# Chapter 2
2D Convex Hulls and Extreme Points

Susan Hert and Stefan Schirra

A subset S 2 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. A point in P is an extreme point (with respect to P) if it is a vertex of the convex hull of P.

CGAL provides functions for computing convex hulls in two dimensions as well as functions for testing if a given set of points is strongly convex or not. There are also a number of functions available for computing particular extreme points in 2D and subsequences of the hull points, such as the lower hull or upper hull of a set of points.

### Assertions

The assertion flags for the convex hull and extreme point algorithms use CH in their names (e.g., CGAL_CH_NO_POSTCONDITIONS). For the convex hull algorithms, the postcondition check tests only convexity (if not disabled), but not containment of the input points in the polygon or polyhedron defined by the output points. The latter is considered an expensive checking and can be enabled by defining CGAL_CH_CHECK_EXPENSIVE .

### Concepts

ConvexHullTraits_2

### Traits Classes

CGAL::Convex_hull_constructive_traits_2<R>
CGAL::Convex_hull_projective_xy_traits_2<Point_3>
CGAL::Convex_hull_projective_xz_traits_2<Point_3>
CGAL::Convex_hull_projective_yz_traits_2<Point_3>
CGAL::Convex_hull_traits_2<R>

### Convex Hull Functions

CGAL::ch_akl_toussaint
CGAL::ch_bykat
CGAL::ch_eddy
CGAL::ch_graham_andrew
CGAL::ch_jarvis
CGAL::ch_melkman
CGAL::convex_hull_2

### Convexity Checking Functions

CGAL::is_ccw_strongly_convex_2
CGAL::is_cw_strongly_convex_2

### Hull Subsequence Functions

CGAL::ch_graham_andrew_scan
CGAL::ch_jarvis_march
CGAL::lower_hull_points_2
CGAL::upper_hull_points_2

### Extreme Point Functions

CGAL::ch_e_point
CGAL::ch_nswe_point
CGAL::ch_n_point
CGAL::ch_ns_point
CGAL::ch_s_point
CGAL::ch_w_point
CGAL::ch_we_point

### Alphabetical Listing of Reference Pages

- ch_akl_toussaint

- ch_bykat

- ch_eddy

- ch_e_point

- ch_graham_andrew

- ch_graham_andrew_scan

- ch_jarvis

- ch_jarvis_march

- ch_melkman

- ch_nswe_point

- ch_ns_point

- ch_n_point

- ch_s_point

- ch_we_point

- ch_w_point

- convex_hull_2

- ConvexHullTraits_2

- Convex_hull_constructive_traits_2<R>

- Convex_hull_projective_xy_traits_2<Point_3>

- Convex_hull_projective_xz_traits_2<Point_3>

- Convex_hull_projective_yz_traits_2<Point_3>

- Convex_hull_traits_2<R>

- is_ccw_strongly_convex_2

- is_cw_strongly_convex_2

- lower_hull_points_2

- upper_hull_points_2