Question

Make super- and subclass for a point. A point $(x,y)$ in the plane can be represented by a class: We can extend the Point class to also contain the representation of the point in polar coordinates. To this end, create a subclass PolarPoint whose constructor takes the polar representation of a point, $(r,\theta )$, as arguments. Store $r$ and $\theta$ as attributes and call the superclass constructor with the corresponding $x$ and $y$ values (recall the relations $x=r\cos \theta$ and $y=r\sin \theta$ between Cartesian and polar coordinates). Add a _-str__- method in class PolarPoint which prints out $r,\theta ,x$, and $y$. Verify the implementation by initializing three points and printing these points. Name of program file: PolarPoint.py. 0

Short answer

Define a `Point` class and extend it with `PolarPoint`, using cosine and sine to convert polar to Cartesian coordinates, and verify by printing.

Step-by-step solution

Define the Base Class

Create the base class `Point` which represents a point in Cartesian coordinates. Include an initializer `__init__` method to take `x` and `y` as parameters and store them as instance variables. Here's how the code will look: ```python class Point: def __init__(self, x, y): self.x = x self.y = y ```

Extend the Class for Polar Coordinates

Define a subclass `PolarPoint` which will extend the `Point` class. This class should include an initializer that accepts polar coordinates `r` and `theta`. Calculate the corresponding `x` and `y` using the relationships: $$x=r\cos \theta$$$$y=r\sin \theta$$Pass these values to the `Point` class initializer.

Implement the Subclass Constructor

Inside the `PolarPoint` subclass, use the calculated `x` and `y` values from the polar coordinates to call the constructor of the superclass `Point`. Store `r` and `theta` as additional instance variables: ```python import math class PolarPoint(Point): def __init__(self, r, theta): x = r * math.cos(theta) y = r * math.sin(theta) super().__init__(x, y) self.r = r self.theta = theta ```

Add a String Representation Method

Implement a `__str__` method in the `PolarPoint` class to print the polar and Cartesian coordinates. This method should return a string that appropriately formats these values: ```python def __str__(self): return f'PolarPoint(r={self.r}, theta={self.theta}, x={self.x}, y={self.y})' ```

Verify the Implementation

Create a Python script named `PolarPoint.py`. In this script, initialize three `PolarPoint` objects with different `r` and `theta` values, then print each object's string representation to verify that all attributes are correctly computed and stored: ```python if __name__ == "__main__": p1 = PolarPoint(1, math.pi/4) p2 = PolarPoint(2, math.pi/2) p3 = PolarPoint(3, math.pi) print(p1) print(p2) print(p3) ```

Key concepts

Polar Coordinates

Polar coordinates are a way to represent the position of a point in a two-dimensional plane using two values: the radius $r$, and the angle $\theta$. The radius is the distance from the origin (0,0) to the point, and the angle is measured from the positive x-axis toward the point. This is particularly useful for situations involving rotation or circular motion.
Key aspects of polar coordinates:

• The radius $r$ is always non-negative.
• The angle $\theta$ is typically in radians in the range $[0,2\pi ]$.
• They can be converted to Cartesian coordinates using the formulas: $x=r\cos \theta$ and $y=r\sin \theta$.

In object-oriented programming, you can represent a point using polar coordinates by creating a dedicated class to handle the conversion between polar and Cartesian systems efficiently.

Inheritance in Python

Inheritance is an essential concept in object-oriented programming that allows a class (known as a subclass or a derived class) to inherit attributes and methods from another class (known as a superclass or a base class). This enables code reusability and a clear hierarchical class structure. In Python, inheritance allows us to extend the functionality of existing code without modifying it.
Here’s how inheritance works:

• The subclass automatically gets all the methods and properties from the superclass.
• Additional properties and methods can be added to the subclass.
• Methods of the subclass can overlay methods of the superclass with the same name, known as method overriding.

In our extension of the `Point` class to `PolarPoint`, inheritance is used to take advantage of the `Point` class's Cartesian capabilities while adding specific polar coordinate features, demonstrating the combination and enhancement of functionalities across different contexts.

Class Constructors

A constructor in Python is a special method that is automatically called when an instance (object) of a class is created. The constructor sets up the initial state of the object using the provided parameters to assign values to instance variables. In Python, the constructor method is usually named `__init__`.
Main features of class constructors include:

• Allow defining the attributes of a new object.
• Can take any number of parameters but must have `self` as the first parameter, which refers to the instance being created.
• For derived classes, the constructor of the base class can be invoked using `super()` to ensure the base class is initialized correctly.

In the exercise, the `PolarPoint` class uses a constructor to initialize the values of $r$, $\theta$, and compute the Cartesian coordinates, which are then used by the `Point` superclass constructor to set further parameters.

Cartesian Coordinates Conversion

The conversion from polar to Cartesian coordinates is essential for seamlessly working with different coordinate systems, particularly in mathematical modeling, graphics, and simulations. Cartesian coordinates use horizontal and vertical axes, making it easier to understand and visualize motion and geometry compared to polar coordinates.
Key conversion formulas:

• To convert polar coordinates $(r,\theta )$ to Cartesian coordinates $(x,y)$:
• $x=r\cos \theta$
• $y=r\sin \theta$
• These transformations allow you to map any point from a circular plane to a rectangular plane, providing versatility in simulations and graphical representations.

This conversion is utilized in the `PolarPoint` class by calculating the values of $x$ and $y$ from $r$ and $\theta$, thus allowing the instantiation of a Point object that is inherently compatible with Cartesian representation and functionality.