Ch 9 Infinity page 1CSC 367 Fractals (9.2) Self similar curves appear identical at every level of detail often created by recursively drawing lines

Ch 9 Infinity page 2CSC 367 Koch Curve infinitely long line within a region of finite area mathematicians love this stuff!

Ch 9 Infinity page 3CSC 367 Polygons regular polygon has equal interior angles how do you generate the points? x = radius * cos angle y = radius * sin angle cos 0 = 1 sin 0 = 0 DrawNgon(center, radius, numSides) slice = 2π / Numsides for I = 1 to numSides angle = (I-1) * slice x2 = radius * cos angle + center.x y2 = radius * sin angle + center.y drawline (x1,y1) to (x2,y2) x1 = x2; y1 = y2;

Ch 9 Infinity page 4CSC 367 Random Fractals (9.8) Given a line segment AB Generate a random point on the perpendicular bisector Distance from the segment is a gaussian value with a mean of zero The range is proportional to the segment length

Ch 9 Infinity page 5CSC 367 Recursively Defined Trees Recursive definition a tree is a branch with several trees attached to the end of the branch Parameters include number of branches, branch length, depth of recursion, fan angle, starting point These parameters can be random

Ch 9 Infinity page 6CSC 367 Iterated Function Systems (9.6) Complex Numbers z = x + iy real part and imaginary part iteratively reapply functions until the values converge F k+1 = F k 2 This function converges with values less than 1 Goes to infinity for numbers greater than 1 Stays at 1 for values that start at 1 Points are in the set if they do not approach infinity Stop iterating after max times if |v| has not exceeded 2

Ch 9 Infinity page 7CSC 367 Mandelbrot Set Interesting Facts has a fractal perimeter the set is connected not matter how distinct various islands appear Perhaps the most complicated object seen in mathematics Area centered around c = i0.0 is most interesting the set of all complex numbers that produce a finite orbit of oi there is only one Mandelbrot set Calculating the set A set of points in the complex plane F k+1 = F k 2 + C Where C is a complex number for the current pixel C = x + iy (for every pixel to be generated) F 0 = 0 + i0

Ch 9 Infinity page 8CSC 367 The algorithm for each pixel i, j x = mapped from i to window coordinate y = mapped from j to window coordinate steps = iterateToInfinity(x, y); if (steps == max) point is in the set color is black else point is not in the set different colors are possible draw pixel with defined color

Ch 9 Infinity page 9CSC 367 Julia Set (9.7) Quite similar to Mandelbrot F k+1 = F k 2 + C Where C is a fixed complex number F 0 = x + iy (for every pixel to be generated) The set of points generated by a given C is called the Julia Set Different values of C produce different pictures There are an infinite number of Julia sets