Voronoi diagrams of semi-algebraic sets

François Anton

Summary1. Introduction2. Voronoi diagrams for curved objects3. Combining symbolic computation and

scientific computation4. The offset to an algebraic curve5. The Delaunay graph predicate for conics6. The Delaunay graph predicate for semi-

algebraic sets7. Conclusions

● Most of the objects we encounter in the real world are curved objects in the three dimensional Euclidean space. Curves are frequently encountered in maps.

● Many problems from different fields involve proximity queries like finding the nearest neighbour, finding all the neighbours, quantifying or qualifying the neighbour-liness of two objects.

● The Voronoi diagram (of semi-algebraic sets) is the solution to those proximity problems.

1.1 The motivation

1.2 The Voronoi diagram

1.3 Semi-algebraic sets

1.4 The Delaunay graph

CIRCLE

CUBIC

1.5 The Delaunay empty circle criterion

1.6 The Delaunay graph predicate

S1S2S3S4Rr(x,y)(x1,y1)(x2,y2)(x3,y3)

1.7 The problem

2. Voronoi diagrams for curved objects

-Voronoi diagrams of general manifolds (Devillers, Meiser, Teillaud 1992)

-Voronoi diagrams of curved objects (Alt and Schwarzkopf 1995)

-Voronoi diagram and medial axis transform for planar domains with curved boundaries (Ramammurthy, Farouki, Johnstone 1994-1999)

3.1 The Delaunay graph predicate for circles

(x,y)rREvaluate the signs of R-r among the points (x,y,r,R) such that (x,y) is at the same distance from the three green circles.

C4

r R

(x,y)

3.2 The Delaunay graph predicate computed from the

original curves

11 minutes11 minutesnot zero-dimesionalnot zero-dimesionalobject too largeobject too largebad resultbad resultnot enough memorynot enough memorysegmentation fault12 hoursALIASMacaulay 2Maple (Bezoutian)Computer Algebra SystemMaximaSparse resultantGB/RSafter arranging equations

without arranging equations

3.3 Linking symbolic and scientific computations

•Using the sparse resultant, we reduced the semi-algebraic computation of the Delaunay graph predicate to a series of eigenvalues computations.

•These eigenvalues computations can be solved in theory with sparse methods.

•Some symbolic precomputations can be used to speed numerical computations.

4.1 The true offset to an algebraic curve

4.2 The generalised offset to an algebraic curve

4.3 The implicit equation of the generalised offset

-p=(x,y) ∈ = ( )C V f- p non singular- ( , )=0f x y- =( , ) q u v∈ ( , , )V f n d- The equation of the

generalised offset is obtained by

eliminating x and y , .from f n and d

5.1 The generalised Voronoi vertex

C1C2C3r(x1,y1)(x2,y2)(x3,y3)

5.2 The Delaunay graph predicate

S1S2S3S4Rr(x,y)(x1,y1)(x2,y2)(x3,y3)

5.3 The geometry-algebra dictionary

Sparse resultant of R-r,o1,o2,o3,o4 Eigenvalues of the matrix of the multiplication map by f in AValues of f on the points of XToric variety constructed from the Newton polytopes of ----->Canonic representative of f in the quotient algebra A=K[x]/I(X)Polynomial function f on a variety X (R-r)Ideal I(X)=<o1,o2,o3,o4>oi r-generalised offset to Cio4 R-generalised offset to C4

Variety X ((x,y,r,R) where (x,y) is a generalised Voronoi vertex of C1, C2, and C3) and r and R are local distances

AlgebraGeometry

5.4 The algebraic precomputations

•The number of solutions of the variety X varies between 756 for four parabolas to 2048 for four ellipses and/or hyperbolas.

•The size of the sparse resultant matrix reaches 7995 (four ellipses/hyperbolas).

•The computation of the multiplication matrix is error-prone because the sparse resultant matrix is ill-conditionned.

6.1 The formulation of the Delaunay graph predicate

for all l=1,..,n: nijkl(xi,q)=0di(xi,q)=0fxi1

...fxin≠0

point q on a set Vi=V({fijk| *ijk is“=”}),

i=1,..,n+2

fijk(xi) *ijk 0point xi on a semi-algebraic set Xi, i=1,..,n+2

point mi on Xi such that q is in the normal space to Vi or

V({fijk}) at mi

For each geometric locus ...fijk(xi) *ijk 0for all l=1,..,n: nijkl(mi,q)=0

di(mi,q)<0

... the following algebraic equations/inequalities

ELLIPSECIRCLE

6.2 An example

HYPERBOLACIRCLEELLIPSE

3 min 41 s2 min 26 s12 min 37 s20 h 17 min 42 s2 h 56 min 10 s6 min 38 soptimised Hessian Solve

optimised Gradient Solve

optimised General Solve

using only generalised offset implicit equations

from the equations of curves, normals,

and distances

Predicate running time (3B bisection)6.3 The results with and without

generalised offsets

7. Conclusions•The certified computation of the Delaunay graph of conics using interval analysis gradient and hessian based solvers can benefit from the use of the implicit equation of the generalised offset to a conic (from 35 to 331 times faster).

•Even though there is no theoretical lower nor upper bound for the interval analysis based solvers, in practice those solvers can compute the predicate faster than the computation of the eigenvalues of the multiplication matrix.

7. Conclusions (continued)•Moreover, the computation of the predicate by interval analysis based solvers is more direct than the computation of the predicate through eigenvalues of the multiplication matrix.

•Finally the computation of the predicate by interval analysis based solvers can be used for semi-algebraic sets of arbitrary degree, which is not possible for algebraic techniques.

Acknowledgments•Dr. David Kirkpatrick (UBC, CS)•Dr. James Carrell (UBC, Maths)•Dr. Anne Condon (UBC, CS)•Dr. Joel Friedman (UBC, Maths & CS)•Dr. Jean-Pierre Merlet (INRIA, COPRIN)•Dr. Ioannis Emiris, Dr. Monique Teillaud, Dr. Bernard Mourrain (INRIA, GALAAD)

•Dr. Jean-Daniel Boissonnat (INRIA, PRISME)•Students of UBC Beta Lab and PRISME and GALAAD projects at INRIA Sophia Antipolis

Acknowledgments (continued)•Technical support staff at UBC Department of Computer Science and UMS Medicis CNRS-Ecole Polytechnique

•Gouvernement Français, CROUS, CNOUS•Institut National de Recherche en Informatique et en Automatique (INRIA)

•Fonds pour la Formation de chercheurs et l’aide à la recherche (FCAR) du Québec

•Natural Sciences and Engineering Research Council of Canada (NSERC)