2D Arrangements in CGAL: Recent Developments

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2D Arrangements in CGALRecent Developments

• CGAL Team
• School of Computer Science
• Tel Aviv University

Eti Ezra, Eyal Flato, Efi Fogel, Dan Halperin,
Shai Hirsch, Eran Leiserowitz, Eli Packer, Tali
Zvi, Ron Wein
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Outline

• Introduction
• The Packages in Brief
• Exploiting the Kernel
• Categorizing the Traits
• Benchmarking
• More Work

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Outline

• Introduction
• The Package in Brief
• Exploiting the Kernel
• Categorizing the Traits
• Benchmarking
• More Work

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Introduction

• Bypasses are devices that allow some people to
  dash from point A to point B very fast while
  other people dash from point B to point A very
  fast. People living at point C, being a point
  directly in between, are often given to wonder
  what's so great about point A that so many people
  from point B are so keen to get there and what's
  so great about point B that so many people from
  point A are so keen to get there. They often wish
  that people would just once and for all work out
  where the hell they wanted to be.
• Douglas Adams

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Definitions

• Planar Maps
• Planar graphs that are embedded
• in the plane

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Definitions (cont.)

• Planar Arrangements
• Given a collection G of planar curves,
• the arrangement A(G) is the partition
• of the plane to vertices, edges and
• faces induced by the curves of G

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Application GIS
Nguyen Dong Ha, et al.
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Application Robot Motion Planning
Flato, Halperin
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Outline

• Introduction
• The Package in Brief
• Exploiting the Kernel
• Categorizing the Traits
• Benchmarking
• More Work

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The Package in Brief

• A common mistake that people make when trying to
  design something completely foolproof is to
  underestimate the ingenuity of complete fools.
• Douglas Adams

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The Package in Brief

• Goal Construct, maintain, modify, traverse,
  query and present subdivisions of the plane
• Exact
• Generic
• Handles all degeneracies
• Efficient

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• Topological_map
• Maintains topological maps of finite edges
• Planar_map_2
• Maintains planar maps of interior-disjoint
  x-monotone curves
• Planar_map_with_intersections_2
• Maintains planar maps of general curves (may
  intersect, may be non-x-monotone)
• Arrangement_2
• Maintains planar maps of intersecting curves
  along with curve history

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The Package in Brief
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Functionality

• Creation Destruction
• I/O
• Save, Load, Print (ASCII streams)
• Draw (graphic streams)
• Flexibility (Adaptable and Extensible, Verbose
  mode, I/O of specific elements)
• Modification
• Insertion, Removal, Split, Merge
• Traversal
• Queries
• Number of Vertices, Halfedges, Faces
• Is Point in Face
• Point Location, Vertical ray shoot

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Traversal

• Element Traversal
• Vertex Iterator
• Face Iterator
• Edge Iterator
• Halfedge Iterator
• Map Traversal
• Connected Component of the Boundary (CCB)
  Halfedge Circulator
• Around Vertex Halfedge Circulator
• Hole Iterator

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Point Location Strategies

• Naive
• No preprocessing, no internal data
• Linear query time
• Walk along a line
• No preprocessing, no internal data
• Linear query time with heuristics
• Trapezoidal decomposition based
• Preprocessing, internal data
• Expected logarithmic query time

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Traits Classes

• Geometric Interface
• Parameter of package
• Defines the family of curves in interest
• Package can be used with any family of curves for
  which a traits class is supplied
• Aggregate
• geometric types (points, curves)
• Operations over types (accessors, predicates,
  constructors)

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Traits Classes

• Supplied Traits Classes
• Segments, Polylines, Circular arcs and Line
  segments, Conics (and line segments).
• Other Known Traits Classes
• Circular arcs, Canonical Parabola, Bezier Curves

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Insertions

• Non intersecting insert

Halfedge_handle non_intersecting_insert(const
X_curve_2 cv,
Change_notification
en NULL)

• Intersecting insert

Halfedge_handle insert(const X_curve_2 cv,
Change_notification en NULL)
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Insertions

• Incremental Insert
• Aggregate Insert
• Often information is known in advance
• Containing face
• Insert in face interior
• Incident vertices
• Insert from vertex, between vertices
• Order around vertex
• Insert from halfedge target, between halfedge
  targets

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Aggregate Insert

• Inserts a container into the map

template ltclass curve_iteratorgt Halfedge_iterator
insert(const curve_iterator begin,
const curve_iterator end,
Change_notification en NULL)

• Two versions
• Simplified - planar map no intersections
• General - planar map with intersections
• Sweep based
• If planar map is not empty, use overlay

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Outline

• Introduction
• The Package in Brief
• Exploiting the Kernel
• Categorizing the Traits
• Benchmarking
• More Work

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Exploiting the Kernel

• Human beings, who are almost unique in having
  the ability to learn from the experience of
  others, are also remarkable for their apparent
  disinclination to do so.
• Douglas Adams

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CGAL Kernel Context

• CGAL consists of three major parts
• Kernel
• Basic geometric data structures and algorithms
• Convex Hull, Planar_map, Arrangement, etc.
• Non-geometric support facilities

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CGAL Kernel

• Encapsulates
• Constant-size non-modifiable geometric primitive
  object representations
• Point, Segments, hopefully Conics, etc
• operations (and predicates) on these objects
• Adaptable and Extensible
• Efficient
• Used as a traits class for algorithms

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Adapting the kernel

• Exchange of representation classes
• Representation classes are parameterized by a
  number type
• Geometric objects are extracted from a
  representation class

template ltclass Kernelgt class Pm_segment_traits_2
public Kernel public typedef typename
KernelPoint_2 Point_2 typedef typename
KernelSegment_2 X_curve_2
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Adapting the kernel

• Functors provide the functionality
• Functor a class that define an appropriate
  operator()
• Object for functors are obtained through access
  member functions

template ltclass Kernelgt class Pm_segment_traits_2
public Kernel Comparison_result
compare_x(const Point_2 p1, const
Point_2 p2) const return compare_x_2_object(
)(p1, p2)
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Adapting the kernel

• Code reduction
• Implementation is simple and concise
• Traits reduction
• Matthias Baesken LEDA Kernel makes the dedicated
  LEDA Traits obsolete

if defined(USE_LEDA_KERNEL) typedef
CGALleda_rat_kernel_traits
Kernel else typedef leda_rational
NT typedef CGALCartesianltNTgt
Kernel endif typedef CGALPm_segment_traits_2
ltKernelgt Traits
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Outline

• Introduction
• The Package in Brief
• Exploiting the Kernel
• Categorizing the Traits
• Benchmarking
• More Work

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Categorizing the Traits

• It is a mistake to think you can solve any major
  problems just with potatoes.
• Douglas Adams

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Categorizing the Traits

• In the past 2 levels of refinements
• Planar map Traits
• Planar map of intersecting curves Traits
• In the future multiple categories
• Each category identifies a behavior
• Multiple Tags
• All categories identify the Traits

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Dispatching Algorithms

• Tailored Algorithms
• Curve category
• Segments, Circular Arcs, Conics

template ltclass Kernelgt class Arr_segment_traits_2
typedef Segment_tag Curve_category templ
ate ltclass Kernelgt class Arr_conic_traits_2
typedef Conic_tag Curve_category
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Dispatching Algorithms

• Trading between efficiency and complexity
• Intersection Category
• Lazy, Efficient

typedef Lazy_intersection_tag Intersection_categor
y Point_2 reflect_point(const Point_2 pt)
const X_curve_2 reflect_curve(const X_curve_2
cv) const Bool nearest_intersection_to_right()
const
typedef Efficient_intersection_tag
Intersection_category Bool nearest_intersection_t
o_right() const Bool nearest_intersection_to_lef
t() const
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Tightening the Traits

• Different operations may have
• Different requirements
• Different preconditions
• Minimal set of requirements
• Sweep has less requirement

bool do_intersect_to_left(c1, c2, pt) bool
do_intersect_to_right(c1, c2, pt) bool
nearest_intersection_to_left(c1, c2, pt, ) bool
nearest_intersection_to_right(c1, c2, pt,
) result curve_compare_at_x_left(cv1, cv2,
pt) result curve_compare_at_x_right(cv1, cv2, pt)
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Specialization

• Caching
• Avoid computations (intersection points)
• Avoid construction (extreme end-points)
• Code Reuse
• Caching of intersection points is currently
  implemented as part of the conic traits
• Requires redefinition of some classes (e.g.,
  halfedge)
• Work in progress

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Outline

• Introduction
• The Package in Brief
• Exploiting the Kernel
• Categorizing the Traits
• Benchmarking
• More Work

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Insert Multiplications
Non intersecting vs. intersecting 2
Incremental vs. aggregate 2
Point location strategies 3
CGAL cartesian parameterized with LEDA rational number type vs. Matthias LEDA Kernel 2
Segments, Conics 2
Traits categories 2
Total 96
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Benchmarks
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Benchmarks
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Outline

• Introduction
• The Package in Brief
• Exploiting the Kernel
• Categorizing the Traits
• Benchmarking
• More Work

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More Work

• Capital letters were always the best way of
  dealing with things you didn't have a good answer
  to.
• Douglas Adams

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More Work

• Consolidate Pm and Pmwx into a unified class
  Planar_map_2
• Introduce more Specialization categories and
  options
• Introduce more Point Location Strategies
• Introduce Traits classes for complex curves
• Move up to higher dimensions

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End