# Naive convex hull from Algorithms in a Nutshell book

I am working through the book Algorithms in a Nutshell by George T. Heineman and trying to translate the pseudocodes into Rust.

Here is a naive convex hull algorithm:

use std::cmp::Ordering::Equal;
use std::cmp::Ordering;
use std::collections::BTreeSet;

#[derive(Debug, Copy, Clone)]
pub struct Point {
x: f64,
y: f64,
}

impl PartialEq for Point {
fn eq(&self, other: &Point) -> bool {
if self.x.is_nan() & other.x.is_nan() & self.y.is_nan() & other.y.is_nan() {
return true;
}
if self.x.is_nan() & other.x.is_nan() {
return self.y == other.y;
}
if self.y.is_nan() & other.y.is_nan() {
return self.x == other.x;
}
(self.x == other.x) & (self.y == other.y)
}
}

impl Eq for Point {}

impl PartialOrd for Point {
fn partial_cmp(&self, other: &Point) -> Option<Ordering> {
Some(self.cmp(other))
}
}

impl Ord for Point {
fn cmp(&self, other: &Point) -> Ordering {
let c = self.x.partial_cmp(&other.x).unwrap();
if c == Equal {
self.y.partial_cmp(&other.y).unwrap()
} else {
c
}
}
}

impl Point {
pub fn new(x: f64, y: f64) -> Point {
if x.is_nan() | y.is_nan() {
panic!("Coordinates cannot be NaN!");
}
Point {
x: x,
y: y,
}
}

// Euclidean distance
fn distance(&self, other: &Point) -> f64 {
((self.x - other.x).powi(2) + (self.y - other.y).powi(2)).sqrt()
}

// Draw a horizontal line through this point, connect this point with the other, and measure the angle between these two lines.
fn angle(&self, other: &Point) -> f64 {
(other.y - self.y).atan2(other.x - self.x)
}
}

type Output = Point;
fn add(self, rhs: Point) -> Point {
Point::new(self.x + rhs.x, self.y + rhs.y)
}
}
impl Sub for Point {
type Output = Point;
fn sub(self, rhs: Point) -> Point {
Point::new(self.x - rhs.x, self.y - rhs.y)
}
}
// dot product
impl Mul for Point {
type Output = f64;
fn mul(self, rhs: Point) -> f64 {
self.x * rhs.x + self.y * rhs.y
}
}

#[test]
fn test_point() {
use std::f64::consts::PI;
let p1 = Point::new(0.0, 0.0);
let p2 = Point::new(0.0, 1.0);
assert_eq!(p1.angle(&p2), PI/2.0);
assert_eq!(p1.distance(&p2), 1.0);
let p1 = Point::new(0.0, 0.0);
let p2 = Point::new(1.0, 1.0);
assert_eq!(p1.angle(&p2), PI/4.0);
assert_eq!(p1.distance(&p2), 2.0f64.sqrt());
let p1 = Point::new(0.0, 0.0);
let p2 = Point::new(1.0, -1.0);
assert_eq!(p1.angle(&p2), -PI/4.0);
assert_eq!(p1.distance(&p2), 2.0f64.sqrt());
}

#[derive(PartialEq,Eq,Debug)]
struct Triangle {
p0: Point,
p1: Point,
p2: Point,
}

impl Triangle {
fn new(p0: Point, p1: Point, p2: Point) -> Triangle {
// Sort by x-coordinate to make sure the first point is the leftmost and lowest.
let mut v: Vec<Point> = vec![p0, p1, p2];
v.sort();
Triangle {
p0: v[0],
p1: v[1],
p2: v[2],
}
}

fn range_x(&self) -> (f64, f64) {
(self.p0.x, self.p2.x)
}
fn range_y(&self) -> (f64, f64) {
let mut v = vec![self.p0, self.p1, self.p2];
v.sort_by(|a, b| {
a.y.partial_cmp(&(b.y)).unwrap_or(Equal)
});
(v[0].y, v[2].y)
}

// Barycentric Technique, check whether point is in triangle, see http://blackpawn.com/texts/pointinpoly/
fn contains(&self, p: Point) -> bool {
let v0 = self.p2 - self.p0;
let v1 = self.p1 - self.p0;
let v2 = p - self.p0;
let dot00 = v0 * v0;
let dot01 = v0 * v1;
let dot02 = v0 * v2;
let dot11 = v1 * v1;
let dot12 = v1 * v2;
let inv_denom = 1.0f64 / (dot00 * dot11 - dot01 * dot01);
let u = (dot11 * dot02 - dot01 * dot12) * inv_denom;
let v = (dot00 * dot12 - dot01 * dot02) * inv_denom;
(u > 0.0) & (v > 0.0) & (u + v < 1.0)
}
}

#[macro_export]
macro_rules! btreeset {
($($x: expr),*) => {{
let mut set = ::std::collections::BTreeSet::new();
$( set.insert($x); )*
set
}}
}

pub fn convex_hull_naive(points: &BTreeSet<Point>) -> BTreeSet<Point> {
// you must have at least 3 points, otherwise there is no hull
assert!(points.len() > 2);
// a fn for removing just one point from the set
let minus_one = |p: &Point| {
points.difference(&(btreeset!(p.clone()))).cloned().collect::<BTreeSet<Point>>()
};
// set of points that are marked as internal
let mut p_internal_set: BTreeSet<Point> = BTreeSet::new();
// check permutations of 4 points, check if the fourth point is contained in the triangle
for p_i in points {
let minus_i = minus_one(&p_i);
for p_j in &minus_i {
let minus_j = minus_one(&p_j);
for p_k in &minus_j {
let minus_k = minus_one(&p_k);
for p_m in &minus_k {
if Triangle::new(p_i.clone(), p_j.clone(), p_k.clone()).contains(p_m.clone()) {
p_internal_set.insert(p_m.clone());
}
}
}
}
}
// set of points that are not internal
let mut hull = points.difference(&p_internal_set).cloned().collect::<Vec<Point>>();
// sort by coordinates so that the first point is the leftmost
hull.sort();
let angle_to_head = |p: &Point| {
0.0
} else {
}
};
// sort by the angle with the first point
// when that is equal, sort by distance to head
hull.sort_by(|a, b| {
angle_a.partial_cmp(&angle_b).unwrap()
});
hull.into_iter().collect::<BTreeSet<Point>>()
//  return vec![Point::new(1.0, 1.0)];
}

#[test]
fn test_convex_hull_naive() {
let points = (0..4).into_iter().flat_map(move |i| {
let i = i as f64;
(0..4).into_iter().map(move |j| {
let j = j as f64;
Point::new(i, j)
})
}).collect::<BTreeSet<Point>>();
//  let mut points: BTreeSet<Point> = BTreeSet::new();
//  for i in 0..4 {
//      let i = i as f64;
//      for j in 0..4 {
//          let j = j as f64;
//          points.insert(Point::new(i, j));
//      }
//  }
assert_eq!((&points).len(), 16);
let hull = convex_hull_naive(&points);
let hull_should_be = btreeset!(
Point::new(0.0, 0.0),
Point::new(1.0, 0.0),
Point::new(2.0, 0.0),
Point::new(3.0, 0.0),
Point::new(3.0, 1.0),
Point::new(3.0, 2.0),
Point::new(3.0, 3.0),
Point::new(2.0, 3.0),
Point::new(1.0, 3.0),
Point::new(0.0, 3.0),
Point::new(0.0, 2.0),
Point::new(0.0, 1.0)
);
assert_eq!(hull, hull_should_be);
}

#[test]
fn test_triangle() {
let p2 = Point::new(0.0, 0.0);
let p1 = Point::new(0.0, 1.0);
let p0 = Point::new(1.0, 1.0);
let t = Triangle::new(p0, p1, p2);
assert_eq!(t.range_x(), (0.0, 1.0));
assert_eq!(t.range_y(), (0.0, 1.0));
assert_eq!(t.p0, p2);
assert_eq!(t.p1, p1);
assert_eq!(t.p2, p0);
// triangle should not contain its vertices
assert!(!t.contains(p0));
assert!(!t.contains(p1));
assert!(!t.contains(p2));
// triangle should contain points on any side
assert!(!t.contains(Point::new(0.5, 0.5)));
assert!(!t.contains(Point::new(0.3, 0.3)));
assert!(!t.contains(Point::new(0.2, 0.2)));
assert!(!t.contains(Point::new(0.1, 0.1)));
assert!(!t.contains(Point::new(0.0, 0.1)));
assert!(!t.contains(Point::new(0.0, 0.2)));
assert!(!t.contains(Point::new(0.1, 1.0)));
assert!(!t.contains(Point::new(0.2, 1.0)));
assert!(!t.contains(Point::new(0.2, 1.1)));
// strictly inside the triangle
assert!(t.contains(Point::new(0.5, 0.51)));
assert!(t.contains(Point::new(0.5, 0.52)));
assert!(t.contains(Point::new(0.5, 0.53)));
let p2 = Point::new(0.0, 0.0);
let p1 = Point::new(0.5, 1.0);
let p0 = Point::new(1.0, 0.5);
let t = Triangle::new(p0, p1, p2);
assert_eq!(t.range_x(), (0.0, 1.0));
assert_eq!(t.range_y(), (0.0, 1.0));
assert_eq!(t.p0, p2);
assert_eq!(t.p1, p1);
assert_eq!(t.p2, p0);
}


All suggestions are welcome.

1. unwrapping the result of partial_cmp means you don't support NaN, but the PartialEq implementation indicates you do. Which is it?

2. The explicit panic saying you don't support NaN also indicates that...

3. My biggest suggestion is to create a wrapper around f64 to ensure that number is never NaN. Controversial change, probably, but helps you highlight where that error class can occur. I left some f64 around, mostly for the test uses, but also because I'm not sure if powi / atan2 can produce NaN.

4. Add / Sub could skip new as we know the value must already be non-NaN.

5. Create a test module

6. Split up the tests; giving them good names that explain why one set of assertions is different from another.

7. Use a space after comma in derive list.

8. No need to allocate a vector to sort; an array works fine.

9. No need to explictly specify the type of the 3 points vector / array.

10. Can use sort_by_key when there's an Ord implementation.

11. There's no reason for contains to take ownership of p; could take reference. Should benchmark to see if it matters.

12. Taking the reciprocal is a method.

13. Strange to use bitwise ANDs (&) instead of logical (&&)...

14. Make assertion match text - >= 3.

15. Comment calls a closure a fn; that's too confusing. Don't worry about the implementation; code can show that.

16. No need to use difference to remove a single value; remove works.

17. No need for an explicit type on p_internal_set.

18. No need to clone; Point is Copy so you can just dereference.

19. No need to iterate over reference to the created BTreeSet.

20. I prefer to specify the collect type on the variable.

21. Let inference handle the type inside the collection with _.

22. Why not make angle return 0 for same point?

23. No need for type hint on collect as last line; type of fn defines it.

24. Why convert from a Vec anyway?

25. Remove commented code.

use std::cmp::Ordering;
use std::collections::BTreeSet;

#[derive(Debug, Copy, Clone)]
struct DefinitelyANumber(f64);

impl DefinitelyANumber {
// TODO: Make this a real error
fn new(v: f64) -> Result<Self, ()> {
if v.is_nan() {
Err(())
} else {
Ok(DefinitelyANumber(v))
}
}

fn to_f64(&self) -> f64 {
self.0
}
}

type Output = DefinitelyANumber;
fn add(self, rhs: DefinitelyANumber) -> DefinitelyANumber {
DefinitelyANumber(self.0 + rhs.0)
}
}

impl Sub for DefinitelyANumber {
type Output = DefinitelyANumber;
fn sub(self, rhs: DefinitelyANumber) -> DefinitelyANumber {
DefinitelyANumber(self.0 - rhs.0)
}
}

impl Mul for DefinitelyANumber {
type Output = DefinitelyANumber;
fn mul(self, rhs: DefinitelyANumber) -> DefinitelyANumber {
DefinitelyANumber(self.0 * rhs.0)
}
}

impl PartialEq for DefinitelyANumber {
fn eq(&self, other: &DefinitelyANumber) -> bool {
self.0 == other.0
}
}

impl PartialEq<f64> for DefinitelyANumber {
fn eq(&self, other: &f64) -> bool {
if other.is_nan() {
return false;
}
self.0 == *other
}
}

impl Eq for DefinitelyANumber {}

impl PartialOrd for DefinitelyANumber {
fn partial_cmp(&self, other: &DefinitelyANumber) -> Option<Ordering> {
Some(self.cmp(other))
}
}

impl Ord for DefinitelyANumber {
fn cmp(&self, other: &DefinitelyANumber) -> Ordering {
self.0.partial_cmp(&other.0).expect("A number that can't be NaN was NaN")
}
}

#[derive(Debug, Copy, Clone, PartialEq, Eq, PartialOrd, Ord)]
pub struct Point {
x: DefinitelyANumber,
y: DefinitelyANumber,
}

impl Point {
pub fn new(x: f64, y: f64) -> Point {
Point {
x: DefinitelyANumber::new(x).expect("X coordinate cannot be NaN!"),
y: DefinitelyANumber::new(y).expect("Y coordinate cannot be NaN!"),
}
}

// Euclidean distance
fn distance(&self, other: &Point) -> f64 {
((self.x - other.x).to_f64().powi(2) + (self.y - other.y).to_f64().powi(2)).sqrt()
}

// Draw a horizontal line through this point, connect this point with the other, and measure the angle between these two lines.
fn angle(&self, other: &Point) -> f64 {
if self == other {
0.0
} else {
(other.y - self.y).to_f64().atan2((other.x - self.x).to_f64())
}
}
}

type Output = Point;
fn add(self, rhs: Point) -> Point {
Point {
x: self.x + rhs.x,
y: self.y + rhs.y,
}
}
}
impl Sub for Point {
type Output = Point;
fn sub(self, rhs: Point) -> Point {
Point {
x: self.x - rhs.x,
y: self.y - rhs.y,
}
}
}
// dot product
impl Mul for Point {
type Output = f64;
fn mul(self, rhs: Point) -> f64 {
(self.x * rhs.x + self.y * rhs.y).to_f64()
}
}

#[derive(PartialEq, Eq, Debug)]
pub struct Triangle {
p0: Point,
p1: Point,
p2: Point,
}

impl Triangle {
fn new(p0: Point, p1: Point, p2: Point) -> Triangle {
// Sort by x-coordinate to make sure the first point is the leftmost and lowest.
let mut v = [p0, p1, p2];
v.sort();
Triangle {
p0: v[0],
p1: v[1],
p2: v[2],
}
}

fn range_x(&self) -> (f64, f64) {
(self.p0.x.to_f64(), self.p2.x.to_f64())
}

fn range_y(&self) -> (f64, f64) {
let mut v = [self.p0, self.p1, self.p2];
v.sort_by_key(|v| v.y);
(v[0].y.to_f64(), v[2].y.to_f64())
}

// Barycentric Technique, check whether point is in triangle, see http://blackpawn.com/texts/pointinpoly/
fn contains(&self, p: Point) -> bool {
let v0 = self.p2 - self.p0;
let v1 = self.p1 - self.p0;
let v2 = p - self.p0;
let dot00 = v0 * v0;
let dot01 = v0 * v1;
let dot02 = v0 * v2;
let dot11 = v1 * v1;
let dot12 = v1 * v2;
let inv_denom = (dot00 * dot11 - dot01 * dot01).recip();
let u = (dot11 * dot02 - dot01 * dot12) * inv_denom;
let v = (dot00 * dot12 - dot01 * dot02) * inv_denom;

(u > 0.0) && (v > 0.0) && (u + v < 1.0)
}
}

#[macro_export]
macro_rules! btreeset {
($($x: expr),*) => {{
let mut set = ::std::collections::BTreeSet::new();
$( set.insert($x); )*
set
}}
}

pub fn convex_hull_naive(points: &BTreeSet<Point>) -> BTreeSet<Point> {
// you must have at least 3 points, otherwise there is no hull
assert!(points.len() >= 3);
// Remove just one point from the set
let minus_one = |p: &Point| {
let mut subset = points.clone();
subset.remove(p);
subset
};
// set of points that are marked as internal
let mut p_internal_set = BTreeSet::new();
// check permutations of 4 points, check if the fourth point is contained in the triangle
for p_i in points {
let minus_i = minus_one(&p_i);
for p_j in minus_i {
let minus_j = minus_one(&p_j);
for p_k in minus_j {
let minus_k = minus_one(&p_k);
for p_m in minus_k {
if Triangle::new(*p_i, p_j, p_k).contains(p_m) {
p_internal_set.insert(p_m);
}
}
}
}
}
// set of points that are not internal
let mut hull: Vec<_> = points.difference(&p_internal_set).cloned().collect();
// sort by coordinates so that the first point is the leftmost
hull.sort();

// sort by the angle with the first point
// when that is equal, sort by distance to head
hull.sort_by(|a, b| {
angle_a.partial_cmp(&angle_b).unwrap()
});
hull.into_iter().collect()
}

#[cfg(test)]
mod test {
use std::collections::BTreeSet;

use super::*;

#[test]
fn test_point() {
use std::f64::consts::PI;
let p1 = Point::new(0.0, 0.0);
let p2 = Point::new(0.0, 1.0);
assert_eq!(p1.angle(&p2), PI / 2.0);
assert_eq!(p1.distance(&p2), 1.0);
let p1 = Point::new(0.0, 0.0);
let p2 = Point::new(1.0, 1.0);
assert_eq!(p1.angle(&p2), PI / 4.0);
assert_eq!(p1.distance(&p2), 2.0f64.sqrt());
let p1 = Point::new(0.0, 0.0);
let p2 = Point::new(1.0, -1.0);
assert_eq!(p1.angle(&p2), -PI / 4.0);
assert_eq!(p1.distance(&p2), 2.0f64.sqrt());
}

#[test]
fn test_convex_hull_naive() {
let points: BTreeSet<_> = (0..4)
.into_iter()
.flat_map(move |i| {
let i = i as f64;
(0..4).into_iter().map(move |j| {
let j = j as f64;
Point::new(i, j)
})
})
.collect();
assert_eq!((&points).len(), 16);
let hull = convex_hull_naive(&points);
let hull_should_be = btreeset!(Point::new(0.0, 0.0),
Point::new(1.0, 0.0),
Point::new(2.0, 0.0),
Point::new(3.0, 0.0),
Point::new(3.0, 1.0),
Point::new(3.0, 2.0),
Point::new(3.0, 3.0),
Point::new(2.0, 3.0),
Point::new(1.0, 3.0),
Point::new(0.0, 3.0),
Point::new(0.0, 2.0),
Point::new(0.0, 1.0));
assert_eq!(hull, hull_should_be);
}

#[test]
fn test_triangle() {
let p2 = Point::new(0.0, 0.0);
let p1 = Point::new(0.0, 1.0);
let p0 = Point::new(1.0, 1.0);
let t = Triangle::new(p0, p1, p2);
assert_eq!(t.range_x(), (0.0, 1.0));
assert_eq!(t.range_y(), (0.0, 1.0));
assert_eq!(t.p0, p2);
assert_eq!(t.p1, p1);
assert_eq!(t.p2, p0);
// triangle should not contain its vertices
assert!(!t.contains(p0));
assert!(!t.contains(p1));
assert!(!t.contains(p2));
// triangle should contain points on any side
assert!(!t.contains(Point::new(0.5, 0.5)));
assert!(!t.contains(Point::new(0.3, 0.3)));
assert!(!t.contains(Point::new(0.2, 0.2)));
assert!(!t.contains(Point::new(0.1, 0.1)));
assert!(!t.contains(Point::new(0.0, 0.1)));
assert!(!t.contains(Point::new(0.0, 0.2)));
assert!(!t.contains(Point::new(0.1, 1.0)));
assert!(!t.contains(Point::new(0.2, 1.0)));
assert!(!t.contains(Point::new(0.2, 1.1)));
// strictly inside the triangle
assert!(t.contains(Point::new(0.5, 0.51)));
assert!(t.contains(Point::new(0.5, 0.52)));
assert!(t.contains(Point::new(0.5, 0.53)));
let p2 = Point::new(0.0, 0.0);
let p1 = Point::new(0.5, 1.0);
let p0 = Point::new(1.0, 0.5);
let t = Triangle::new(p0, p1, p2);
assert_eq!(t.range_x(), (0.0, 1.0));
assert_eq!(t.range_y(), (0.0, 1.0));
assert_eq!(t.p0, p2);
assert_eq!(t.p1, p1);
assert_eq!(t.p2, p0);
}
}

• What do you mean "to make this a real error"?
– qed
Sep 19, 2016 at 6:58
• @qed something that implements std::error::Error. Result<T, ()> is a lazy way of saying "the operation can fail but I'm not going to tell you how or why". Not very nice to end users of that method. Sep 19, 2016 at 12:45