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2D Convex Hull in

2D Convex Hull in. Pierre Alliez. http://www.cgal.org. Outline. Definitions Convex hull in CGAL Interface Extreme points Subsequences of hull points Convexity Checking Exercise. Definitions.

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2D Convex Hull in

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  1. 2D Convex Hull in Pierre Alliez http://www.cgal.org

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

  3. 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 Pis a convex polygon with vertices in P.

  4. 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.

  5. 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

  6. 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)

  7. 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).

  8. 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

  9. Default... #include <CGAL/Cartesian.h> #include <CGAL/convex_hull_2.h> #include <list> typedef CGAL::Cartesian<double> kernel; typedefkernel::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()));

  10. Graham-Andrew #include <CGAL/Cartesian.h> #include <CGAL/graham_andrew.h> #include <list> typedef CGAL::Cartesian<double> kernel; typedefkernel::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()));

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

  12. SubSequences of Hull Points 2 • Lower Hull • Upper Hull 1 4 1 3 2

  13. Convexity Checking #include <CGAL/convexity_check_2.h> template <class ForwardIterator, class Traits> boolis_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.

  14. Exercise

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