Union of Balls and -Complexes Jean-Daniel Boissonnat Geometrica, - PowerPoint PPT Presentation

Union of Balls and α -Complexes Jean-Daniel Boissonnat Geometrica, INRIA http://www-sop.inria.fr/geometrica Winter School, University of Nice Sophia Antipolis January 26-30, 2015 Winter School 2 Weighted Delaunay Complexes Sophia Antipolis 1 / 38

Laguerre geometry Power distance of two balls or of two weighted points. → weigthed point ( p 1 , r 2 1 ) ∈ R d ball b 1 ( p 1 , r 1 ) , center p 1 radius r 1 ← → weigthed point ( p 2 , r 2 2 ) ∈ R d ball b 2 ( p 2 , r 2 ) , center p 2 radius r 2 ← π ( b 1 , b 2 ) = ( p 1 − p 2 ) 2 − r 2 1 − r 2 2 Orthogonal balls ⇒ ( p 1 − p 2 ) 2 ≤ r 2 1 + r 2 b 1 , b 2 closer ⇐ ⇒ π ( b 1 , b 2 ) < 0 ⇐ 2 ⇒ ( p 1 − p 2 ) 2 = r 2 1 + r 2 b 1 , b 2 orthogonal ⇐ ⇒ π ( b 1 , b 2 ) = 0 ⇐ 2 ⇒ ( p 1 − p 2 ) 2 ≤ r 2 1 + r 2 b 1 , b 2 further ⇐ ⇒ π ( b 1 , b 2 ) > 0 ⇐ 2 Winter School 2 Weighted Delaunay Complexes Sophia Antipolis 2 / 38

Laguerre geometry Power distance of two balls or of two weighted points. → weigthed point ( p 1 , r 2 1 ) ∈ R d ball b 1 ( p 1 , r 1 ) , center p 1 radius r 1 ← → weigthed point ( p 2 , r 2 2 ) ∈ R d ball b 2 ( p 2 , r 2 ) , center p 2 radius r 2 ← π ( b 1 , b 2 ) = ( p 1 − p 2 ) 2 − r 2 1 − r 2 2 Orthogonal balls ⇒ ( p 1 − p 2 ) 2 ≤ r 2 1 + r 2 b 1 , b 2 closer ⇐ ⇒ π ( b 1 , b 2 ) < 0 ⇐ 2 ⇒ ( p 1 − p 2 ) 2 = r 2 1 + r 2 b 1 , b 2 orthogonal ⇐ ⇒ π ( b 1 , b 2 ) = 0 ⇐ 2 ⇒ ( p 1 − p 2 ) 2 ≤ r 2 1 + r 2 b 1 , b 2 further ⇐ ⇒ π ( b 1 , b 2 ) > 0 ⇐ 2 Winter School 2 Weighted Delaunay Complexes Sophia Antipolis 2 / 38

Power distance of a point wrt a ball If b 1 is reduced to a point p : π ( p, b 2 ) = ( p − p 2 ) 2 − r 2 2 Normalized equation of bounding sphere : t m ′ p ∈ ∂b 2 ⇐ ⇒ π ( p, b 2 ) = 0 n ′ p ∈ int b 2 ⇐ ⇒ π ( p, b ) < 0 m n p ∈ ∂b 2 ⇐ ⇒ π ( p, b ) = 0 p p 2 p �∈ b 2 ⇐ ⇒ π ( p, b ) > 0 Tangents and secants through p π ( p, b ) = pt 2 = pm · pm ′ = pn · pn ′ Winter School 2 Weighted Delaunay Complexes Sophia Antipolis 3 / 38

Radical Hyperplane The locus of point ∈ R d with same power distance to balls b 1 ( p 1 , r 1 ) and b 2 ( p 2 , r 2 ) is a hyperplane of R d ( x − p 1 ) 2 − r 2 1 = ( x − p 2 ) 2 − r 2 π ( x, b 1 ) = π ( x, b 2 ) ⇐ ⇒ 2 − 2 p 1 x + p 2 1 − r 2 1 = − 2 p 2 x + p 2 2 − r 2 ⇐ ⇒ 2 2( p 2 − p 1 ) x + ( p 2 1 − r 2 1 ) − ( p 2 2 − r 2 ⇐ ⇒ 2 ) = 0 A point in h 12 is the center of a ball orthogonal to b 1 and b 2 Winter School 2 Weighted Delaunay Complexes Sophia Antipolis 4 / 38

Radical Hyperplane The locus of point ∈ R d with same power distance to balls b 1 ( p 1 , r 1 ) and b 2 ( p 2 , r 2 ) is a hyperplane of R d ( x − p 1 ) 2 − r 2 1 = ( x − p 2 ) 2 − r 2 π ( x, b 1 ) = π ( x, b 2 ) ⇐ ⇒ 2 − 2 p 1 x + p 2 1 − r 2 1 = − 2 p 2 x + p 2 2 − r 2 ⇐ ⇒ 2 2( p 2 − p 1 ) x + ( p 2 1 − r 2 1 ) − ( p 2 2 − r 2 ⇐ ⇒ 2 ) = 0 A point in h 12 is the center of a ball orthogonal to b 1 and b 2 Winter School 2 Weighted Delaunay Complexes Sophia Antipolis 4 / 38

Power Diagrams also named Laguerre diagrams or weighted Voronoi diagrams Sites : n balls B = { b i ( p i , r i ) , i = 1 , . . . n } Power distance: π ( x, b i ) = ( x − p i ) 2 − r 2 i Power Diagram: Vor ( B ) One cell V ( b i ) for each site V ( b i ) = { x : π ( x, b i ) ≤ π ( x, b j ) . ∀ j � = i } Each cell is a polytope V ( b i ) may be empty p i may not belong to V ( b i ) Winter School 2 Weighted Delaunay Complexes Sophia Antipolis 5 / 38

Weighted Delaunay triangulations B = { b i ( p i , r i ) } a set of balls Del ( B ) = nerve of Vor ( B ) : B τ = { b i ( p i , r i ) , i = 0 , . . . k }} ⊂ B ⇒ � B τ ∈ Del ( B ) ⇐ b i ∈ B τ V ( b i ) � = ∅ To be proved (next slides): under a general position condition on B , B τ − → τ = conv ( { p i , i = 0 . . . k } ) embeds Del ( B ) as a triangulation in R d , called the weighted Delaunay triangulation Winter School 2 Weighted Delaunay Complexes Sophia Antipolis 6 / 38

Characteristic property of weighted Delaunay complexes � τ ∈ Del ( B ) ⇐ ⇒ V ( b i ) � = ∅ b i ∈ B τ ∃ x ∈ R d s . t . ⇐ ⇒ ∀ b i , b j ∈ B τ , b l ∈ B \ B τ π ( x, b i ) = π ( x, b j ) < π ( x, b l ) ⇐ ⇒ ∃ ball b ( x, ω ) s . t . ∀ b i ∈ B τ , b l ∈ B \ B τ 0 = π ( b, b i ) < π ( b, b l ) Winter School 2 Weighted Delaunay Complexes Sophia Antipolis 7 / 38

The space of spheres P b ( p, r ) ball of R d h ( σ ) → point φ ( b ) ∈ R d +1 φ ( b ) = ( p, s = p 2 − r 2 ) → polar hyperplane h b = φ ( b ) ∗ ∈ R d +1 x ∈ R d +1 : x d +1 = x 2 } P = { ˆ x ∈ R d +1 : x d +1 = 2 p · x − s } σ h b = { ˆ Balls will null radius are mapped onto P h p is tangent to P at φ ( p ) . The vertical projection of h b ∩ P onto x d +1 = 0 is ∂b Winter School 2 Weighted Delaunay Complexes Sophia Antipolis 8 / 38

The space of spheres P b ( p, r ) ball of R d h ( σ ) → point φ ( b ) ∈ R d +1 φ ( b ) = ( p, s = p 2 − r 2 ) → polar hyperplane h b = φ ( b ) ∗ ∈ R d +1 x ∈ R d +1 : x d +1 = x 2 } P = { ˆ x ∈ R d +1 : x d +1 = 2 p · x − s } σ h b = { ˆ Balls will null radius are mapped onto P h p is tangent to P at φ ( p ) . The vertical projection of h b ∩ P onto x d +1 = 0 is ∂b Winter School 2 Weighted Delaunay Complexes Sophia Antipolis 8 / 38

The space of spheres P b ( p, r ) ball of R d h ( σ ) → point φ ( b ) ∈ R d +1 φ ( b ) = ( p, s = p 2 − r 2 ) → polar hyperplane h b = φ ( b ) ∗ ∈ R d +1 x ∈ R d +1 : x d +1 = x 2 } P = { ˆ x ∈ R d +1 : x d +1 = 2 p · x − s } σ h b = { ˆ Balls will null radius are mapped onto P h p is tangent to P at φ ( p ) . The vertical projection of h b ∩ P onto x d +1 = 0 is ∂b Winter School 2 Weighted Delaunay Complexes Sophia Antipolis 8 / 38

The space of spheres φ ∗ ( b ) b ( p, r ) ball of R d → point φ ( b ) ∈ R d +1 φ ( b ) = ( p, s = p 2 − r 2 ) → polar hyperplane h b = φ ( b ) ∗ ∈ R d +1 x ∈ R d +1 : x d +1 = 2 p · x − s } h b = { ˆ b x x = ( x, x 2 ) and h b is equal to The vertical distance between ˆ x 2 − 2 p · x + s = π ( x, b ) The faces of the power diagram of B are the vertical projections onto x d +1 = 0 of the faces of the polytope V ( B ) = � i h + b of R d +1 Winter School 2 Weighted Delaunay Complexes Sophia Antipolis 9 / 38

The space of spheres φ ∗ ( b ) b ( p, r ) ball of R d → point φ ( b ) ∈ R d +1 φ ( b ) = ( p, s = p 2 − r 2 ) → polar hyperplane h b = φ ( b ) ∗ ∈ R d +1 x ∈ R d +1 : x d +1 = 2 p · x − s } h b = { ˆ b x x = ( x, x 2 ) and h b is equal to The vertical distance between ˆ x 2 − 2 p · x + s = π ( x, b ) The faces of the power diagram of B are the vertical projections onto x d +1 = 0 of the faces of the polytope V ( B ) = � i h + b of R d +1 Winter School 2 Weighted Delaunay Complexes Sophia Antipolis 9 / 38

Power diagrams, weighted Delaunay triangulations and polytopes D ( B ) = conv − ( ˆ V ( B ) = ∩ i φ ( b i ) ∗ + P ) Winter School 2 Weighted Delaunay Complexes Sophia Antipolis 10 / 38

Proof of the theorem B τ ⊂ B, | B τ | = d + 1 , τ = conv ( { p i , b i ( p i , r i ) ∈ B τ } ) , φ ( τ ) = conv ( { φ ( b i ) , b i ∈ B τ } ) ∃ b ( p, r ) s.t. h b = φ ( b ) ∗ = aff ( { φ ( b i ) , b i ∈ B τ } ) conv − ( { φ ( b i ) } ) φ ( τ ) ∈ D ( B ) = ∀ b j �∈ B τ , φ ( b j ) ∈ h ∗ + ⇐ ⇒ ∀ b i ∈ B τ , φ ( b i ) ∈ h b b ⇐ ⇒ ∀ b i ∈ B τ , π ( b, b i ) = 0 ∀ b j �∈ B τ , π ( b, b j ) > 0 � ⇐ ⇒ p ∈ V ( b i ) b i ∈ B τ ⇐ ⇒ τ ∈ Del ( B ) Winter School 2 Weighted Delaunay Complexes Sophia Antipolis 11 / 38

Delaunay’s theorem extended B = { b 1 , b 2 . . . b n } is said to be in general position wrt spheres if � ∃ x ∈ R d with equal power to d + 2 balls of B P = { p 1 , ..., p n } : set of centers of the balls of B Theorem If B is in general position wrt spheres, the simplicial map f : vert(Del( B )) → P provides a realization of Del( B ) Del( B ) is a triangulation of P ′ ⊆ P called the Delaunay triangulation of B Winter School 2 Weighted Delaunay Complexes Sophia Antipolis 12 / 38

Delaunay’s theorem extended B = { b 1 , b 2 . . . b n } is said to be in general position wrt spheres if � ∃ x ∈ R d with equal power to d + 2 balls of B P = { p 1 , ..., p n } : set of centers of the balls of B Theorem If B is in general position wrt spheres, the simplicial map f : vert(Del( B )) → P provides a realization of Del( B ) Del( B ) is a triangulation of P ′ ⊆ P called the Delaunay triangulation of B Winter School 2 Weighted Delaunay Complexes Sophia Antipolis 12 / 38

Power diagrams, Delaunay triangulations and polytopes If B is a set of balls in general position wrt spheres : duality V ( B ) = h + b 1 ∩ . . . ∩ h + D ( B ) = conv − ( { φ ( b 1 ) , . . . , φ ( b n ) } ) − → b n ↑ ↓ nerve Voronoi Diagram of B − → Delaunay Complex of B Winter School 2 Weighted Delaunay Complexes Sophia Antipolis 13 / 38