Lecture03 p1

CS 543: Computer Graphics
Lecture 3 (Part I): Fractals
Emmanuel Agu

What are Fractals?
n Mathematical expressions
n Approach infinity in organized way
n Utilizes recursion on computers
n Popularized by Benoit Mandelbrot (Yale university)
n Dimensional:
n Line is one-dimensional
n Plane is two-dimensional
n Defined in terms of self-similarity

Fractals: Self-similarity
n Level of detail remains the same as we zoom in
n Example: surface roughness or profile same as we zoom in
n Types:
n Exactly self-similar
n Statistically self-similar

Examples of Fractals
n Clouds
n Grass
n Fire
n Modeling mountains (terrain)
n Coastline
n Branches of a tree
n Surface of a sponge
n Cracks in the pavement
n Designing antennae (www.fractenna.com)

Example: Fractal Terrain
Courtesy: Mountain 3D
Fractal Terrain software

Example: Fractal Art
Courtesy: Internet
Fractal Art Contest

Application: Fractal Art
Courtesy: Internet
Fractal Art Contest

Koch Curves
n Discovered in 1904 by Helge von Koch
n Start with straight line of length 1
n Recursively:
n Divide line into 3 equal parts
n Replace middle section with triangular bump with sides of
length 1/3
n New length = 4/3

Koch Snowflakes
n Can form Koch snowflake by joining three Koch curves
n Perimeter of snowflake grows as:
where Pi is the perimeter of the ith snowflake iteration
n However, area grows slowly and S∞ = 8/5!!
n Self-similar:
n zoom in on any portion
n If n is large enough, shape still same
n On computer, smallest line segment > pixel spacing
( )i
iP
3
43=

Koch Snowflakes
Pseudocode, to draw Kn:
If (n equals 0) draw straight line
Else{
Draw Kn-1
Turn left 60°
Draw Kn-1
Turn right 120°
Draw Kn-1
Turn left 60°
Draw Kn-1
}

Iterated Function Systems (IFS)
n Recursively call a function
n Does result converge to an image? What image?
n IFS’s converge to an image
n Examples:
n The Fern
n The Mandelbrot set

Mandelbrot Set
n Based on iteration theory
n Function of interest:
n Sequence of values (or orbit):
cszf += 2
)()(
ccccsd
cccsd
ccsd
csd
++++=
+++=
++=
+=
2222
4
222
3
22
2
2
1
))))((((
)))(((
))((
)(

Mandelbrot Set
n Orbit depends on s and c
n Basic question,:
n For given s and c,
• does function stay finite? (within Mandelbrot set)
• explode to infinity? (outside Mandelbrot set)
n Definition: if |d| < 1, orbit is finite else inifinite
n Examples orbits:
n s = 0, c = -1, orbit = 0,-1,0,-1,0,-1,0,-1,…..finite
n s = 0, c = 1, orbit = 0,1,2,5,26,677…… explodes

Mandelbrot Set
n Mandelbrot set: use complex numbers for c and s
n Always set s = 0
n Choose c as a complex number
n For example:
• s = 0, c = 0.2 + 0.5i
n Hence, orbit:
• 0, c, c2, c2+ c, (c2+ c)2 + c, ………
n Definition: Mandelbrot set includes all finite orbit c

Mandelbrot Set
n Some complex number math:
n For example:
n Modulus of a complex number, z = ai + b:
n Squaring a complex number:
1* −=ii
63*2 −=ii
22
baz +=
ixyyxyix )2()( 22
+−=+
Im
Re
Argand
diagram

Mandelbrot Set
n Calculate first 4 terms
n with s=2, c=-1
n with s = 0, c = -2+i

Mandelbrot Set
n Calculate first 3 terms
n with s=2, c=-1, terms are
n with s = 0, c = -2+i
575124
2415
512
2
2
2
=−
=−
=−
( ) iii
iii
ii
510)2(31
31)2()2(
2)2(0
2
2
−−=+−+−
−=+−++−
+−=+−+

Mandelbrot Set
n Fixed points: Some complex numbers converge to certain
values after x iterations.
n Example:
n s = 0, c = -0.2 + 0.5i converges to –0.249227 + 0.333677i
after 80 iterations
n Experiment: square –0.249227 + 0.333677i and add
-0.2 + 0.5i
n Mandelbrot set depends on the fact the convergence of
certain complex numbers

Mandelbrot Set
n Routine to draw Mandelbrot set:
n Cannot iterate forever: our program will hang!
n Instead iterate 100 times
n Math theorem:
n if number hasn’t exceeded 2 after 100 iterations, never will!
n Routine returns:
n Number of times iterated before modulus exceeds 2, or
n 100, if modulus doesn’t exceed 2 after 100 iterations
n See dwell( ) function in Hill (figure A4.5, pg. 755)

Mandelbrot dwell( ) function (pg. 755)
int dwell(double cx, double cy)
{ // return true dwell or Num, whichever is smaller
#define Num 100 // increase this for better pics
double tmp, dx = cx, dy = cy, fsq = cx*cx + cy*cy;
for(int count = 0;count <= Num && fsq <= 4; count++)
{
tmp = dx; // save old real part
dx = dx*dx – dy*dy + cx; // new real part
dy = 2.0 * tmp * dy + cy; // new imag. Part
fsq = dx*dx + dy*dy;
}
return count; // number of iterations used
}

Mandelbrot Set
n Map real part to x-axis
n Map imaginary part to y-axis
n Set world window to range of complex numbers to
investigate. E.g:
n X in range [-2.25: 0.75]
n Y in range [-1.5: 1.5]
n Choose your viewport. E.g:
n Viewport = [V.L, V.R, V.B, V.T]= [60,380,80,240]
n Do window-to-viewport mapping

Mandelbrot Set
n So, for each pixel:
n Compute corresponding point in world
n Call your dwell( ) function
n Assign color <Red,Green,Blue> based on dwell( ) return
value
n Choice of color determines how pretty
n Color assignment:
n Basic: In set (i.e. dwell( ) = 100), color = black, else color =
white
n Discrete: Ranges of return values map to same color
• E.g 0 – 20 iterations = color 1
• 20 – 40 iterations = color 2, etc.
n Continuous: Use a function

Mandelbrot Set
Use continuous function

FREE SOFTWARE
n Free fractal generating software
n Fractint
n FracZoom
n Astro Fractals
n Fractal Studio
n 3DFract

Lecture03 p1

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Lecture03 p1