 80u3v + 252u2v4 − 504u2v3 + 324u2v2 − 72u2v − 56uv4 + 112uv3 − 72uv2 + 16uv.

For a function h of total degree d, P has total degree at most 15d − 22. P is a bivariate polynomial total degree 84, degree 43 in u and v, 1907 terms, coefficients with up to 53 digits.

Differential Geometry Problem General Algebraic Problem

Topology and Some Geometry of Real Algebraic Plane Curves

Curve: f(x, y) = 0 with f ∈ Q[x, y] Isotopic approximation of the curve by a straight line graph give results in the original coordinate system of the plane. In addition, identify and localize

extreme points, singular points, vertical asymptotes.

Differential Geometry Problem General Algebraic Problem

Notation

Curve : square free polynomial f ∈ Q[x, y]. A point p = (α, β) ∈ C2 is (x-)critical if f(p) = fy(p) = 0, in addition it is

singular if fx(p) = 0 (x-)extreme if fx(p) = 0 (i.e. x-critical and non-singular).

Differential Geometry Problem General Algebraic Problem

Previous Work

Mainly 2 approaches Subdivision Only guaranty the drawing up to some precision Need to go up to the theoretical separation bound to be certified Or need to be coupled with an exact 2d solver. Cylindrical Algebraic Decomposition based with sub-resultant and lifting Several variants: use Sturm-Habitch sequences or just principal SH coeff use generic position assumption use several projections shear and shear back

Differential Geometry Problem General Algebraic Problem

CAD based method

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Projection Compute x-coordinates critical points: αi.

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Lifting Compute intersection points between the curve and the fiber x = αi. Compute with polynomial with algebraic coefficients.

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Adjacencies Count the number of branches connected to the left and right May require generic position.

Differential Geometry Problem General Algebraic Problem

General Idea

Replace Sub-resultant sequences + computations with algebraic coefficient polynomials by GB + RUR Identify local topology at critical points using multiplicities and refinement Compute adjacencies with a vertical rectangular decomposition using multiplicities

Differential Geometry Problem General Algebraic Problem

Our Algorithm

Based on Incremental work upon [WS05] and [CFPR08] Groebner basis and Rational Univariate Representation of critical points. Specifications: Compute the exact topology (output a straight line graph) Do not require any generic position asumption Give results in the original coordinate system (identifies critical points and vertical asymptotes)

[WS05] N. Wolpert and R. Seidel. On the Exact Computation of the Topology of Real Algebraic Curves. SoCG05. [CFPR08] F. Cazals, J.-C. Faugère, M. Pouget, and F. Rouillier. Ridges and Umbilics of Polynomial Parametric Surfaces, in Geometric Modeling and Algebraic Geometry, Springer.

Differential Geometry Problem General Algebraic Problem

Algorithm Outline

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Compute isolating boxes for critical points Easily refinable with the RUR

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Topology at extreme points:

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Topology at singular points.

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Topology in non critical cells of the induced vertical rectangular decomposition of the plane.

Differential Geometry Problem General Algebraic Problem

Algebraic Tools

Univariate root isolation for polynomial with rational coefficients: Descartes algorithm. Solve zero dimensional systems with Rational Univariate Representation (RUR) preserve

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Real roots

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Multiplicities

Interval arithmetic.

Differential Geometry Problem General Algebraic Problem

Solve Zero Dimensional Systems

Idea: Multivariate case − → univariate one. Rational Univariate Representation (RUR) V(I)(∩Rn) ≈ V(ft)(∩R) α = (α1, . . . , αn) → t(α) (

gt,X1(t(α)) gt,1(t(α)) , . . . , gt,Xn (t(α)) gt,1(t(α)) )

← t(α)

Zero Dimensional Multivariate System I =< p1, ..., pn > Univariate Polynomial p(t) = 0

p1 p2 x1 x2 x

Differential Geometry Problem General Algebraic Problem

Topology at Extreme points

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Isolate the extreme system Ie = I(f, fy, fx = 0) = I(f, fy, Tfx − 1) ∩ Q[x, y]

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Refine boxes to get 2 crossings on the border.

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Store the multiplicities in the system Ie for the connection step ... see later

Differential Geometry Problem General Algebraic Problem

Topology at Singularities

What do I mean?: Ideas: Compute multiplicities in fibers Rolle’s Theorem: isolate roots of P by those of P’

Differential Geometry Problem General Algebraic Problem

Application of Rolle’s Theorem

Theorem If βi is a root of P(y) of multiplicity k, then P(k)(y) vanishes between βi and the other roots of P.

βi βi+1 βi−1 P P (k)

Apply to P(y) = f(αi, y) for the singular point (αi, βj).

Differential Geometry Problem General Algebraic Problem

Multiplicity in the Fiber

The multiplicity k of a singular point p = (α, β) in its fiber is the univariate multiplicity of β in f(α, y)

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Use saturation: k = min j such that p is no longer solution of Is,k =< f, fx, fy, fy2, ..., fyk >

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Teissier’s formula

Theorem (Teissier) k = Mult(p, < f, fy >) − Mult(p, < fx, fy >) + 1 IMPORTANT: RUR maps roots of a system to roots of a univariate polynomial with the same multiplicity.

Differential Geometry Problem General Algebraic Problem

Topology at Singularities, Summary

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Isolate singular points in boxes

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Compute multiplicities k in fibers

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Refine the box to avoid the curve fyk := ∂kf

∂yk

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Refine the box to avoid top/bottom crossings

Differential Geometry Problem General Algebraic Problem

Rectangle decomposition of the plane

The topology is known inside critical boxes. Compute a vertical decomposition of the plane wrt these boxes Compute intersections of the curve with the decomposition

Differential Geometry Problem General Algebraic Problem

Greedy Connection Algorithm Using multiplicities

Overlapping of extreme point boxes: need parity of multiplicity in fiber

Extreme point

Extreme point

• dd
• dd

even even

Differential Geometry Problem General Algebraic Problem

Complexity Analysis

Theorem The algorithm runs in OB(Rd4(dτs + s2)), where R = nb of real critical points, d = degree of the polynomial f, τ = maximum bitsize of coeff of f, s = maximum bitsize of

the separation bound of Ic, and the distance between a singular point and its isolating

• curve. (Worst case s = d3τ).

Differential Geometry Problem General Algebraic Problem

Experiments

Isotop: Maple implementation with packages: FGB (Groebner basis) RS (RUR and isolation) Faster on non generic and high degree curves.

degree Isotop AlciX Top 16 37.8 13.1 52.0 25 1121.1 416.0 > 1800

Table: Running times in seconds for resultant of two surfaces

degree Isotop AlciX ratio AlciX/Isotop Top 12 3.3 6.9 2.1 8.7 or > 600 15 14.4 35.2 2.5 39.5 or > 600 18 49.7 191.8 3.8 > 600 21 177.4 729.3 4.1 > 1200 24 503.0 2310.0 4.6 > 2400

Table: Running times in seconds for non generic curves

Differential Geometry Problem General Algebraic Problem

Conclusion

Replace Sub-resultant sequences + computations with algebraic coefficient polynomials by GB + RUR Identify local topology at critical points using multiplicities and refinement Compute adjacencies with a vertical rectangular decomposition using multiplicities