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ConvexHull-GrahamScan

Description

The convex hull of a set of points is the set of all possible convex combinations formed by the points. Graham Scan is an algorithm for finding the convex hull of a finite set of points. The algorithm is performed through the following steps:

  1. A starting point is found by finding the point in the set with the smallest y value. If there is a tie, the one with the smallest x value will be taken.
  2. The rest of the points are ranked according to their polar angle with respect to the starting point $(x_{1}, y_{1})$. In the result of a tie, the farthest point is taken and the rest are removed. The polar angle is found by the equation: $$\theta =\arctan\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$$
  3. Points are considered sequentially. Points which form a left-handed or counter-clockwise turn will be added to the convex hull perimeter but those which form a right-handed or clockwise turn will be removed. A left-hand turn is defined for 3 points when: $$( x_{2} - x_{1} )( y_{3} - y_{1} ) - ( y_{2} - y_{1} )( x_{3} - x_{1} ) > 0$$ The Graham Scan algorithm has time complexity O(n log n)

Organization

Tree

code
   |-- convexhullvis.py
   |-- runtimevis.py
results
   |-- n5_convexhull.png
   |-- n50_convexhull.png
   |-- n500_convexhull.png
   |-- runtime.png

Running the project

To run the project download the repository and run:

python3 convexhullvis.py

You will then be prompted to enter the desired number of points to be randomly generated and visualized. Once picked the program will run and generate the visualization in a separate window.

To generate an analysis on the runtime complexity of the algorithm download the repository and run:

python3 runtimevis.py

Results

n = 5 n = 50 n = 500