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### In this assignment, you will write a program that plot ASCII text approximations of the Mandelbrot set.

INSTRUCTIONS TO CANDIDATES

Assignment 2

In this assignment, you will write a program that plot ASCII text approximations of the Mandelbrot set.

Problem 1

Consider the function𝑃(𝑥,𝑦)defined as follows

𝑃(𝑥,𝑦)(𝑢, 𝑣) = (𝑢2− 𝑣2+ 𝑥, 2𝑢𝑣 + 𝑦)

We define the orbit𝑂(𝑥, 𝑦)of a point(𝑥, 𝑦)to be an infinite list of items

𝑂(𝑥, 𝑦) = {(0, 0), 𝑃(𝑥,𝑦)(0, 0), 𝑃(𝑥,𝑦)(𝑃(𝑥,𝑦)(0, 0)), 𝑃(𝑥,𝑦)(𝑃(𝑥,𝑦)(𝑃(𝑥,𝑦)(0, 0))), ...}

In other words, the nth entry of the list𝑂(𝑥, 𝑦)is the𝑃(𝑥,𝑦)function composedwith itself n times and then applied to(0, 0)

Define a F# functionorbitthat takes a single point(𝑥, 𝑦)and index𝑖asan arguments and returns an𝑖𝑡ℎelement of the infinite list corresponding to𝑂(𝑥, 𝑦).

You may want to define a helper function corresponding to𝑃(𝑥,𝑦).

## Problem 2

Define a recursive function disp that takes two arguments: - a number d, and - a list of tuples pairs.

Every tuple in this input list consists of a number followed by a character, and you can assume the input list is always in ascending order. For example, a possible input list might be:

[(0.15, '#'), (0.5, 'x'), (1.0, '.')]

The function disp should return the character from the list that corresponds to the smallest number on the list that is greater than the input d, and if d is larger than all the number in the list, disp should return a space character, ’ ’. For example,

disp 0.01 [(0.15, '#'), (0.5, 'x'), (1.0, '.')] = '#'

disp 0.4 [(0.15, '#'), (0.5, 'x'), (1.0, '.')]     = 'x'

disp 100 [(0.15, '#'), (0.5, 'x'), (1.0, '.')]     = ' '

## Problem 3

One way to approximate the Mandelbrot set is to consider a certain element within the orbit of every point on the plane (such as the 12th element) and to check whether that element is within a a certain distance from the origin; if it is within this distance, then a non-blank character should be printed, else it should be left blank. You should use the following function to calculate distances of points from the origin:

let norm (x,y) = x*x + y*y

These distance values can then be used with disp and orbit to turn points on the plane into appropriate ASCII characters within an ASCII plot of the Mandelbrot set.

Define a function mandelbrot that takes three arguments: - the resolution of the approximation, r (used with the plane function), - the index of the elements, i, to check in the orbit lists of the points, and - the formatting list, l (to be used with the disp function).

This function should return a list of characters that corresponds to a picture approximating the shape of the Mandelbrot set on the plane.

You will need to combine the split, plane, disp, and orbit functions appro- priately; list comprehensions are allowed.

Once you’ve defined the function mandelbrot, you can generate an ASCII ver- sion of an approximation of the Mandelbrot set by evaluating the expression:

mandelbrot1712disp_symbols |> printfn"%s";;

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