A friend has asked me to publish his algorithm in hopes to secure it for public use. He recommends this algorithm can be used as a new approach to creating 2 dimensional, rectilinear grids which can be extended into 3d. These grids can be calculated using a simple interpolation method. Computation physics problems that can be ran in parallel allow vectors to be computed on a GPU for massive speed gains. This algorithm has applications in finite difference analysis (FD) and computational physics, potentially simplifying the solvers and reducing required computing power without sacrificing accuracy.
Sean would like to offer his work for public availability and use, securing it from those who may desire to keep it private for profit or other motives. Please find this useful.
have van will travel 
Grab a sample set of dot product sums vs. scaling coefficients.
Compute the means (mean sum and mean scaling coefficient).
Compute the following terms:
C_n = sig_i(|x_i – x_mean|) {if you can’t figure out what x and y represent, quit now}
C_i = |x_i – x_mean|/C_n
m = sig_i(C_i*[y_i – y_mean]/[x_i – x_mean]) {slope times weighting coefficient}
b = y_mean – m*x_mean
{This is an asymptotic approximation for the minimum residual line that is analytically accurate for a set with infinite data samples}
This gives an indication of the ‘downward direction’ given the fact that the dot product sums are not smooth (ruling out the Finite Difference Newton Method). You then move ‘downwardly’ as far as desired. You can cycle through the coefficents you want to adjust more than once.
For instance: (n points on:) S scale, T scale, S derivative, T derivative, Twist, repeat (m times)…
You can vary the number of samples per slope calculation (n).
You can vary the number of times you cycle through every coefficient (m).
Continuously update the minimum dot product sum found.
Eventually, you will have reduced it to a value that will vary in accuracy. For symmetric grids it will be very nearly orthogonal. For asymmetric grids the results are less pleasing. Two important measurements to keep track of are the ‘maximum dot product’ as well as the
‘mean dot product’ (sum divided by grid point count). There are methodologies in existence that more accurately describe the end results. I don’t know all of these methodologies. I trained myself in everything I know concerning this algorithm. It is based on the fact that
one can ‘sketch with a pencil’ what a mobius transfrom would ‘look like’ given a set of ‘bounding curves’ that meet at right angles. The algorithm simply emulates this process numerically. I haven’t explored acute/obtuse bounding corners. I have not advanced this method to higher orders as there are many more coefficient sets to cycle through and my computer is completely defective (thank you ‘sadistic mathematicians’/’hackers’).
Euclid’s Fifth Postulate has already been proven. If you don’t know that, don’t complain to anyone about why you don’t think any of this is mathematically valid, because you are already wrong.
I don’t have any money. No one will ever pay me for this. The previous statement is how I’ve learned this world actually works.
If understood and adopted, it will remove the Finite Element Method from Engineering practice because we can compute potential functions faster using Finite Differences. This method is automatable. If you don’t understand what that means, it’s why we spend all our time doing nothing useful.
Often times an Engineer will be happy with an accuracy as low as the hundredths (relative to desired measurement). That would be 2 decimal places. I can’t count to infinity so close enough is good enough given the errors introduced when processing physical materials. If you don’t believe this is true, learn how to design and manufacture an entire machine to the level desired by a ‘potential customer’ and find out if I am right or wrong.
I could rephrase this another way: go and figure out how to become an ‘Engineer’.
If my grammar upsets you, forget Engineering forever. I never care about commas. They don’t build bridges, much less the object you’re reading this text through.
If they say they did it first, I called it here.
~Sean~
side note: Godel’s incompleteness theorem => never have to write a proof for you besides showing you an accurate computation (i consider observed convergence part of the spiel) as well as the 5th Postulate at the same time.