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Page 1: 2D Convex Hull in Pierre Alliez

2D Convex Hull in2D Convex Hull in

http://www.cgal.orghttp://www.cgal.org

Pierre AlliezPierre Alliez

Page 2: 2D Convex Hull in Pierre Alliez

http://http://www.cgal.orgwww.cgal.org

OutlineOutline

• DefinitionsDefinitions• Convex hull in CGALConvex hull in CGAL

– InterfaceInterface– Extreme pointsExtreme points– Subsequences of hull pointsSubsequences of hull points– Convexity CheckingConvexity Checking

• ExerciseExercise

Page 3: 2D Convex Hull in Pierre Alliez

http://http://www.cgal.orgwww.cgal.org

DefinitionsDefinitions

• A subset A subset S S IRIR22 is is convexconvex if for any if for any two points two points p p and and q q in the set the in the set the line segment with endpoints line segment with endpoints p p and and q q is contained in is contained in SS. .

• The The convex hullconvex hull of a set of a set S S is the is the smallest convex set containing smallest convex set containing SS. .

• The The convex hull of a set of points convex hull of a set of points PP is a convex polygon with vertices in is a convex polygon with vertices in PP. .

Page 4: 2D Convex Hull in Pierre Alliez

http://http://www.cgal.orgwww.cgal.org

DefinitionsDefinitions

• A point in A point in P P is an is an extremeextreme point (with point (with respect to respect to PP) if it is a vertex of the ) if it is a vertex of the convex hull of convex hull of PP. .

• A point set is said to be A point set is said to be strongly convexstrongly convex if it consists of only extreme points.if it consists of only extreme points.

Page 5: 2D Convex Hull in Pierre Alliez

http://http://www.cgal.orgwww.cgal.org

Convex Hull in CGALConvex Hull in CGAL

5 algorithms to compute the 5 algorithms to compute the counterclockwisecounterclockwise sequence of sequence of extreme pointsextreme points for a 2D point for a 2D point set.set. 11

22

33 44

55

66

Page 6: 2D Convex Hull in Pierre Alliez

http://http://www.cgal.orgwww.cgal.org

Convex Hull in CGALConvex Hull in CGAL

nn input points with input points with hh extreme points extreme points

Algorithms:Algorithms:• [Bykat 78][Bykat 78] O( O(nhnh) output-sensitive) output-sensitive• [Akl & Toussaint][Akl & Toussaint] O( O(nlognnlogn) worst ) worst

casecase• [Graham-Andrew][Graham-Andrew] O( O(nlognnlogn))• [Jarvis 73][Jarvis 73] O( O(nhnh))• [Eddy][Eddy] O( O(nhnh))

Page 7: 2D Convex Hull in Pierre Alliez

http://http://www.cgal.orgwww.cgal.org

InterfaceInterface

template <class InputIterator, class OutputIterator>template <class InputIterator, class OutputIterator>

OutputIterator OutputIterator convex_hull_2convex_hull_2(InputIterator first,(InputIterator first, InputIterator beyond, InputIterator beyond, OutputIterator result, OutputIterator result, Traits ch_traits = Traits ch_traits = Default_traits) Default_traits)

generates the counterclockwise sequence of extreme points of the points in the range [first,beyond). The resulting sequence is placed starting at position result, and the past-the-end iterator for the resulting sequence is returned (it is not specified at which point the cyclic sequence of extreme points is cut into a linear sequence).

Page 8: 2D Convex Hull in Pierre Alliez

http://http://www.cgal.orgwww.cgal.org

InterfaceInterface

template <class InputIterator, class OutputIterator>

OutputIterator convex_hull_2(InputIterator first, InputIterator beyond, OutputIterator result, Traits ch_traits = Default_traits)

InputIterator::value_type

OutputIterator::value_type

Traits contains types and functions:

Traits::Point_2, Traits::Less_xy_2, Traits::Less_yx_2, Traits::Leftturn_2.

Traits::Point_2Traits::Point_2

Page 9: 2D Convex Hull in Pierre Alliez

http://http://www.cgal.orgwww.cgal.org

Default...Default...#include#include <CGAL/Cartesian.h> <CGAL/Cartesian.h>

#include#include <CGAL/convex_hull_2.h> <CGAL/convex_hull_2.h>

#include#include < <listlist>>

typedeftypedef CGAL::Cartesian< CGAL::Cartesian<doubledouble> > kernelkernel;;

typedeftypedef kernelkernel::Point_2 ::Point_2 PointPoint;;

std::liststd::list<<PointPoint> points;> points;

std::liststd::list<<PointPoint> hull;> hull;

points.push_back(points.push_back(PointPoint(0,0)); (0,0));

points.push_back(points.push_back(PointPoint(0,1));(0,1));

// ...// ...

CGAL::convex_hull_2CGAL::convex_hull_2(points.begin(),(points.begin(),

points.end(),points.end(),

std::inserterstd::inserter(hull,hull.begin()));(hull,hull.begin()));

Page 10: 2D Convex Hull in Pierre Alliez

http://http://www.cgal.orgwww.cgal.org

Graham-AndrewGraham-Andrew#include#include <CGAL/Cartesian.h> <CGAL/Cartesian.h>

#include#include <CGAL/graham_andrew.h> <CGAL/graham_andrew.h>

#include#include < <listlist>>

typedeftypedef CGAL::Cartesian< CGAL::Cartesian<doubledouble> > kernelkernel;;

typedeftypedef kernelkernel::Point_2 ::Point_2 PointPoint;;

std::liststd::list<<PointPoint> points;> points;

std::liststd::list<<PointPoint> hull;> hull;

points.push_back(points.push_back(PointPoint(0,0)); (0,0));

points.push_back(points.push_back(PointPoint(0,1));(0,1));

// ...// ...

CGAL::ch_graham_andrewCGAL::ch_graham_andrew(points.begin(),(points.begin(),

points.end(),points.end(),

std::inserterstd::inserter(hull,hull.begin()));(hull,hull.begin()));

Page 11: 2D Convex Hull in Pierre Alliez

http://http://www.cgal.orgwww.cgal.org

Extreme PointsExtreme Points

• North, South, East, West and North, South, East, West and combinations.combinations.

NN

WW

SS

EE

Page 12: 2D Convex Hull in Pierre Alliez

http://http://www.cgal.orgwww.cgal.org

SubSequences of Hull SubSequences of Hull PointsPoints

• Lower HullLower Hull• Upper HullUpper Hull

22

11

22 33

44

11

Page 13: 2D Convex Hull in Pierre Alliez

http://http://www.cgal.orgwww.cgal.org

Convexity CheckingConvexity Checking

#include <CGAL/convexity_check_2.h>

template <class ForwardIterator, class Traits>

bool is_ccw_strongly_convex_2 (

ForwardIterator first,

ForwardIterator beyond,

Traits ch_traits = Default_traits)

returns true iff the point elements in [first,beyond) form a counterclockwise-oriented strongly convex polygon.

Page 14: 2D Convex Hull in Pierre Alliez

http://http://www.cgal.orgwww.cgal.org

ExerciseExercise


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