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December 30th, 2007, 08:08 AM   #1
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Curve fitting.

Hoping someone can help me here. I've been fumbling around for ages with no success. I don't know how to post a picture which would probably help things a lot.

I have 5 points and each point has different constraints. I need to find an equation (or equations) that will link the consecutive points with one smooth curve. The curve can only change curvature(inflect form +ve curvature to -ve curvatre) at points 2 and 4.

Points 1 & 5 (2 constraints each)
X,Y known
Gradient = 0

Point 2 & 4 (2 constraints each)
X known
Curvature = 0

Point 3 (2 constraints)
Y known
Gradient = 0

That's 10 constraints. From what I believe I need a 10 variable equation to achieve it in a single equation or something like one 4th order polynomial and 2 cubic splines to find a solution. Is this right?

Hope this is somewhat understandable. If someone can help me I will email excel files to show the track I was on.
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January 12th, 2008, 10:12 AM   #2
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At first glance, I was going to say that you are seeking a spline; however, most spline routines I know about use just the (x,y) data pairs, and don't allow you to specify curvature or gradient at specific points. Incidentally, I am not clear about your terminology: I assume when you say Gradient=0 you mean the first derivative of the line is zero (i.e. - a horizontal line). However, when you say Curvature=0, what do you mean? Second derivative is horizontal?

Are you looking for a theoretical solution?

Or do you have real data pairs? You only have five data pairs, why not post them, so we can see what you are working with. As a guess, it sounds like a cosine curve: horizontal at 0, inflection point at 90 degrees, horizontal at 180 degrees, inflection point at 270 degrees, horizontal at 360 degrees.


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