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 a r c t a n ( 𝑥 / 𝑦 ) or a r c t a n ( 𝑠 / 𝑡 ) , the reciprocal of their normal usage for compatibility with Draves’ large existing library of nonlinear fractals.
(a)
FunctionForward 𝑉 ( 𝑥 , 𝑦 ) , with 𝑟 = √ 𝑥 2 + 𝑦 2 Inverse 𝑉 − 1 ( 𝑠 , 𝑡 ) , with √ 𝑟 = 𝑠 2 + 𝑡 2 Jacobian
1. Sinusoidal ⎡ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎦ s i n 𝑥 s i n 𝑦 ⎡ ⎢ ⎢ ⎣ s i n − 1 𝑠 s i n − 1 𝑡 ⎤ ⎥ ⎥ ⎦ plus multiples of 𝜋 c o s 𝑥 c o s 𝑦
2. Spherical ⎡ ⎢ ⎢ ⎣ 𝑥 𝑦 ⎤ ⎥ ⎥ ⎦ / 𝑟 2 ⎡ ⎢ ⎢ ⎣ 𝑠 𝑡 ⎤ ⎥ ⎥ ⎦ / 𝑟 2 1 / 𝑟 4
3. Swirl ⎡ ⎢ ⎢ ⎣ 𝑥 s i n 𝑟 2 − 𝑦 c o s 𝑟 2 𝑥 c o s 𝑟 2 + 𝑦 s i n 𝑟 2 ⎤ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎣ 𝑠 s i n 𝑟 2 + 𝑡 c o s 𝑟 2 − 𝑠 c o s 𝑟 2 + 𝑡 s i n 𝑟 2 ⎤ ⎥ ⎥ ⎦ 1
4. Horseshoe ⎡ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎦ ( 𝑥 − 𝑦 ) ( 𝑥 + 𝑦 ) 2 𝑥 𝑦 / 𝑟 √ 𝑦 = ± ( 𝑟 2 − 𝑠 𝑟 ) / 2 ; 𝑥 = 𝑦 ( 𝑠 + 𝑟 ) / 𝑡 2
5. Polar ⎡ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎦ 𝜃 / 𝜋 𝑟 − 1 ⎡ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎦ ( 𝑡 + 1 ) s i n ( 𝜋 𝑠 ) c o s ( 𝜋 𝑠 ) 1 / ( 𝜋 𝑟 )
7. Heart 𝑟 ⎡ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎦ s i n ( 𝜃 𝑟 ) − c o s ( 𝜃 𝑟 ) 𝑟 ⎡ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎦ s i n ( ( a t a n 2 ( 𝑠 , − 𝑡 ) + 2 𝜋 𝑘 ) / 𝑟 ) c o s ( ( a t a n 2 ( 𝑠 , − 𝑡 ) + 2 𝜋 𝑘 ) / 𝑟 ) 𝑟
8. Disc ⎡ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎦ 𝜃 / 𝜋 s i n 𝜋 𝑟 c o s 𝜋 𝑟 ⎡ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎦ ( a t a n 2 ( 𝑠 , 𝑡 ) / 𝜋 + 2 𝑘 ) s i n 𝜋 𝑟 c o s 𝜋 𝑟 𝜃 / ( 𝜋 𝑟 )
10. Hyperbolic ⎡ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎣ ( s i n 𝜃 ) / 𝑟 𝑟 c o s 𝜃 𝑥 / 𝑟 2 𝑦 ⎤ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎣ √ ( 1 ± 1 − 4 𝑠 2 𝑡 2 𝑡 ⎤ ⎥ ⎥ ⎦ ) / 2 𝑠 c o s ( 2 𝜃 ) / 𝑟 2
13. Julia ± √ 𝑟 ⎡ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎦ c o s ( 𝜃 / 2 ) s i n ( 𝜃 / 2 ) ⎡ ⎢ ⎢ ⎣ √ = ± √ ( 𝑟 + 𝑦 ) / 2 𝑥 / | 𝑥 | ⎤ ⎥ ⎥ ⎦ ( 𝑟 − 𝑦 ) / 2 𝑟 2 ⎡ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎦ s i n ( 2 𝜃 ) c o s ( 2 𝜃 ) = ⎡ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎦ 2 𝑠 𝑡 ( 𝑠 − 𝑡 ) ( 𝑠 + 𝑡 ) 1 / ( 4 𝑟 )
14. Bent ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ [ 𝑥 𝑦 ] 𝑇 𝑖 𝑓 𝑥 ≥ 0 , 𝑦 ≥ 0 [ 2 𝑥 𝑦 ] 𝑇 𝑖 𝑓 𝑥 < 0 , 𝑦 ≥ 0 [ 𝑥 𝑦 / 2 ] 𝑇 𝑖 𝑓 𝑥 ≥ 0 , 𝑦 < 0 [ 2 𝑥 𝑦 / 2 ] 𝑇 𝑖 𝑓 𝑥 < 0 , 𝑦 < 0 ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ [ 𝑠 𝑡 ] 𝑇 𝑖 𝑓 𝑠 ≥ 0 , 𝑡 ≥ 0 [ 𝑠 / 2 𝑡 ] 𝑇 𝑖 𝑓 𝑠 < 0 , 𝑡 ≥ 0 [ 𝑠 2 𝑡 ] 𝑇 𝑖 𝑓 𝑠 ≥ 0 , 𝑡 < 0 [ 𝑠 / 2 2 𝑡 ] 𝑇 𝑖 𝑓 𝑠 < 0 , 𝑡 < 0 ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 2 1 1 / 2
16. Fisheye ⎡ ⎢ ⎢ ⎣ 𝑦 𝑥 ⎤ ⎥ ⎥ ⎦ 2 / ( 𝑟 + 1 ) ⎡ ⎢ ⎢ ⎣ 𝑡 𝑠 ⎤ ⎥ ⎥ ⎦ 1 / ( 2 − 𝑟 ) 4 / ( 1 + 𝑟 ) 3
18. Exponential 𝑒 𝑥 − 1 ⎡ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎦ c o s 𝜋 𝑦 s i n 𝜋 𝑦 ⎡ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎦ l o g ( 𝑟 ) + 1 . 0 a t a n 2 ( 𝑡 , 𝑠 ) / 𝜋 𝜋 𝑒 2 𝑥 − 2
19. Power 𝑟 𝑥 / 𝑟 − 1 ⎡ ⎢ ⎢ ⎣ 𝑦 𝑥 ⎤ ⎥ ⎥ ⎦ 𝑟 𝑟 / 𝑠 − 1 ⎡ ⎢ ⎢ ⎣ 𝑠 𝑡 ⎤ ⎥ ⎥ ⎦ 𝑟 2 𝑥 / 𝑟 − 2 𝑥 / 𝑟
(b)
FunctionForward 𝑉 ( 𝑥 , 𝑦 ) InverseJacobian determinant
6.  Handkerchief 𝑟 ⎡ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎦ s i n ( 𝜃 + 𝑟 ) c o s ( 𝜃 − 𝑟 ) Nonelementary c o s 2 𝑟 + ( 2 𝑥 𝑦 / 𝑟 ) − 𝑟 s i n 2 𝑟
9. Spiral ⎡ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎦ c o s 𝜃 + s i n 𝑟 c o s 𝜃 − s i n 𝑟 / 𝑟 Nonelementary ( 1 − 𝑟 c o s ( 𝑟 − 𝜃 ) + s i n ( 𝑟 − 𝜃 ) ) / 𝑟 2
11. Diamond ⎡ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎦ s i n 𝜃 c o s 𝑟 c o s 𝜃 s i n 𝑟 32 root families ( c o s ( 2 𝑟 ) + 2 𝑦 2 / 𝑟 2 − 1 ) / 2 𝑟
12. Ex 𝑟 ⎡ ⎢ ⎢ ⎣ s i n 3 ( 𝜃 + 𝑟 ) + c o s 3 ( 𝜃 − 𝑟 ) 3 s i n 3 ( 𝜃 + 𝑟 ) − c o s 3 ( 𝜃 − 𝑟 ) 3 ⎤ ⎥ ⎥ ⎦ Nonelementary ( 6 𝑥 𝑦 + 𝑟 c o s 2 𝑟 − 3 𝑟 2 s i n 2 𝑟 ) ∗ ( − 3 / ( 2 𝑟 ) ) ∗ ( s i n 2 𝑟 + 𝑥 𝑦 / 𝑟 2 ) 2
15. Waves ⎡ ⎢ ⎢ ⎣ 𝑥 + 𝑎 1 s i n ( 𝑎 2 𝑦 ) 𝑦 + 𝑎 3 s i n ( 𝑎 4 ⎤ ⎥ ⎥ ⎦ 𝑥 ) Nonelementary 1 − 𝑎 1 𝑎 2 𝑎 3 𝑎 4 c o s ( 𝑎 4 𝑥 ) c o s ( 𝑎 2 𝑦 )
17. Popcorn ⎡ ⎢ ⎢ ⎣ 𝑥 + 𝑎 1 s i n ( t a n 3 𝑦 ) 𝑦 + 𝑎 2 ⎤ ⎥ ⎥ ⎦ s i n ( t a n 3 𝑥 ) Nonelementary 1 − 9 𝑎 1 𝑎 2 c o s ( t a n 3 𝑥 ) c o s ( t a n 3 𝑦 ) + s e c 2 ( 3 𝑥 ) s e c 2 ( 3 𝑦 )
20. Cosine ⎡ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎦ c o s ( 𝜋 𝑥 ) c o s h 𝑦 − s i n ( 𝜋 𝑥 ) s i n h ( 𝑦 ) 16 root families 𝜋 / 2 ( − c o s 2 𝜋 𝑥 + c o s h 2 𝑦 )