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Assignment 2

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

Problem 1

Consider the functionπ(π₯,π¦)defined as follows

π(π₯,π¦)(π’, π£) = (π’2− π£2+ π₯, 2π’π£ + π¦)

We define the orbitπ(π₯, π¦)of a point(π₯, π¦)to be an infinite list of items

π(π₯, π¦) = {(0, 0), π(π₯,π¦)(0, 0), π(π₯,π¦)(π(π₯,π¦)(0, 0)), π(π₯,π¦)(π(π₯,π¦)(π(π₯,π¦)(0, 0))), ...}

In other words, the nth entry of the listπ(π₯, π¦)is theπ(π₯,π¦)function composedwith itself n times and then applied to(0, 0)

Define a F# functionorbitthat takes a single point(π₯, π¦)and indexπasan arguments and returns anππ‘βelement of the infinite list corresponding toπ(π₯, π¦).

You may want to define a helper function corresponding toπ(π₯,π¦).

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, '.')] = ' '

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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