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 Pouget 1 1 LORIA, INRIA Nancy - Grand Est, VEGAS JNCF’08 Luminy, CIRM, 20-24, Oct. 2008

Differential Geometry Problem General Algebraic Problem Outline Differential Geometry Problem 1 General Algebraic Problem 2

Differential Geometry Problem General Algebraic Problem Local Differential Formulation Locally in a surface can be parameterized by a height function : z = 1 2 ( k 1 x 2 + k 2 y 2 ) + 1 6 ( b 0 x 3 + 3 b 1 x 2 y + 3 b 2 xy 2 + b 3 y 3 ) + . . . k 1 is the maximal principal curvature : blue curvature. k 2 is the minimal principal curvature : red curvature. Umbilics are characterized by k 1 = k 2 Taylor expansion of the blue curvature along the blue curvature line going through the origin : k 1 ( x ) = k 1 + b 0 x + . . . 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 k 1 along the blue line d 1 : P 1 2 ( k 1 − k 2 ) x 2 + . . . P 1 = 3 b 2 1 +( k 1 − k 2 )( c 0 − 3 k 3 k 1 ( x ) = k 1 + b 0 x + 1 ) . A blue ridge point is characterized by b 0 = < ▽ k 1 , d 1 > = 0. elliptic if P 1 < 0 then the blue curvature is maximal along its line; hyperbolic if P 1 > 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 : R 2 − → R 3 Define an implicit curve in the parametric domain P : R 2 − → 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 S 3 R = { p 2 = P = P u = P v = 0 , δ ( P 3 ) > 0 } 1-ridge umbilics S 1 R = { p 2 = P = P u = P v = 0 , δ ( P 3 ) < 0 } Purple points S p = { a = b = a ′ = b ′ = 0 , δ ( P 2 ) > 0 , p 2 � = 0 } δ ( P 2 ) ( δ ( P 3 ) ) is the discriminant of the quadratic (cubic) form of the 2 nd (3 rd ) 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 ) = 116 u 4 v 4 − 200 u 4 v 3 + 108 u 4 v 2 − 24 u 4 v − 312 u 3 v 4 + 592 u 3 v 3 − 360 u 3 v 2 + 80 u 3 v + 252 u 2 v 4 − 504 u 2 v 3 + 324 u 2 v 2 − 72 u 2 v − 56 uv 4 + 112 uv 3 − 72 uv 2 + 16 uv . For a function h of total degree d , P has total degree at most 15 d − 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 ) ∈ C 2 is (x-)critical if f ( p ) = f y ( p ) = 0, in addition it is singular if f x ( p ) = 0 (x-)extreme if f x ( 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 Projection 1 Compute x -coordinates critical points: α i . Lifting 2 Compute intersection points between the curve and the fiber x = α i . Compute with polynomial with algebraic coefficients. Adjacencies 3 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 Compute isolating boxes for critical points 1 Easily refinable with the RUR Topology at extreme points: 2 Topology at singular points. 3 Topology in non critical cells of the induced vertical rectangular 4 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 Real roots 1 Multiplicities 2 Interval arithmetic.

Differential Geometry Problem General Algebraic Problem Solve Zero Dimensional Systems Idea: Multivariate case − → univariate one. Rational Univariate Representation (RUR) V ( I )( ∩ R n ) ≈ V ( f t )( ∩ R ) α = ( α 1 , . . . , α n ) → t ( α ) g t , X 1 ( t ( α )) g t , 1 ( t ( α )) , . . . , g t , Xn ( t ( α )) ( g t , 1 ( t ( α )) ) ← t ( α ) x 2 p 1 p 2 x x 1 Univariate Polynomial Zero Dimensional Multivariate p ( t ) = 0 System I = < p 1 , ..., pn >

Differential Geometry Problem General Algebraic Problem Topology at Extreme points Isolate the extreme system 1 I e = I ( f , f y , f x � = 0 ) = I ( f , f y , Tf x − 1 ) ∩ Q [ x , y ] Refine boxes to get 2 crossings on the border. 2 Store the multiplicities in the system I e for the connection 3 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 β i is a root of P ( y ) of multiplicity k, then P ( k ) ( y ) vanishes between β i and the other roots of P. P ( k ) P β i − 1 β i β i +1 Apply to P ( y ) = f ( α i , y ) for the singular point ( α i , β j ) .

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 ) Use saturation: 1 k = min j such that p is no longer solution of I s , k = < f , f x , f y , f y 2 , ..., f y k > Teissier’s formula 2 Theorem (Teissier) k = Mult ( p , < f , f y > ) − Mult ( p , < f x , f y > ) + 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 Isolate singular points in boxes 1 Compute multiplicities k in fibers 2 Refine the box to avoid the curve f y k := ∂ k f 3 ∂ y k Refine the box to avoid top/bottom crossings 4

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 odd even point Extreme even odd point