2D Convex Hull in http://www.cgal.org Pierre Alliez http://www.cgal.org.

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Presentation transcript:

2D Convex Hull in http://www.cgal.org Pierre Alliez http://www.cgal.org

Outline Definitions Convex hull in CGAL Exercise Interface Extreme points Subsequences of hull points Convexity Checking Exercise

Definitions A subset S  IR2 is convex if for any two points p and q in the set the line segment with endpoints p and q is contained in S. The convex hull of a set S is the smallest convex set containing S. The convex hull of a set of points P is a convex polygon with vertices in P.

Definitions A point in P is an extreme point (with respect to P) if it is a vertex of the convex hull of P. A point set is said to be strongly convex if it consists of only extreme points.

Convex Hull in CGAL 5 algorithms to compute the counterclockwise sequence of extreme points for a 2D point set. 1 6 5 2 4 3

Convex Hull in CGAL n input points with h extreme points Algorithms: [Bykat 78] O(nh) output-sensitive [Akl & Toussaint] O(nlogn) worst case [Graham-Andrew] O(nlogn) [Jarvis 73] O(nh) [Eddy] O(nh)

Interface template <class InputIterator, class OutputIterator> OutputIterator convex_hull_2(InputIterator first, InputIterator beyond, OutputIterator result, Traits ch_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).

Interface 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_2

Default... #include <CGAL/Cartesian.h> #include <CGAL/convex_hull_2.h> #include <list> typedef CGAL::Cartesian<double> kernel; typedef kernel::Point_2 Point; std::list<Point> points; std::list<Point> hull; points.push_back(Point(0,0)); points.push_back(Point(0,1)); // ... CGAL::convex_hull_2(points.begin(), points.end(), std::inserter(hull,hull.begin()));

Graham-Andrew #include <CGAL/Cartesian.h> #include <CGAL/graham_andrew.h> #include <list> typedef CGAL::Cartesian<double> kernel; typedef kernel::Point_2 Point; std::list<Point> points; std::list<Point> hull; points.push_back(Point(0,0)); points.push_back(Point(0,1)); // ... CGAL::ch_graham_andrew(points.begin(), points.end(), std::inserter(hull,hull.begin()));

Extreme Points North, South, East, West and combinations. N E W S

SubSequences of Hull Points Lower Hull Upper Hull 2 1 4 1 3 2

Convexity 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.

Exercise