convex hull Algorithm
As well as for finite point sets, convex hulls have also been study for simple polygons, Brownian movement, space curves, and epigraphs of functions. related structures include the orthogonal convex hull, convex layers, Delaunay triangulation and Voronoi diagram, and convex skull.
The lower convex hull of points in the airplane looks, in the form of a Newton polygon, in a letter from Isaac Newton to Henry Oldenburg in 1676.The term" convex hull" itself looks as early as the work of Garrett Birkhoff (1935), and the corresponding term in German looks earlier, for case in Hans Rademacher's review of Kőnig (1922).By 1938, according to Lloyd Dines, the term" convex hull" had become standard;
Dines adds that he finds the term unfortunate, because the colloquial meaning of the word" hull" would suggest that it refers to the surface of a shape, whereas the convex hull includes the interior and not exactly the surface.
use std::cmp::Ordering::Equal;
fn sort_by_min_angle(pts: &[(f64, f64)], min: &(f64, f64)) -> Vec<(f64, f64)> {
let mut points: Vec<(f64, f64, (f64, f64))> = pts
.iter()
.map(|x| {
(
((x.1 - min.1) as f64).atan2((x.0 - min.0) as f64),
// angle
((x.1 - min.1) as f64).hypot((x.0 - min.0) as f64),
// distance (we want the closest to be first)
*x,
)
})
.collect();
points.sort_by(|a, b| a.partial_cmp(b).unwrap_or(Equal));
points.into_iter().map(|x| x.2).collect()
}
// calculates the z coordinate of the vector product of vectors ab and ac
fn calc_z_coord_vector_product(a: &(f64, f64), b: &(f64, f64), c: &(f64, f64)) -> f64 {
(b.0 - a.0) * (c.1 - a.1) - (c.0 - a.0) * (b.1 - a.1)
}
/*
If three points are aligned and are part of the convex hull then the three are kept.
If one doesn't want to keep those points, it is easy to iterate the answer and remove them.
The first point is the one with the lowest y-coordinate and the lowest x-coordinate.
Points are then given counter-clockwise, and the closest one is given first if needed.
*/
pub fn convex_hull_graham(pts: &[(f64, f64)]) -> Vec<(f64, f64)> {
if pts.is_empty() {
return vec![];
}
let mut stack: Vec<(f64, f64)> = vec![];
let min = pts
.iter()
.min_by(|a, b| {
let ord = a.1.partial_cmp(&b.1).unwrap_or(Equal);
match ord {
Equal => a.0.partial_cmp(&b.0).unwrap_or(Equal),
o => o,
}
})
.unwrap();
let points = sort_by_min_angle(pts, &min);
if points.len() <= 3 {
return points;
}
for point in points {
while stack.len() > 1
&& calc_z_coord_vector_product(&stack[stack.len() - 2], &stack[stack.len() - 1], &point)
< 0.
{
stack.pop();
}
stack.push(point);
}
stack
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn empty() {
assert_eq!(convex_hull_graham(&vec![]), vec![]);
}
#[test]
fn not_enough_points() {
let list = vec![(0f64, 0f64)];
assert_eq!(convex_hull_graham(&list), list);
}
#[test]
fn not_enough_points1() {
let list = vec![(2f64, 2f64), (1f64, 1f64), (0f64, 0f64)];
let ans = vec![(0f64, 0f64), (1f64, 1f64), (2f64, 2f64)];
assert_eq!(convex_hull_graham(&list), ans);
}
#[test]
fn not_enough_points2() {
let list = vec![(2f64, 2f64), (1f64, 2f64), (0f64, 0f64)];
let ans = vec![(0f64, 0f64), (2f64, 2f64), (1f64, 2f64)];
assert_eq!(convex_hull_graham(&list), ans);
}
#[test]
// from https://codegolf.stackexchange.com/questions/11035/find-the-convex-hull-of-a-set-of-2d-points
fn lots_of_points() {
let list = vec![
(4.4, 14.),
(6.7, 15.25),
(6.9, 12.8),
(2.1, 11.1),
(9.5, 14.9),
(13.2, 11.9),
(10.3, 12.3),
(6.8, 9.5),
(3.3, 7.7),
(0.6, 5.1),
(5.3, 2.4),
(8.45, 4.7),
(11.5, 9.6),
(13.8, 7.3),
(12.9, 3.1),
(11., 1.1),
];
let ans = vec![
(11., 1.1),
(12.9, 3.1),
(13.8, 7.3),
(13.2, 11.9),
(9.5, 14.9),
(6.7, 15.25),
(4.4, 14.),
(2.1, 11.1),
(0.6, 5.1),
(5.3, 2.4),
];
assert_eq!(convex_hull_graham(&list), ans);
}
#[test]
// from https://codegolf.stackexchange.com/questions/11035/find-the-convex-hull-of-a-set-of-2d-points
fn lots_of_points2() {
let list = vec![
(1., 0.),
(1., 1.),
(1., -1.),
(0.68957, 0.283647),
(0.909487, 0.644276),
(0.0361877, 0.803816),
(0.583004, 0.91555),
(-0.748169, 0.210483),
(-0.553528, -0.967036),
(0.316709, -0.153861),
(-0.79267, 0.585945),
(-0.700164, -0.750994),
(0.452273, -0.604434),
(-0.79134, -0.249902),
(-0.594918, -0.397574),
(-0.547371, -0.434041),
(0.958132, -0.499614),
(0.039941, 0.0990732),
(-0.891471, -0.464943),
(0.513187, -0.457062),
(-0.930053, 0.60341),
(0.656995, 0.854205),
];
let ans = vec![
(1., -1.),
(1., 0.),
(1., 1.),
(0.583004, 0.91555),
(0.0361877, 0.803816),
(-0.930053, 0.60341),
(-0.891471, -0.464943),
(-0.700164, -0.750994),
(-0.553528, -0.967036),
];
assert_eq!(convex_hull_graham(&list), ans);
}
}
