Guangzh Gu zhou Discrete Mathematics Seminar GEOMETRICAL PROPERTIES OF WITH APPLICATIONS Supanut Chaidee Department of Mathematics, Faculty of Science Chiang Mai University, Thailand
Mo Moti tivati ation Pa Patterns on the fruit skins There are many phenomena • Patterns on (approximated) related to polygonal net problems. spherical surface ? Can we use mathematical concepts to model or understand the pattern formation? 2
Tessella Tes ellatio ion Pattern erns Polygons Points Tessellation Problem Formulation Geometrical Viewpoint Understanding phenomena A skin pattern of “Jackfruit” Computational Geometry Artocarpus Heterophyllus the study of algorithms which can be stated in terms of geometry. 3
Voronoi Diagram Vo Some problems are related to the space partitioning. Pos ost Offic ice Prob oble lem Suppose that a city has a set of post offices. We need to determine which houses will be operated by which office. A resident needs to send a letter at a post office near his home! A subdivision of a plane into these regions is called Voronoi diagram . Let be a set of sites over . R 2 P = { p 1 , p 2 , ..., p n } n The Vor Voron onoi oi region ion of the site is defined by p i ∈ S V ( p i ) V ( p i ) = { x 2 R 2 | d ( x, p i ) d ( x, p j ) for i 6 = j } where denotes the Euclidean distance between d ( x, y ) points and in the plane. x y 4
Co Considering on the Re Real-wo world Problem.. Is ordinary Voronoi diagram enough for modeling the pattern? Jackfruit skin pattern Lychee skin pattern ? To model this kind of tessellation, weig ights of each generator is important due to real-world phenomena. 5
Co Considering on the Re Real-wo world Problem.. 6
Vo Voronoi Di Diag agram am Voronoi Diagram V(space/generator/distance) K. Sugihara, Journal for Geometry and Graphics (2002) Laguerre Voronoi Diagram on the Sphere 7
Vo Voronoi Di Diag agram am c i ` r i P P i Each generator comes with its circle. d L ( P, c i ) = d ( P, P i ) 2 − r 2 i Laguerre Voronoi Diagram Voronoi Diagram V(space/generator/distance) K. Sugihara, Journal for Geometry and Graphics (2002) Laguerre Voronoi Diagram on the Sphere 8
Vo Voronoi Di Diag agram am c i ` r i P P i Q Each generator comes with its circle. R i P d L ( P, c i ) = d ( P, P i ) 2 − r 2 i P i ˜ Laguerre Voronoi Diagram c i A spherical circle on U corresponding to P i is O c i = { Q ∈ U | ˜ ˜ d ( P i , Q ) = R i } where . 0 ≤ R i /R < π / 2 The Laguerre Proximity ⇣ ⌘ ˜ cos d ( P, P i ) /R ˜ d L ( P, ˜ c i ) = cos( R i /R ) V(sphere/points/Laguerre) Spherical Laguerre Voronoi Diagram (SLVD) K. Sugihara, Journal for Geometry and Graphics (2002) Laguerre Voronoi Diagram on the Sphere 9
Research Scopes Re ? to construct mathematical models for understanding the polygonal tessellation on the fruit skins We use the spheric ical l Laguerre Vor Voron onoi oi dia iagram as a main tool for solving the problem. t=0 t=1 ? Inverse Voronoi Voronoi-based Diagram Problem Modeling Properties of the spheric ical l Laguerre Vor Voron onoi oi dia iagram 10
Co Construction of SLVD VD ` ij P ∗ i = ( x i /t i , y i /t i , z i /t i ) ˜ c i ˜ c i ˜ c j P i ( x i , y i , z i ) P j P i 1 r i t i O O π (˜ c i ) π (˜ c i ) π (˜ c j ) Spherical Laguerre Spherical Laguerre Voronoi diagram Delaunay diagram 11
Co Corresponding Structures Spherical Laguerre Delaunay Diagram Spherical Laguerre Voronoi diagram [1] K. Sugihara. Laguerre Voronoi Diagram on the Sphere , Journal for Geometry and Graphics, 6 :1, 69–81 (2002). [2] S. Chaidee, K. Sugihara. Recognition of Spherical Laguerre Voronoi Diagram , submitted Convex polyhedron 12 Voronoi-based Model for Generating the Tessellation Patterns of the Fruit Skins
Co Correspondence between SLVD VD and Pol Polyhedr hedra By definition and construction algorithms Spherical Laguerre Voronoi diagram Pr Proposition ℒ is a SLVD if and only if there is a convex polyhedron " containing the center of the sphere whose central projection coincides with ℒ . Convex polyhedron 13
Correspondence between SLVD Co VD and Pol Polyhedr hedra Polyhedron transformation ℒ is a SLVD. v a = ( t a , x a , y a , z a ) ∈ P 3 ( R ) For a point in the homogeneous coordinate system, define a map such that f : P 3 ( R ) → P 3 ( R ) α β γ δ 0 0 0 η f ( v a ) = v a 0 0 η 0 0 0 0 η Theorem The There exists a transformation satisfying the projection preservation properties. We We proposed ed algorithms for constructing a polyhedron with respect to SLVD. 14
Tes Tessella ellatio ion Analy lysis is ? i n v e r s e SLVD Recognition Problem find the SLVD which best recover the generators fits to the given tessellation and their weights. Approximation Problem S. Chaidee and K. Sugihara, Recognition of the S. Chaidee and K. Sugihara (2018), Spherical Spherical Laguerre Voronoi Diagram , preprint. Laguerre Voronoi Diagram Approximation of Tessellation without Generators , Graphical Models 95, pp. 1 – 13 Convex Spherical Tessellation ! ={ T 1 , …, T n } 15
SL SLVD R Recogni gnition P n Probl blem The Theorem There are exactly four degrees of freedom in the choice of a polyhedron ! with respect to the given SLVD. “Any choice of the initial pair of planes is sufficient to recognize the SLVD.” ` i,j P i,j ! Spherical circle radius r i Alignment of the plane $ π (˜ c j ) Spherical circle center π (˜ c j ) coordinates x i , y i Û e i,j " # π (˜ c i ) ˜ c j ˜ P j,k v i,j,k c i j i P i,k ` j,k ` i,k Û Û e i,k e j,k k 16
SLVD A SL Appr pproxi ximation P n Probl blem Voronoi Approximation of Voronoi Approximation of the Spike-containing Objects the Objects without Spikes S. Chaidee and K. Sugihara (2017), S. Chaidee, K. Sugihara (2016), Pattern Analysis and Applications Discrete and Computational Geometry and Graphs (LNCS 9943) Approximation of Fruit Skin Patterns Using Fitting Spherical Laguerre Voronoi Diagrams Spherical Voronoi Diagram to Real World Tessellations Using Planar Photographic Images 17
Obj Object ect Classifica cation on Spheric ical l Te Tessella llation ion Object 1. The object is a convex Objects surface which can be Without approximated by a sphere. Spikes 2. There exists a polygonal net on the surface. Sp Spike-co containing Object ect Spike- 1. The object is spherical containing tessellation object. 2. Each unit of the Objects polygonal net contains exactly one spike. 3. The heights of spikes are approximately uniform. 18
Vo Voronoi Ap Approxima mation Problem ? Find the spherical (Laguerre) Voronoi diagram which best fits to the given tessellation. ‘ Dis Discrepa panc ncy ’ is defined as the ratio of sum of different areas to sum of total areas. T or T V or L T: (projected) tessellation on the plane : spherical tessellation on the unit sphere T V: (projected) spherical Voronoi diagram on the plane : spherical Laguerre Voronoi diagram L ’ ≡ the be minimized ‘Dis Discrepa panc ncy’ best fit itted d Vo Voronoi dia diagr gram 19
Voronoi Approximation of the Spike-containing Objects 20
Mai Main Fram amewo work Tessellation Fitting using ordinary spherical Voronoi diagram The discrepancy depends on the sphere radius R , the spike height h , and the sphere center position ( x , z ). The parameters for obtaining the best fit spherical Voronoi diagram Claim min D ( x , z , R , h ) The discrepancy function D ( x , z , R , h ) for obtaining the appropriate with respect to the variables x , z , R , h x, z, R, h We consider the optimization problem by constructing an iterated (decreasing) sequence tending to the minimum. Fix R , h and optimize D ( x , z ) Fix x , z and optimize D ( R , h ) 0.3 0.03 0.2 0.02 0.10 0.1 0.50 0.01 0.0 0.15 0.30 0.00 0.45 20 0.25 0.20 19 0.40 0.20 18 0.25 0.35 0.15 17 0.30 0.10 0.30 16 The Method of Steepest Descent The Circular Search 21
Weig Weight Appro roxim imatio ion Tessellation Fitting using spherical Laguerre Voronoi diagram From the fitting result using an ordinary spherical Voronoi diagram, we approximate weight of each generator. The tessellation edges of the given tessellation on the plane are projected onto the sphere. • For each pair, compute that geodesic lengths d i , d j and minimize the sum of square of the residual ✓ R j ◆ ✓ R i ◆ A i − cos A j . cos v ij 1 R R Q ji ˜ ! d i d ( P i , Q i ) where P i A i = cos d j R Q ij P j The approximation is done v ij 2 using the fact of SLVD 22
Ex Experimental Results Fitting with the ordinary Fitting with the spherical spherical Voronoi diagram Laguerre Voronoi diagram Fitting with the ordinary Fitting with the spherical spherical Voronoi diagram Laguerre Voronoi diagram 23
Spherical Laguerre Voronoi Diagram Approximation Problem (Objects without Spikes) S. Chaidee, K. Sugihara (2018), Graphical Models Spherical Laguerre Voronoi Diagram Approximation to Tessellations without Generators 24
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