Fractal Find : Explore Fractal and Quantum Variations

APPENDIX B
Julia Unit Circle Variation Bows

Julia Unit Circle Variation Bows use polar coordinates and trigonometric functions where the angle, θ, is doubled each iteration.
The three-dimensional effect was created with pixel offset for a number of arbitrary iterations.

Julia Unit Circle with Pseudocode

for (int i = 0; i ≤ 360000; i++) {
	θ = i*0.001;
	for (k = 1; k ≤ 16; k++) {
		θ = 2.0 * θ;
		x = cos(θ);
		y = sin(θ);
	}
	PlotPoint(x, y, color);
}

Modified Julia Unit Circle with Pseudocode

for (i = 0; i ≤ 180000; i++) {
	x = 0.0;
	y = 0.0;
	θ = i*0.001;
	for (k = 1; k ≤ 2; k++) {
		θ = 2.0 * θ;
		xnew = cos(θ) + x;
		ynew = sin(θ) + y;
		x = xnew;
		y = ynew;
		PlotPoint(x, y, color);
	}
}

Bow	Map	Build (f(θ,x,y), g(θ,x,y))	Escape k > value	Plot
Unit Circle	θ	(cos(θ), sin(θ))	k > 16	Point
Modified Unit Circle	θ	(cos(θ) + x, sin(θ) + y)	k > 2	Point
Bow #1	θ	(cos(θ) + 0.5*x, sin(θ) + y)	k > 5	Point
Bow #2	θ	(cos(θ) + sin(θ) + 0.5*x, sin(θ) + y)	k > 5	Point
Bow #3	θ	(cos(θ) + sin(θ), sin(θ) + y)	k > 5	Point
Bow #4	θ	(sin(θ) + x, cos(θ) + y)	k > 4	Point
Bow #5	θ	(sin(θ) + cos(x), cos(θ) + y)	k > 6	Point
Bow #6	θ	(sin(θ) + x*y, cos(θ) + y)	k > 4	Point
Bow #7	θ	(sin(θ) + xy, cos(θ) + 1.20x*y)	k > 5	Point
Bow #8	θ	(sin(θ) + x*y, cos(θ) + sin(θ) + y)	k > 4	Point
Bow #9	θ	(sin(θ) - x + y, cos(θ) + y)	k > 4	Point
Bow #10	θ	(sin(θ) - x, cos(θ) + y)	k > 5	Point
Bow #11	θ	(sin(θ) - y, cos(θ) - x)	k > 5	Point
Bow #12	θ	(cos(θ) + x - y*y, sin(θ) + y)	k > 5	Point
Bow #13	θ	(cos(θ + y) + x, sin(θ + x) - y)	k > 4	Point
Bow #14	θ	(sin(θ)*cos(θ) + x, sin(θ + x) - y)	k > 4	Point
Bow #15	θ	(sin(θ)*cos(θ) - x, sin(θ) + y)	k > 5	Point
Bow #16	θ	(sin(θ)*cos(θ) - x, cos(θ) + y)	k > 5	Point
Bow #17	θ	(sin(θ)*cos(θ) - x, sin(θ) + y - x)	k > 5	Point
Bow #18	θ	(cos(θ) + y, sin(θ) - x	k > 3	Point
Bow #19	θ	(cos(θ) + y, x*sin(θ) - x)	k > 5	Point
Bow #20	θ	(cos(θ) * x, sin(θ) + y)	k > 3	Point
Bow #21	θ	(y*cos(θ) + y, sin(θ) + x)	k > 4	Point
Bow #22	θ	(cos(θ), y*sin(θ) + x)	k > 4	Point
Bow #23	θ	(cos(θ) - y, y*sin(θ) + x)	k > 4	Point
Bow #24	θ	(cos(θ) - sin(θ)y, ysin(θ) + x)	k > 4	Point