Week 6: Exercises

1. Rectangle Class

Write a class Rectangle representing a rectangle in 2 dimensions. The class should support these operations:

• Rectangle(p, q) – make a rectangle with corners at Points p and q

• r.area() - return the area of a rectangle

• r.perimeter() - return the perimeter of a rectangle

• r.contains(p) – return true if point p is inside the rectangle r

• r.intersects(s) – return true if rectangle r intersects rectangle s. (Be warned: this is a bit tricky. As a hint, you might first write a helper function that determines whether two line segments intersect.)

2. Circle Class

Write a class Circle representing a circle in 2 dimensions. The class should support these operations:

• Circle(p, r) – make a circle whose center is at p, with radius r

• c.area() - return the area of a circle

• c.circumference() – return the circumference of a circle

• c.contains(p) – return true if point p is inside the circle c

3. Vector Class

Write a class Vector that represents a vector in n-dimensional space. (We partially implemented this class in the lecture.)

• Provide an initializer that takes a list of numbers, yielding a vector with those components.

• v.add(w) should add two vectors.

• v.mul(w) should multiply a scalar times a vector, yielding a new vector.

• v.dot(w) method should compute the dot product of two vectors, yielding a scalar.

• v.length() should compute the length of a vector.

• v.angle(w) should computes the angle between two vectors in radians.

• A vector should have a string representation such as "[3.25 5.02 -6.12]", with two digits after the decimal point in each component.

4. Date Class

Write a class Date representing a month and day. The class should support these operations:

• Date(m, d) – make a Date representing the given month (1 ≤ m ≤ 12) and day (1 ≤ d ≤ 31)

• d.next() - return the Date after d

• d.prev() - return the Date before d

• d.add(n) – add n days to the given Date, wrapping past Dec 31 to Jan 1 if necessary

• A Date object's string representation should look like this: "May 15".

5. Matrix Sum

Write a function that computes the sum of two matrices A and B, represented as lists of lists. Assert that the matrices have the same dimensions.

6. Matrix Product

Write a function that computes the product of two matrices A and B, represented as lists of lists. Assert that the matrices have dimensions that are compatible for multiplication.

7. Cycle

Write a function cycle that takes a list of length N representing a permutation of the integers 0, …, N – 1. The function should return True if the permutation is a single cycle. For example:

• cycle([2, 3, 1, 0]) is True because the permutation

  • 0 -> 2
    1 -> 3
    2 -> 1
    3 -> 0

  is a cycle (0 -> 2 -> 1 -> 3 -> 0).

• cycle([2, 3, 0, 1]) is False because the permutation

  • 0 -> 2
    1 -> 3
    2 -> 0
    3 -> 1

  is not a cycle (0 -> 2 -> 0, 1 -> 3 -> 1).