new computational geometryacg.cs.tau.ac.il/courses/computational-geometry/spring... · 2020. 5....

Computational Geometry

Efi Fogel

Tel Aviv University

Computational Geometry Algorithm LibraryMay. 11th, 2020

Outline

1 CgalIntroductionContentLiteratureGeometry FactoryDetails

2 ArrangementMinimum Area TriangleSpherical Gaussian Map

Computational Geometry 2

Outline




Cgal: Mission

“Make the large body of geometric algorithms developed in the field ofcomputational geometry available for industrial applications”

Cgal Project Proposal, 1996


Cgal Facts

A collection of software packages written in C++

Adheres the generic programming paradigmDevelopment started in 1995An open source librarySeveral active contributor sitesHigh search-engine ranking for www.cgal.org

Used in a diverse range of domainse.g., computer graphics, scientific visualization, computer aided designand modeling, additive manufacturing, geographic information systems,molecular biology, medical imaging, and VLSI

The de-facto standard in applied Computational Geometry


www.cgal.org

Cgal in Numbers

600,000 lines of C++ code10,000 downloads per year not including Linux distributions4,500 manual pages (user and reference manual)1,000 subscribers to user mailing list200 commercial users120 packages30 active developers6 months release cycle2 licenses: Open Source and commercial


Cgal HistoryYear Version Released Other Milestones1996 Cgal founded1998 July 1.11999 Work continued after end of European support2001 Aug 2.3 Editorial Board established2002 May 2.42003 Nov 3.0 Geometry Factory founded...

2008 CMake2009 Jan 3.4, Oct 3.52010 Mar 3.6, Oct 3.7 Google Summer of Code (GSoC) 20102011 Apr 3.8, Aug 3.9 GSoC 20112012 Mar 4.0, Oct 4.1 GSoC 20122013 Mar 4.2, Oct 4.3 GSoC 2013, Doxygen2014 Apr 4.4, Oct 4.5 GSoC 20142015 Apr 4.6, Oct 4.7 GitHub, HTML5, Main repository made public2016 Apr 4.8, Sep 4.9 Only headers, 20th anniversary2017 May 4.10, Sep 4.11 CTest, GSoC 20172018 Apr 4.12 GSoC 20182019 Nov 5.0 C++14, GSoC 20192020 5.1 GSoC 2020


http://www.geometryfactory.com/

https://cmake.org/

http://code.google.com/soc/

http://www.google-melange.com/



http://www.stack.nl/~dimitri/doxygen/


https://github.com/

https://github.com/CGAL/cgal-public-dev

https://cmake.org/Wiki/CMake/Testing_With_CTest

https://summerofcode.withgoogle.com/archive/2017/projects/

https://summerofcode.withgoogle.com/



Cgal Properties

ReliabilityExplicitly handles degeneraciesFollows the Exact Geometric Computation (EGC) paradigm

EfficiencyDepends on leading 3rd party libraries

F e.g., Boost, Gmp, Mpfr, Qt, Eigen, Tbb, and CoreAdheres to the generic-programming paradigm

F Polymorphism is resolved at compile time

The best of both worlds


http://www.boost.org/

http://gmplib.org/

http://www.mpfr.org/

http://www.trolltech.com/products/qt

https://eigen.tuxfamily.org/

https://www.threadingbuildingblocks.org/

http://cs.nyu.edu/exact/core_pages/

Cgal Properties, Cont

FlexibilityAdaptable, e.g., graph algorithms can directly be applied to Cgal datastructuresExtensible, e.g., data structures can be extended

Ease of UseHas didactic and exhaustive ManualsFollows standard concepts (e.g., C++ and Stl)Has a modular structure, e.g., geometry and topology are separatedCharacterizes with a smooth learning-curve


Outline




2D Algorithms and Data Structures

Triangulations Mesh Generation Polyline Simplification Voronoi Diagrams

Arrangements Boolean Operations Neighborhood Queries Minkowski Sums Straight Skeleton


3D Algorithms and Data Structures

Triangulations Mesh Generation Polyhedral Surface Deformation Boolean Operations

Mesh Simplification Skeleton Segmentation Classification Hole Filling


Outline




Cgal Bibliography I

The Cgal Project.Cgal User and Reference Manual.Cgal Editorial Board, 4.4 edition, 2014. http://doc.cgal.org/4.2/CGAL.CGAL/html/index.html .Efi Fogel, Ron Wein, and Dan Halperin.Cgal Arrangements and Their Applications, A Step-by-Step Guide.Springer, 2012.

Mario Botsch, Leif Kobbelt, Mark Pauly, Pierre Alliez, and Bruno Levy.Polygon Mesh Processing.CRC Press, 2010.

A. Fabri, G.-J. Giezeman, L. Kettner, S. Schirra, and S. Schönherr.On the design of Cgal a computational geometry algorithms library.Software — Practice and Experience, 30(11):1167–1202, 2000. Special Issue on Discrete AlgorithmEngineering.

A. Fabri and S. Pion.A generic lazy evaluation scheme for exact geometric computations.In 2nd Library-Centric Software Design Workshop, 2006.

M. H. Overmars.Designing the computational geometry algorithms library Cgal.In Proceedings of ACM Workshop on Applied Computational Geometry, Towards GeometricEngineering, volume 1148, pages 53–58, London, UK, 1996. Springer.

Many Many Many papers


http://doc.cgal.org/4.2/CGAL.CGAL/html/index.html

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Some Cgal Commercial Customers


Cgal Commercial Customers, Geographic Segmentation


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Cgal Structure

Basic LibraryAlgorithms and Data Structurese.g., Triangulations, Surfaces, and Arrangements

KernelElementary geometric objectsElementary geometric computations on them

Support Library

Configurations, Assertions,...

VisualizationFilesI/O

Number TypesGenerators


Cgal Kernel Concept

Geometric objects of constant size.Geometric operations on object of constant size.

Primitives 2D, 3D, dD OperationsPredicates Constructions

point comparison intersectionvector orientation squared distancetriangle containment . . .iso rectangle . . .circle

. . .


Cgal Kernel Affine Geometry

point - origin → vectorpoint - point → vectorpoint + vector → point

point + point ← Illegalmidpoint(a,b) = a+1/2× (b−a)


Cgal Kernel Classification

Dimension: 2, 3, arbitraryNumber types:

Ring: +,−,×Euclidean ring (adds integer division and gcd) (e.g., CGAL : : Gmpz).Field: +,−,×,/ (e.g., CGAL : : Gmpq).Exact sign evaluation for expressions with roots (F i e l d_w i th_sq r ).

Coordinate representationCartesian—requires a field number type or Euclidean ring if noconstructions are performed.Homegeneous—requires Euclidean ring.

Reference countingExact, Filtered


Cgal Kernels and Number Types

Cartesian representation Homogeneous representation

point∣∣∣∣ x = hx

hwy = hy

hwpoint

∣∣∣∣∣∣hxhyhw

Intersection of two lines{a1x +b1y + c1 = 0a2x +b2y + c2 = 0

{a1hx +b1hy + c1hw = 0a2hx +b2hy + c2hw = 0

(x ,y) = (hx ,hy ,hw) =∣∣∣∣∣ b1 c1

b2 c2

∣∣∣∣∣∣∣∣∣∣ a1 b1a2 b2

∣∣∣∣∣,−

∣∣∣∣∣ a1 c1a2 c2

∣∣∣∣∣∣∣∣∣∣ a1 b1a2 b2

∣∣∣∣∣

(∣∣∣∣ b1 c1b2 c2

∣∣∣∣ ,− ∣∣∣∣ a1 c1a2 c2

∣∣∣∣ , ∣∣∣∣ a1 b1a2 b2

∣∣∣∣)

Field operations Ring operations


Cgal Numerical Issues� �#i f 1

t y p ed e f CORE : : Expr NT;t y p ed e f CGAL : : Ca r t e s i a n <NT> Kerne l ;NT sq r t 2 = CGAL : : s q r t (NT( 2 ) ) ;

#e l s et y p ed e f doub l e NT;t y p ed e f CGAL : : Ca r t e s i a n <NT> Kerne l ;NT sq r t 2 = s q r t ( 2 ) ;

#e n d i f

Ke rne l : : Point_2 p (0 , 0 ) , q ( sq r t2 , s q r t 2 ) ;Ke rne l : : C i r c l e_2 C(p , 4 ) ;a s s e r t (C . has_on_boundary ( q ) ) ;� �

OK if NT supports exact sqrt.Assertion violation otherwise.

p

qC


Cgal Pre-defined Cartesian Kernels

Support construction of points from doub l e Cartesian coordinates.Support exact geometric predicates.Handle geometric constructions differently:

CGAL : : E x a c t_p r e d i c a t e s_ i n e x a c t_ c on s t r u c t i o n s_k e r n e lF Geometric constructions may be inexact due to round-off errors.F It is however more efficient and sufficient for most Cgal algorithms.

CGAL : : E x a c t_p r e d i c a t e s_e x a c t_con s t r u c t i o n s_k e r n e lCGAL : : E x a c t_p r ed i c a t e s_e xa c t_con s t r u c t i o n s_ke r n e l_w i t h_ sq r t

F Its number type supports the exact square-root operation.


Cgal Special Kernels

Filtered kernels2D circular kernel3D spherical kernel

Refer to Cgal’s manual for more details.


Cgal Basic Library

Generic data structures are parameterized with TraitsSeparates algorithms and data structures from the geometric kernel.

Generic algorithms are parameterized with iterator rangesDecouples the algorithm from the data structure.


Cgal Components Developed at Tel Aviv University

2D Arrangements2D Regularized Boolean Set-Operations2D Minkowski Sums2D Envelopes3D Envelopes2D Snap Rounding2D Set Movable Seperability (2D Casting)3D Set Movable Seperability (3D Casting)Inscribed Areas / 2D Largest empty iso rectangleCGAL Python bindings for the above


Outline




2D Arrangements

Definition (Arrangement)Given a collection C of curves on a surface, the arrangement A(C) is thepartition of the surface into vertices, edges and faces induced by thecurves of C

An arrangement of cir-cles in the plane

An arrangement of lines in theplane

An arrangement ofgreat-circle arcs on asphere


Arrangement_2<Tra i t s , Dcel>

Is the main component in the 2D Arrangements package.An instance of this class template represents 2D arrangements.The representation of the arrangements and the various geometricalgorithms that operate on them are separated.The topological and geometric aspects are separated.

The T r a i t s template-parameter must be substituted by a model of ageometry-traits concept, e.g., ArrangementBasicTraits_2.

F Defines the type X_monotone_curve_2 that represents x -monotonecurves.

F Defines the type Point_2 that represents two-dimensional points.F Supports basic geometric predicates on these types.

The Dce l template-parameter must be substituted by a model of theArrangementDcel concept, e.g., A r r_de f au l t_dce l <Tra i t s >.


Arrangement Background

Arrangements have numerous applicationsrobot motion planning, computer vision, GIS, optimization,computational molecular biology

A planar map of the Boston area showing the top of the arm of cape cod.Raw data comes from the US Census 2000 TIGER/line data files


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Point-Line Duality Transform

Points and lines are transformed into lines and points, respectively.Primal Plane Dual Planethe point p : (a,b) the line p∗ : y = ax −bthe line l : y = cx +d the point l∗ : (c,−d)

This duality transform does not handle vertical lines!

The transform is incidencepreserving.The transform preserves theabove/below relation.The transform preserves thevertical distance between apoint and a line.

k

`pq

r

Primalp∗

q∗ r ∗k∗

`∗ Dual


Application: Minimum-Area Triangle

Application (Minimum-Area Triangle)Given a set P = {p1,p2, . . . ,pn} of n points in the plane, find three distinctpoints pi ,pj ,pk ∈ P such that the area of the triangle 4pipjpk is minimalamong all other triangles defined by three distinct points in P.

p1

p2

p3

p4

p5

A naive algorithm requires O(n3) time.It is possible to compute in O(n2)time.

The analysis of the algorithmtime-complexity uses the zonecomplexity theorem.




p1

p2

p3

p4

p5A naive algorithm requires O(n3) time.

It is possible to compute in O(n2)time.





p1

p2

p3

p4

p5A naive algorithm requires O(n3) time.It is possible to compute in O(n2)time.



Minimum Area Triangle: DualityP∗—the set of lines dual to the input points.

P∗ does not contain vertical lines.

p∗i ,p∗j ∈ P∗

`∗ij—the point of intersection betweenp∗i and p∗j .ìj—the line that contains pi and pj .pk—the point that defines theminimum-area triangle with pi and pj .pk is the closest point to ìj =⇒pk is the closest point to ìj in thevertical distance.p∗k—the line immedialy above orbelow the point `∗ij .

pi

pjpk

lij

Primal

l∗ij

p∗i p∗j

p∗k

Dual


Outline




Gaussian Map (Normal Diagram) in 2DDefinition (Gasusian map or normal diagram)The Gaussian map of a convex polygon P is thedecomposition of S into maximal connected arcs so that theextremal point of P is the same for all directions within onearc

~d2

~d1

An arc of directionsbetween ~d1 and ~d2.

⇐⇒

⇐⇒

Generalizes to higher dimensionsComputational Geometry 40

Carl Friedrich Gauss1777–1855

Gasusian Map (Normal Diagram) in 3DDefinition (Gasusian map or normal diagram in 3D)The Gaussian map of a convex polytope P in IR3, denoted as GM(∂P), isthe decomposition of S2 into maximal connected regions so that theextremal point of P is the same for all directions within one region

GM is a set-valued function from ∂P to S2. GM(p ∈ ∂P) = the set ofoutward unit normals to support planes of P at p.

v ,e, f—a vertex, an edge, a facet of PGM(f ) = outward unit normal to fGM(e) = geodesic segmentGM(v) = spherical polygonGM(P) is an arrangement on S2

GM(P) is unique.

If each face GM(v) is extended with v , ⇒ GM−1(GM(P)) = PComputational Geometry 41

new computational geometryacg.cs.tau.ac.il/courses/computational-geometry/spring... · 2020. 5....

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