Convex Hulls

The algorithm I’m going to showcase is taken from the book “Computational Geometry algorithms and applications” by Berg, Cheong, Kreveld, and Overmars. They give an excellent analysis of the algorithm in the book so I’ll only go over the highlights and show my implementation in Javascript using the html5 canvas.

The problem statement is given a set of points, S = \{p_1, p_2, \ldots, p_n\} in R^2 compute the convex hull of S. The input of the algorithm is the set, S, of points and the output of the algorithm is the set of points that are the vertices of the convex hull of S.

Our algorithm will use a standard design technique to generate what’s called an incremental algorithm. That is we compute the solution for the first p_1, \ldots, p_i points then add the point p_{i+1} and compute the new solution using the previous solution.

Algorithm ConvexHull(P) (the pseudocode is borrowed from “Computational Geometry algorithms and applications”)
Input. A set P of points in the plane
Output. A list containing the vertices of \mathcal{CH}(P) in clockwise order.
1. Sort the points by x-coordinate, resulting in a sequence p_1,\ldots, p_n.
2. Put the points p_1 and p_2 in a list \mathcal{L}_\textrm{upper}, with p_1 as the first point.
3. for i \leftarrow 3 to n
4.        do Append p_i to \mathcal{L}_\textrm{upper}.
5.                        while \mathcal{L}_\textrm{upper} contains more than two points and the last three points in \mathcal{L}_\textrm{upper} do not make a right turn
6.                           do Delete the middle of the last three points from \mathcal{L}_\textrm{upper}.
7. Put the points p_n and p_{n-1} in a list \mathcal{L}_\textrm{lower}, with p_n as the first point.
8. for i \leftarrow n -2 downto 1
9.           do Append p_i to \mathcal{L}_\textrm{lower}.
10.                     while \mathcal{L}_\textrm{lower} contains more than 2 points and the last three points in \mathcal{L}_\textrm{lower} do not make a right turn
11.                     do Delete the middle of the last three points from \mathcal{L}_\textrm{lower}.
12. Remove the first and last point from \mathcal{L}_\textrm{lower}. to ovoid duplication of the points where the upper and lower hull meet.
13. Append \mathcal{L}_\textrm{lower} to \mathcal{L}_\textrm{upper}, and call the resulting list \mathcal{L}.
14. return \mathcal{L}.

You can view an implementation using Javascript and the html5 canvas element, here. To add a new point simply click the left mouse button and it’ll add a point where the mouse is.

Advertisements

Right and Left Turns

In my previous post on line segment intersection I introduced the two dimensional cross product as v1 \times v2 = {v1}_x \cdot {v2}_y - {v2}_x \cdot {v1}_y. The cross product can also be used to determine if a set of three points p_1, p_2, \textrm{ and } p_3 make a right turn.

First note that if v_1 \times v_2 > 0 then the angle between v_1 and v_2 is strictly less than \pi. For example the two vectors (1, 0) \times (0, 1) = 1 \cdot 1 - 0 = 1 > 0. Similarly if v_1 \times v_2 < 0 then the angle between them is strictly greater than \pi. In the below image I've shown an example where the three points p_1, p_2, \textrm{ and } p_3 make a right turn.

Points p1, p2, p3 making a right turn

To mathematically determine if the points make a right turn we let v_1 = p_1 - p_2 \textrm{ and } v_2 = p_3 - p_2. Then taking the cross product of v_1 \textrm{ and } v_2, we have v_1 \times v_2 > 0 which implies that the angle between them is less than \pi that is they make a right turn.

This method for determining whether the points make a right or left turn is very useful for determining the convex hull of a set of points.

I’ve posted an example webpage, here, where using javascript the lines change color depending on whether the turn is to the right or the left. Please note that the page uses the html5 canvas element so it will not work in internet explorer 7, or 6.