CSC 3102 Programming Assignment 3
Convex Hull Problem
Due: November 17, Tuesday (by 11:55 pm)
Points: 5 plus a bonus of 2.5
Penalty: You will lose one point per delayed day
Submit your work as instructed towards the end of this document. Contact TA Di Lin by e-
mail to [email protected] or me by e-mail to [email protected] for help.
Objective: Implement an algorithm to find a convex hull for given sets of points on a plane.
Background: The convex-hull problem is the problem of finding the smallest convex
polygon (that is, convex hull) that contains a given set of n points in the plane. In the class,
we have discussed two algorithms for solving the problem. The brute-force algorithm
(section 3.3 of the text book and the lecture note) is based on the fact that a line segment
connecting any two points of a given set of n points is a part of the convex hull’s boundary if
and only if all the other points of the set lie on the same side of the straight line through these
two points. This requires checking whether the expression (ax + by – c) has the same sign
(including zero) at each of these points. On the other hand, the quickhull algorithm is based
on the divide-and-conquer approach (see section 4.6 of the text book and the lecture note).
The algorithm assumes that the points are sorted in an increasing order of their x-coordinates
and then y-coordinates. The line joining the leftmost and rightmost points (P1Pn) separates
the points into two sets: The set of points to the left of the line results in the upper boundary
called upper hull and the other set of points to the right of the line results in the lower
boundary called lower hull. Both hulls are constructed using recursive approach as discussed
in the book and the lecture note. One can check whether a point lies to the left of a line
determined by two other points by computing the determinant formed by three points.
Project task: Implement either the brute-force algorithm or the quickhull algorithm to solve
the convex hull problem using the language of your choice. This is for 5 points. Implement
both the algorithms to get a bonus of 2.5 points. You can use a simple array data structure.
The first input array contains a set of n = 15 points on the xy-plane as follows:
Data1(x, y): (1, 4), (2, 5), (2, 6), (3, 14), (4, 1), (5, 3), (5, 6), (7, 2), (7, 5), (7, 10), (9, 3), (9,
12), (12, 1), (12, 7), (14, 6)
The second input array contains a set of n = 30 points on the xy-plane as follows:
Data2(x, y): (2, 1), (2, 4), (3, 6), (3, 9), (4, 3), (6, 5), (6, 6), (6, 14), (7, 5), (7, 11), (8, 6), (9,
4), (12, 4), (12, 9), (15, 2), (16, 1), (16, 14), (18, 8), (18, 21), (19, 3), (19, 15), (21, 6), (22, 6),
(22, 20), (23, 3), (23, 16), (25, 13), (26, 7), (26, 19), (28, 4)
Output: Your program output should list the points (their x- and y- coordinates) in pairs of A
and B, which define the line segments of the convex hull’s boundary you are constructing.
For Data1, the convex hull is made up of six line segments:
A B
(1, 4) (3, 14)
(1, 4) (4, 1)
(3, 14) (9, 12)
