Approximate model of a convex/concave surface

Question

I have a set of measurements in 3d that yields a concave surface of a function $f(x,y)$ that I don't know its expression. I am thinking to approximate the function to a model where any point from the surface described as a vector $\eta = \bigl(x,~ y, ~f(x,y)\bigr)^T$ could be written in terms of its three surrounding neighbors ($\eta$ in the center) using convex hull property or any other convexity property. So my question what is the best equality or model for this problem knowing that I have the measurements $f(x,y)$ and their corresponding $x$ and $y$.

Answer 1

Given a set of points in 3 dimensions there are multiple surfaces which might approximate the model. You could use one of these methods:

• Taylor series
• Divided differences
• Hermite interpolation
• Cubic spline interpolation
• Parametric Curves
• Bezier Curves

I think that all of these have 3d versions although I have only used the 2d versions. What your maximum will be depends on which method you use. Your maximum will be at least as large as the largest $f(x,y)$ though so that gives you a lower bound.
Taylor series needs a derivative so that may not work for you.
The polynomial interpolation methods tend to fluctuate wildly farther away from the center which could give you false maximums if you don't check if your answers for reasonable.
The interpolation methods will give you quite accurate results but may require a lot of calculations if you have more than a few points.
The parametric and Bezier curves are easy to use but they are piece-wise so it can be hard to find maximums and minimums.
Here are some links:

http://en.wikipedia.org/wiki/Taylor_series

http://mathworld.wolfram.com/DividedDifference.html

http://en.wikipedia.org/wiki/Hermite_interpolation

http://mathworld.wolfram.com/CubicSpline.html

http://en.wikipedia.org/wiki/B%C3%A9zier_curve

http://en.wikipedia.org/wiki/B%C3%A9zier_surface