FRACTAL FIND
APPENDIX I
Polar Variations
Polar Variations use Unit Circle variations with polar coodinates and trigonometric functions.
See Appendix B for a different aspect of this configuration type and further discussion.
for (int i = 0; i ≤ 100000; i++)
{
x = 0.0;
y = 0.0;
for (k = 3; k ≤ 6; k++)
{
θ = (2.0k * (i / 1000.0));
xnew = y + 2.0 * cos(θ);
ynew = x + 2.0 * sin(θ);
x = xnew;
y = ynew;
}
PlotPoint(x * scale, y * scale, color);
}
| Polar | Build: (f(θ,x,y), g(θ,x,y)) | kmin, kmax |
|---|---|---|
| Example | (y + 2.0 * cos(θ), x + 2.0 * sin(θ)) | 3, 6 |
| Polar #1 | (cos(θ + k) + x, sin(θ + k) + y) | 9, 13 |
| Polar #2 | (cos(θ) + x * 0.5, sin(θ) - y * 0.5) | 6, 9 |
| Polar #3 | (cos(θ) + y * 0.75, sin(θ) - x * 0.5) | 6, 9 |
| Polar #4 | (cos(θ + k) + x * y, sin(θ + k) + x * y) | 9, 13 |
| Polar #5 | (y + cos(y + θ), x + sin(x + θ)) | 3, 6 |
| Polar #6 | (y + cos(θ) - 0.25 * x, x + sin(θ) + 0.25 * y) | 3, 7 |
| Polar #7 | (cos(θ * k) + x * 0.5, sin(θ * k) - y * 0.5) | 9, 10 |
| Polar #8 | (cos(θ * k) + x * 0.5, sin(θ * k) - y * 0.5) | 2, 5 |
| Polar #9 | (cos(θ * k) - x * 0.5, sin(θ * k) - x*y * 0.5) | 2, 4 |
| Polar #10 | (y * cos(θ * k) - x * 0.5, sin(θ * k) - y * 0.5) | 2, 4 |
| Polar #11 | (cos(θ * k) - x * 0.5, (cos(θ * k) - x * 0.5) * sin(θ * k) - y * 0.5) | 2, 5 |
| Polar #12 | (x * y + cos(θ * k) - x * 0.5, sin(θ * k) - y * 0.5) | 2, 4 |
