Geometry of Surfaces : Umbilics and Ridges Implicit equation of ridges of a parametric surface Topology of ridges of a polynomial parametric surface Ridges and umbilics of polynomial parametric surfaces Frédéric Cazals 1 Jean-Charles Faugère 2 Marc Pouget 1 Fabrice Rouillier 2 1 INRIA Sophia, Project GEOMETRICA 2 INRIA Rocquencourt, Project SALSA Computational Methods for Algebraic Spline Surfaces II September 14th-16th 2005 Centre of Mathematics for Applications at the University of Oslo, Norway F . Cazals, J-C. Faugère, M. Pouget, F. Rouillier Ridges and umbilics of polynomial parametric surfaces
Geometry of Surfaces : Umbilics and Ridges Implicit equation of ridges of a parametric surface Topology of ridges of a polynomial parametric surface Outline Geometry of Surfaces : Umbilics and Ridges 1 Curvatures and beyond Implicit equation of ridges of a parametric surface 2 The ridge curve and its singularities Topology of ridges of a polynomial parametric surface 3 Introduction Algorithm F . Cazals, J-C. Faugère, M. Pouget, F. Rouillier Ridges and umbilics of polynomial parametric surfaces
Geometry of Surfaces : Umbilics and Ridges Implicit equation of ridges of a parametric surface Curvatures and beyond Topology of ridges of a polynomial parametric surface Outline Geometry of Surfaces : Umbilics and Ridges 1 Curvatures and beyond Implicit equation of ridges of a parametric surface 2 The ridge curve and its singularities Topology of ridges of a polynomial parametric surface 3 Introduction Algorithm F . Cazals, J-C. Faugère, M. Pouget, F. Rouillier Ridges and umbilics of polynomial parametric surfaces
Geometry of Surfaces : Umbilics and Ridges Implicit equation of ridges of a parametric surface Curvatures and beyond Topology of ridges of a polynomial parametric surface Principal curvatures and directions k 1 and d 1 maximal principal curvature and direction (Blue). k 2 and d 2 minimal principal curvature and direction (Red). k i and d i are eigenvalues and eigenvectors of the Weingarten map W = I − 1 II . d 1 and d 2 are orthogonal. F . Cazals, J-C. Faugère, M. Pouget, F. Rouillier Ridges and umbilics of polynomial parametric surfaces
Geometry of Surfaces : Umbilics and Ridges Implicit equation of ridges of a parametric surface Curvatures and beyond Topology of ridges of a polynomial parametric surface Umbilics and curvature lines A curvature line is an integral curve of the principal direction field. Umbilics are singularities of these fields, k 1 = k 2 F . Cazals, J-C. Faugère, M. Pouget, F. Rouillier Ridges and umbilics of polynomial parametric surfaces
Geometry of Surfaces : Umbilics and Ridges Implicit equation of ridges of a parametric surface Curvatures and beyond Topology of ridges of a polynomial parametric surface Ridges A blue (red) ridge point is a point where k 1 ( k 2 ) has an extremum along its curvature line. < ▽ k 1 , d 1 > = 0 ( < ▽ k 2 , d 2 > = 0 ) (1) Ridge points form lines going through umbilics. Umbilics, ridges, and principal blue foliation on the ellipsoid F . Cazals, J-C. Faugère, M. Pouget, F. Rouillier Ridges and umbilics of polynomial parametric surfaces
Geometry of Surfaces : Umbilics and Ridges Implicit equation of ridges of a parametric surface Curvatures and beyond Topology of ridges of a polynomial parametric surface Orientation of principal directions Principal directions d 1 ( d 2 ) are not globally orientable. The sign of < ▽ k 1 , d 1 > is not well defined. < ▽ k 1 , d 1 > = 0 cannot be a global equation of blue ridges. The principal field is not orientable around an umbilic F . Cazals, J-C. Faugère, M. Pouget, F. Rouillier Ridges and umbilics of polynomial parametric surfaces
Geometry of Surfaces : Umbilics and Ridges Implicit equation of ridges of a parametric surface Curvatures and beyond Topology of ridges of a polynomial parametric surface Singularities of the ridge curve 3-ridge umbilic 1-ridge umbilic Purple point F . Cazals, J-C. Faugère, M. Pouget, F. Rouillier Ridges and umbilics of polynomial parametric surfaces
Geometry of Surfaces : Umbilics and Ridges Implicit equation of ridges of a parametric surface Curvatures and beyond Topology of ridges of a polynomial parametric surface Difficulties of ridge extraction Need third order derivatives of the surface. Singularities: Umbilics and Purple points Orientation problem. F . Cazals, J-C. Faugère, M. Pouget, F. Rouillier Ridges and umbilics of polynomial parametric surfaces
Geometry of Surfaces : Umbilics and Ridges Implicit equation of ridges of a parametric surface Curvatures and beyond Topology of ridges of a polynomial parametric surface Illustrations: ridges and crest lines F . Cazals, J-C. Faugère, M. Pouget, F. Rouillier Ridges and umbilics of polynomial parametric surfaces
Geometry of Surfaces : Umbilics and Ridges Implicit equation of ridges of a parametric surface The ridge curve and its singularities Topology of ridges of a polynomial parametric surface Outline Geometry of Surfaces : Umbilics and Ridges 1 Curvatures and beyond Implicit equation of ridges of a parametric surface 2 The ridge curve and its singularities Topology of ridges of a polynomial parametric surface 3 Introduction Algorithm F . Cazals, J-C. Faugère, M. Pouget, F. Rouillier Ridges and umbilics of polynomial parametric surfaces
Geometry of Surfaces : Umbilics and Ridges Implicit equation of ridges of a parametric surface The ridge curve and its singularities Topology of ridges of a polynomial parametric surface Problem statement The surface is parametrerized: Φ : ( u , v ) ∈ R 2 − → Φ( u , v ) ∈ R 3 Find a well defined function P : ( u , v ) ∈ R 2 − → P ( u , v ) ∈ R such that P = 0 is the ridge curve in the parametric domain. F . Cazals, J-C. Faugère, M. Pouget, F. Rouillier Ridges and umbilics of polynomial parametric surfaces
Geometry of Surfaces : Umbilics and Ridges Implicit equation of ridges of a parametric surface The ridge curve and its singularities Topology of ridges of a polynomial parametric surface Solving the orientation problem Consider blue and red ridges together < ▽ k 1 , d 1 > × < ▽ k 2 , d 2 > is orientation independant. Find two vector fields v 1 and w 1 orienting d 1 such that: v 1 = w 1 = 0 characterizes umbilics. Note: each vector field must vanish on some curve joining umbilics v 1 and w 1 are computed from the two dependant equations of the eigenvector system for d 1 . F . Cazals, J-C. Faugère, M. Pouget, F. Rouillier Ridges and umbilics of polynomial parametric surfaces
Geometry of Surfaces : Umbilics and Ridges Implicit equation of ridges of a parametric surface The ridge curve and its singularities Topology of ridges of a polynomial parametric surface Some technicallities p 2 = ( k 1 − k 2 ) 2 = 0 characterize umbilics. It is a smooth function of the second derivatives of Φ . Define a , a ′ , b , b ′ such that: � Numer ( ▽ k 1 ) , v 1 � = a √ p 2 + b and � Numer ( ▽ k 1 ) , w 1 � = a ′ √ p 2 + b ′ . These are smooth function of the derivatives of Φ up to the third order. F . Cazals, J-C. Faugère, M. Pouget, F. Rouillier Ridges and umbilics of polynomial parametric surfaces
Geometry of Surfaces : Umbilics and Ridges Implicit equation of ridges of a parametric surface The ridge curve and its singularities Topology of ridges of a polynomial parametric surface Main result The ridge curve has equation P = ab ′ − a ′ b = 0. For a point of this set one has: If p 2 = 0, the point is an umbilic. If p 2 � = 0 then If ab � = 0 or a ′ b ′ � = 0 then the sign of one these non-vanishing products gives the color of the ridge point. Otherwise, a = b = a ′ = b ′ = 0 and the point is a purple point. F . Cazals, J-C. Faugère, M. Pouget, F. Rouillier Ridges and umbilics of polynomial parametric surfaces
Geometry of Surfaces : Umbilics and Ridges Implicit equation of ridges of a parametric surface The ridge curve and its singularities Topology of ridges of a polynomial parametric surface Singularities of the ridge curve 1-ridge umbilics S 1 R = { p 2 = P = P u = P v = 0 , δ ( P 3 ) < 0 } 3-ridge umbilics S 3 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 } F . Cazals, J-C. Faugère, M. Pouget, F. Rouillier Ridges and umbilics of polynomial parametric surfaces
Geometry of Surfaces : Umbilics and Ridges Implicit equation of ridges of a parametric surface The ridge curve and its singularities Topology of ridges of a polynomial parametric surface Example For the degree 4 Bezier 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 + 324 u 2 v 2 − 72 u 2 v − 56 uv 4 + 112 uv 3 − 72 uv 2 + 16 uv . P is a bivariate polynomial total degree 84, degree 43 in u and v , 1907 terms, coefficients with up to 53 digits. F . Cazals, J-C. Faugère, M. Pouget, F. Rouillier Ridges and umbilics of polynomial parametric surfaces
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