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Liam M

Minimum Curvature Algorithm: Need some Help Intepreting some Math

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hey guys, I'm trying to implement a minimum curvature algorithm, for interpolating surfaces, as shown here:

http://idlastro.gsfc.nasa.gov/idl_html_help/MIN_CURVE_SURF.html

Now, looking at the equations shown, for 3 sample points, solving for b0, b1, b2, a0, a1, and a2, I end up with 3 equations to solve 6 unknowns, however the article says "n+3 simultaneous equations", and instead I have n equations. is there something I'm missing? If I had n + 3, then the algorithm would be over-defined, which would be great. Instead, it's under-defined, which is not so great.

So did I make a mistake, or god forbid, am I right? If I am right, could someone suggest a good algorithm for finding a solution to the algorithm? My math is very, very rusty.

Regards,

Liam

Edit: My variables were very, very wrong, silly mistake, but the question remains the same.

[Edited by - Liam M on October 2, 2010 12:26:50 AM]

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A bit of a followup, here's a spreadsheet I prepared while working out the algorithm, it explains what I was thinking a bit more:



Essentially, one equation per sample point, a c value for each distance, the distance from itself being zero. Not being square, I can't simply solve it by inverting the matrix, which would make life a lot easier. Any ideas, comments, corrections? The only way I can work this out is to have more unknowns than equations, and unfortunately, it's really hard to find any sort of documentation on this algorithm.

Thanks fellas.

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This document shows the details for Thin Plate Splines. It is easy enough to modify for the other cases mentioned in your link. (And I have an implementation at my web site.)

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Quote:
Original post by Dave Eberly
This document shows the details for Thin Plate Splines. It is easy enough to modify for the other cases mentioned in your link. (And I have an implementation at my web site.)


Thanks for the link, I've just had a brief look, it looks like it may be what I'm looking for, I'll have a thorough look and see what I can do.

PS, 3D Game Engine Architecture, one of my go-to books for various topics, thanks for writing that one.

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The key is that the minimization of the binding energy integral has infinitely many solutions; that is, the linear term b0+b1*x+b2*y in f(x,y) is irrelevant when minimizing Integral(fxx^2+2*fxy^2+fyy^2), because the second derivatives of the linear term are zero. You have to supply more constraints, so the typical one is Transpose(B)*a = 0 (using the notation of my PDF).

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