Description

HomeWork 4 – CS5007
Thierry Petit tpetit@WPI.edu
Learning objectives
Pyplot
Graphs
Algorithm
Dates
Write your code and comments on a file FIRSTNAME-LASTNAME-HW4.py that you will create from scratch (there is no template). Upload it on the course website (no delay accepted).
1 Archimedean spiral (35 points)
Using Pyplot and Numpy, state a function arc that draws a green Archimedean spiral according to the following parameterized equation:
x(t) = t.cos(t) y(t) = t.sin(t)
t should vary in [0,5π] (step 0.01) and the function should call plt.plot with arguments x and y, as well as plt.show() function at the end. Call the function.
2 Heart (35 points)
Using Pyplot and Numpy, state a function heart that draws in dashed red the following parameterized equation:
x(t) = 16sin3(t) y(t) = 13cos(t) − 5cos(2t) − 2.5cos(3t) − cos(4t)
t should vary in [0,2π] (step 0.01) and the function should call plt.plot with arguments x and y, as well as plt.show() function at the end. Call the function.
To see the result you should first close the window of part 1 (Archimedean spiral).
3 Graphs (20 points)
We wish to use graph theory in order to solve the following problem. A company should carry different chemical products P1, P2, …, Pk from the factory to a city. For security reasons, some products should not be carried in the same truck: ∀i,0 < i < k,Pi is not compatible with Pi+1. Moreover Pk is not compatible with P1.
1) Write a comment: how can we state an undirected graph that visually represents this problem: 1. What are the vertices, what are the edges? 2. What is the specific property of this graph?
2) Write an algorithm that, given an integer k > 1, returns the minimum number of trucks necessary to carry all the products.
4 Algorithm (10 points)
In ancient times, Egyptians did not use multiplication tables. Instead, they transformed any calculation into multiplication by two, and additions. The principle of Egyptian multiplication is the following: (x ∗ y) = 2 ∗ (x ∗ (y/2)) if y is even, and
(x ∗ y) = x + (x ∗ (y − 1)) if y is odd. For instance,
(7 ∗ 10) = (2 ∗ (7 ∗ 5)) = (2 ∗ (7 + (7 ∗ 4))) = (2 ∗ (7 + (2 ∗ (7 ∗ 2)))).
Write a recursive Python function that, given two strictly positive integers x and y (the arguments of the function), returns the result of x ∗ y obtained using the Egyptian method. Of course, directly multiplying x by y is allowed in your code if and only if x ≤ 2 or y ≤ 2.