Question

Assignment 3

In this assignment, you will write a program that plot ASCII text approximations of the Mandelbrot set. Problems Problem 1 Consider the function ?(?,?) defined as follows: ?(?,?)(?, ?) = (?2 − ?2 + ?, 2?? + ?) We define the orbit ?(?, ?) of a point (?, ?) to be an infinite list of items: ?(?, ?) = {(0, 0), ?(?,?)(0, 0), ?(?,?)(?(?,?)(0, 0)), ?(?,?)(?(?,?)(?(?,?)(0, 0))), …} In other words, the nth entry of the list ?(?, ?) is the ?(?,?) function composed with itself n times and then applied to (0, 0). Define a F# function orbit that takes a single point (?, ?) and index ? as an arguments and returns an ? ?ℎ element of the infinite list corresponding to ?(?, ?). • You may want to define a helper function corresponding to ?(?,?).

2)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, 1

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 appropriately; list comprehensions are allowed. Once you’ve defined the function mandelbrot, you can generate an ASCII version of an approximation of the Mandelbrot set by evaluating the expression: mandelbrot 17 12 disp_symbols |> printfn "%s";;

Fill the missing parts of the functions (see below)

Driver Source Core let disp_symbols = [(0.15, '#'); (0.5, 'x'); (1.0, '.')];; let str chs = List.fold (fun str x -> str + x.ToString()) "" chs;; let plane r = [ for y in -1.0 .. 1.0/r .. 1.0 do for x in -2.0 .. 1.0/r .. 1.0 do yield (x, y) ];; let norm (x,y) :float = x*x + y*y;; let split n ch xs = ... let rec orbit (x,y) i = ... let disp xs (d:float) = ... let mandelbrot r i = ... // Example of the function call mandelbrot 17 12 |> printfn "%s";;