The generalized formula for the monkey saddle surfaces is given here:
https://www.researchgate.net/publication/256808897_Monkey_Starfish_and_Octopus_Saddles
The binomial coefficients (N K) (vertical), for each order N saddle, are given in a row in pascal's triangle, using alternate entries.
For example, the Monkey saddle, with 3 dips, has coefficients 1 and 3.
The Octopus saddle, with 8 arm dips, has coefficients 1, 28, 70, 28, and 1.
Or the binomial coefficients can be easily calculated with factorials.
The formula for each order N, uses the Even values of K = 0, 2, ... (N for even N, N-1 for odd N).
The formula is the sum from K=0 to K=N of the binomial coefficient (N K) * x to the power(N-K) * y to the power(K),
* even powers of i, for a +/- factor.
Note that the .nod files use ConvertPts2 node, which can be downloaded elsewhere on the forum.
Network can be done on the curves, and the surfaces may be trimmed at a +/- z value.
The smelt petal is a trimmed Monkey saddle, with the Monkey saddle z-axis passing through the origin, and (1,1,1)
The z = f(u,v) in MathPoints may be easily done for 4, 6, and 7 arm_leg_tail dip surfaces.
- Brian
