Fractals

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Fractals
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Outline

• Prerequisites
• Recursion
• Simple pictures
• Objectives
• Drawing Fractal images
• Examples
• Mandelbrot set
• Koch fractal
• Approximating circles
• Reference
• HTDP Section 27.1

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Background

• Informally, fractals are endlessly repeating
  geometric figures.
• Formal Definition Fractal a geometrical figure
  consisting of an identical motif that repeats
  itself on an ever decreasing scale.
• Fractals come very close to capturing the
  richness and form found in nature.
• Fractals possess structural self-similarity on
  multiple scales, i.e. a piece of a fractal will
  often look like the whole

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Well known example Mandelbrot
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Goofy Question

• What is the distance around the shore of Lake
  Lanier?
• Normal Answer about 1,000 miles
• Mathematical Answer it depends on the thickness
  of the line you draw
• The 1,000 mile answer assumes a line thickness of
  about 10 ft.
• Who knows what the answer is with a 1/100 inch
  line.
• What about a line thickness in the molecular
  range?

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Koch Fractal

• Simple to draw
• Replace any line with 4 lines as follows
• Repeat with the new lines
• Stop when the line length is smaller than a
  specified limit.

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Koch Fractal Result
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Approximating Circles

• Drawing circles efficiently is not simple
• Relative pixel size changes the number of lines
  necessary to draw an effective circle
• Consider the difference between these circles
• Pixel relatively small - Pixel
  relatively large

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Calculations

• All circles drawn on a X-Y grid are
  approximations
• Each line drawn is an expensive computation
• We need an algorithm that draws the best circle
  for a given radius relative to the pixel size.

1. Start with a chord from one pixel outside the
circle
4. Continue until the mid-point distance is
closer than a pixel
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fractal
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Questions?
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Code Used 1

• (define WIDTH 400)
• (define HEIGHT 400)
• (define PI 3.14159265)
• (define center (make-posn (/ WIDTH 2) (/ HEIGHT
  2)))
• (define DEBUG false)
• circle posn number color -gt picture
• start with the center, radius and color and
  draw an
• approximation to a circle
• (define (circle center radius color)
• (let ((pa (offset center (make-posn (- (
  radius 1)) 0)))
• (pb (offset center (make-posn ( radius
  1) 0))))
• (and (chord pa pb center radius color)
• (chord pb pa center radius color))))

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Code Used 2

• chord posn posn posn color -gt picture
• given two lines and the circle center,
  approximate the chord
• (define (chord pa pb center radius color)
• (begin (if DEBUG
• (begin (display "from ")
• (show-posn pa)
• (display " to ")
• (show-posn pb)
• (display " center ")
• (show-posn center)))
• (if (close-enough? pa pb center radius)
• (draw-solid-line pa pb color)
• (let ((pc (new-location pa pb center
  radius)))
• (and (chord pa pc center radius
  color)
• (chord pc pb center radius
  color))))))

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Code Used 3

• new-location posn posn posn -gt posn
• compute the point on the bisector of the line
  AB that is
• (radius 1) from the center.
• (define (new-location a b center radius)
• (let ((alpha (get-alpha a b center))
• (beta (get-beta a c)))
• (get-point center ( radius 1) ( alpha
  beta))))
• get-alpha posn posn posn -gt number
• compute the angle ACM where M is the mid-point
  of AB
• (define (get-alpha a b c)
• (let ((m (mid-pt a b)))
• (atan (distance a m)
• (distance m c))))

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Code Used 4

• get-beta posn posn -gt number
• compute the angle ACX where X is the pt (Ax
  Cy)
• (define (get-beta a c)
• (atan (- (posn-y c) (posn-y a))
• (- (posn-x c) (posn-x a))))
• get-point posn number number -gt posn
• calculate the point located from c a distance
  r at angle a
• (define (get-point c r a)
• (make-posn (- (posn-x c)
• ( r (cos a)))
• (- (posn-y c)
• ( r (sin a)))))

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Code Used 5

• close-enough? posn posn posn number -gt boolean
• is the mid-point of the chord from pa to pb
  for the circle
• centered at c with radius r at or beyond
  (radius - 1) from
• the center?
• (define (close-enough? pa pb c r)
• (begin
• (if DEBUG
• (begin
• (display "Check ")
• (show-posn (mid-pt pa pb))
• (show-posn c)
• (display (distance (mid-pt pa pb) c))
• (newline)))
• (gt (distance (mid-pt pa pb) c)
• (- r 1))))

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Code Used 6

• mid-pt posn posn -gt posn
• (define (mid-pt a b)
• (make-posn (/ ( (posn-x a) (posn-x b)) 2)
• (/ ( (posn-y a) (posn-y b)) 2)))
• distance posn posn -gt number
• (define (distance a b)
• (sqrt ( (dsq posn-x a b)
• (dsq posn-y a b))))
• dsq (posn -gt number) posn posn -gt number
• (define (dsq f a b)
• (square (- (f a) (f b))))

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Code Used 7

• show-posn posn -gt display
• diagnostic display of a posn
• (define (show-posn p)
• (begin (display "")
• (display (posn-x p))
• (display "' ")
• (display (posn-y p))
• (display "")
• (newline)))
• offset posn posn -gt posn
• add the coordinates of the parameters to make
  a new position
• (define (offset a b)
• (make-posn ( (posn-x a)
• (posn-x b))
• ( (posn-y a)
• (posn-y b))))
• (start WIDTH HEIGHT)
• (circle center radius 'red)

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Final Thought

• Writing a novel is a lot like drawing a fractal
• Start with a plot outline
• Divide the plot into chapters
• Divide each chapter into sections
• Divide each section into paragraphs
• stop dividing when you have a single idea to
  express
• Flesh out that idea in words
• The fractal detail adds real-world character to
  the bare story line

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Summary

• You should now know
• Drawing Fractal images
• Examples
• Mandelbrot set
• Koch fractal
• Approximating circles

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