GPU-Accelerated Rendering of Unbounded Nonlinear Iterated Function System Fixed Points

Table 2

Functions and inverses are discussed in Section 5.2 and used for per-vertex and per-pixel rendering, respectively. The Jacobian determinants are used in Section 6 for attractor density; only the forward direction is shown for the Jacobians. 𝜃 stands for arctan(𝑥/𝑦) or arctan(𝑠/𝑡), the reciprocal of their normal usage for compatibility with Draves’ large existing library of nonlinear fractals.

(a)

Function	Forward 𝑉(𝑥,𝑦), with 𝑟=√𝑥^2+𝑦^2	Inverse 𝑉^−1(𝑠,𝑡), with 𝑟=√𝑠^2+𝑡^2	Jacobian
1. Sinusoidal	⎡⎢⎢⎣ssiinn𝑥𝑦⎤⎥⎥⎦	⎡⎢⎢⎣ssiinn^−1−1𝑠𝑡⎤⎥⎥⎦plus multiples of 𝜋	cos𝑥cos𝑦
2. Spherical	⎡⎢⎢⎣𝑥𝑦⎤⎥⎥⎦/𝑟2	⎡⎢⎢⎣𝑠𝑡⎤⎥⎥⎦/𝑟2	1/𝑟^4
3. Swirl	⎡⎢⎢⎣𝑥sin𝑟^2−𝑦cos𝑟^2𝑥cos𝑟^2+𝑦sin𝑟^2⎤⎥⎥⎦	⎡⎢⎢⎣𝑠sin𝑟^2+𝑡cos𝑟^2−𝑠cos𝑟^2+𝑡sin𝑟^2⎤⎥⎥⎦	1
4. Horseshoe	⎡⎢⎢⎣(𝑥−2𝑦𝑥)𝑦(𝑥+𝑦)⎤⎥⎥⎦/𝑟	𝑦=±√(𝑟^2−𝑠𝑟)/2; 𝑥=𝑦(𝑠+𝑟)/𝑡	2
5. Polar	⎡⎢⎢⎣𝑟𝜃−/𝜋1⎤⎥⎥⎦	(𝑡+1)⎡⎢⎢⎣scions((𝜋𝜋𝑠𝑠))⎤⎥⎥⎦	1/(𝜋𝑟)
7. Heart	𝑟⎡⎢⎢⎣−scions((𝜃𝜃𝑟𝑟))⎤⎥⎥⎦	𝑟⎡⎢⎢⎣scions((((aattaann22((𝑠𝑠,,−−𝑡𝑡))++22𝜋𝜋𝑘𝑘))//𝑟𝑟))⎤⎥⎥⎦	𝑟
8. Disc	𝜃/𝜋⎡⎢⎢⎣scions𝜋𝜋𝑟𝑟⎤⎥⎥⎦	(atan2(𝑠,𝑡)/𝜋+2𝑘)⎡⎢⎢⎣scions𝜋𝜋𝑟𝑟⎤⎥⎥⎦	𝜃/(𝜋𝑟)
10. Hyperbolic	⎡⎢⎢⎣(𝑟sicno𝜃s)𝜃/𝑟⎤⎥⎥⎦=⎡⎢⎢⎣𝑥/𝑟^2𝑦⎤⎥⎥⎦	⎡⎢⎢⎣(𝑡1±√1−4𝑠^2𝑡^2)/2𝑠⎤⎥⎥⎦	cos(2𝜃)/𝑟^2
13. Julia	±√𝑟⎡⎢⎢⎣csions((𝜃𝜃//22))⎤⎥⎥⎦ =±⎡⎢⎢⎣√𝑥/|(𝑥𝑟|√+𝑦)/2(𝑟−𝑦)/2⎤⎥⎥⎦	𝑟^2⎡⎢⎢⎣scions((22𝜃𝜃))⎤⎥⎥⎦ =⎡⎢⎢⎣2(𝑠𝑠𝑡−𝑡)(𝑠+𝑡)⎤⎥⎥⎦	1/(4𝑟)
14. Bent	⎧⎪⎪⎪⎨⎪⎪⎪⎩[[𝑥2𝑥𝑦]𝑦^𝑇]𝑇𝑖𝑓𝑥≥0,𝑦≥0[𝑥𝑦/2]𝑇𝑖𝑓𝑥<0,𝑦≥0[2𝑥𝑦/2]𝑇𝑖𝑓𝑥≥0,𝑦<0𝑖𝑓𝑥<0,𝑦<0	⎧⎪⎪⎪⎨⎪⎪⎪⎩[[𝑠𝑠/2𝑡]^𝑇𝑡]𝑇𝑖𝑓𝑠≥0,𝑡≥0[𝑠2𝑡]𝑇𝑖𝑓𝑠<0,𝑡≥0[𝑠/22𝑡]𝑇𝑖𝑓𝑠≥0,𝑡<0𝑖𝑓𝑠<0,𝑡<0	⎧⎪⎪⎪⎨⎪⎪⎪⎩1211/2
16. Fisheye	2/(𝑟+1)⎡⎢⎢⎣𝑦𝑥⎤⎥⎥⎦	1/(2−𝑟)⎡⎢⎢⎣𝑡𝑠⎤⎥⎥⎦	4/(1+𝑟)^3
18. Exponential	𝑒^𝑥−1⎡⎢⎢⎣csions𝜋𝜋𝑦𝑦⎤⎥⎥⎦	⎡⎢⎢⎣laotagn(2𝑟()𝑡+,𝑠1)/.𝜋0⎤⎥⎥⎦	𝜋𝑒^2𝑥−2
19. Power	𝑟^𝑥/𝑟−1⎡⎢⎢⎣𝑦𝑥⎤⎥⎥⎦	𝑟^𝑟/𝑠−1⎡⎢⎢⎣𝑠𝑡⎤⎥⎥⎦	𝑟^2𝑥/𝑟−2𝑥/𝑟

(b)

Function	Forward 𝑉(𝑥,𝑦)	Inverse	Jacobian determinant
6.  Handkerchief	𝑟⎡⎢⎢⎣scions((𝜃𝜃+−𝑟𝑟))⎤⎥⎥⎦	Nonelementary	cos2𝑟+(2𝑥𝑦/𝑟)−𝑟sin2𝑟
9. Spiral	⎡⎢⎢⎣ccooss𝜃𝜃+−ssiinn𝑟𝑟⎤⎥⎥⎦/𝑟	Nonelementary	(1−𝑟cos(𝑟−𝜃)+sin(𝑟−𝜃))/𝑟^2
11. Diamond	⎡⎢⎢⎣scions𝜃𝜃csoins𝑟𝑟⎤⎥⎥⎦	32 root families	(cos(2𝑟)+2𝑦^2/𝑟^2−1)/2𝑟
12. Ex	𝑟⎡⎢⎢⎣sin^3(𝜃+𝑟)+cos^3(𝜃−𝑟)^3sin^3(𝜃+𝑟)−cos^3(𝜃−𝑟)^3⎤⎥⎥⎦	Nonelementary	(6𝑥𝑦+𝑟cos2𝑟−3𝑟^2sin2𝑟)∗(−3/(2𝑟))∗(sin2𝑟+𝑥𝑦/𝑟^2)^2
15. Waves	⎡⎢⎢⎣𝑦𝑥++𝑎𝑎13sin(𝑎2𝑦)sin(𝑎4𝑥)⎤⎥⎥⎦	Nonelementary	1−𝑎1𝑎2𝑎3𝑎4cos(𝑎4𝑥)cos(𝑎2𝑦)
17. Popcorn	⎡⎢⎢⎣𝑥𝑦++𝑎𝑎12sin(tan3𝑦)sin(tan3𝑥)⎤⎥⎥⎦	Nonelementary	1−9𝑎1𝑎2cos(tan3𝑥)cos(tan3𝑦)+sec^2(3𝑥)sec^2(3𝑦)
20. Cosine	⎡⎢⎢⎣−csions((𝜋𝜋𝑥𝑥))scionshh(𝑦𝑦)⎤⎥⎥⎦	16 root families	𝜋/2(−cos2𝜋𝑥+cosh2𝑦)