$3.($Graham scan in Python$)$ In this question, you will implement the Graham scan algorithm for convex

hull in $2-$space.

$($a$)$ Write a Python function for computing the convex hull of a point cloud in $2-$space. It has

one input parameter: a column$-$major data matrix representing the point cloud. It returns an

NumPy array: the column indices of the input that define the convex hull. These indices are

sorted counterclockwise around the hull, starting with the lowest point.

$($b$)$ Write a Python function that renders a point cloud and its convex hull. It takes two input

parameters, a column$-$major data matrix of the point cloud and an array of the column indices

of its convex hull. Use matplotlib to render. $($As a challenge, you could try implementing using

pyglet.$)$

$($c$)$ To test the two functions on random point clouds, write a Python function with three input

parameters: the size of the point cloud, the bottom left corner of the square containing the point

cloud, the size of the square. It should return a column$-$major data matrix representing a random

point cloud of this size that lies inside this square.

$($d$)$ Use the three functions to test your code by drawing the convex hull of random point clouds.