Ellipse Perimeter Equation

There exists no equation for the perimeter of an ellipse (excluding infinite series). This project uses gradient descent to find an approximate equation of the form area = pi * (wa + vb - sqrt(xa^2 + yab + zb^2)) where w,v,x,y,z are the weights generated in this project.

Example

A full example of running and finding an equation can be seen in main.cpp, although most of the necessary code is repeated in the instructions below.

Usage

Running the equation

To simply run the equation using the default equation, use the method ellipse_perimeter:

std::cout << "Perimeter of ellipse with radii 3 and 2 using default equation: " << ellipse_perimeter(3, 2);

Finding an equation

To run the gradient descent algorithm in order to try and find a better equation, use the gradient_descent method. You will need to pass in a std::vector<long double> object to write the results to.

For example,

//initialise all weights to a random integer between 1 and 100 inclusive
std::vector<long double> weights;
srand(0);
for (int i = 0; i < 5; i++) {
    long double r = rand() % 100 + 1;
    weights.push_back(r);
}

//run gradient descent
gradient_descent(&weights);

There are also a number of optional arguments to gradient_descent. In this order, they are:

• bool show_results - Whether to print the results of the algorithm, defaults to false
• int iterations - The number of iterations to run through, defaults to 1000
• double learning_rate - The learning rate to use (affects how far to adjust the weights each iteration), defaults to 0.01
• double learning_rate_modifier - How much to decrease learning_rate by after each iteration, e.g 0.99 decreases learning_rate by 1% each iteration. Can help to avoid local minima, defaults to 1
• double min_learning_rate - The minimum value for learning_rate, used to stop learning_rate_modifier from decreasing learning_rate too far, defaults to 0

For example, the parameters used to generate the default are as follows:

gradient_descent(&weights, true, 1000000, 0.023, 0.9995, 0.001);

Running the equation with custom weights

Once you have generated new weights with gradient_descent, you can run the equation using them as follows:

std::cout << "Perimeter of ellipse with radii 3 and 2 using default equation: " << ellipse_perimeter(3, 2, &weights);

Notes

• More accurate approximations exist than the one found in this project, and can be found here.
• ellipse_lengths.csv is a slightly modified version of an list of true perimeter lengths by radii ratios created by Matt Parker. You can find the original version here.