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To understand this question, please first understand the question and answer here: Create a formula that creates a curve between two points

We are essentially transcending a 2d problem into a 3d problem.

I have 4 points arranged as the corners of a 1 unit square on the x-y-plane. Each point also has an arbitrary Z value, a x-gradient, a y-gradient.

To be clear, by "x-gradient" I mean that you draw a tangent line on the X-Z-Plane from the point and get (the change in z) divided by (the change in X).

First describe a surface that exists at those four points and has the correct gradients at those points.

Secondly, create a function that returns the Z value for any two values of X and Y between 0 and 1.

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Lorry Laurence mcLarry
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  • Do you only want it smooth & continuous at the 4 corners? Most applications like this would want to define the entire edge and have that smooth & continuous or you won't be able to join two squares next to each other together. – Ian Miller Mar 13 '16 at 10:52
  • I would want it so that a different square that shares the same two corners would get the same result along that edge. I'm not sure wether that is possible when each square is calculated separately? This might take some experimentation on my part. For now just try to solve the problem of creating the one square. – Lorry Laurence mcLarry Mar 13 '16 at 10:57
  • Same result along the edge for both value and gradient? My feeling is this is significantly harder than the 2d version. – Ian Miller Mar 13 '16 at 14:28
  • The Hermite bicubic patch mentioned in my answer below will give you matching of both position and slope across the "seams" between adjacent squares. – bubba Mar 13 '16 at 14:49

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Am I right if I say that you have 3 constraints at each "corner": its $z$ (height) its $x$ gradient and its $y$ gradient ? i.e., 12 informations.

Why don't you try surfaces depending on (at least) 12 parameters, for example

$$z=ax^4+by^4+cx^3y+dxy^3+ex^2y^2+fx^3+gy^3+hx^2y+ixy^2+jx+ky+l$$

Imposing the different constraints will give you a system of 12 linear equations in 12 unknowns.

It can be also a surface of the form $z=f(x)g(y)$, $f$ and $g$ depending resp. on 6 and 7 parameters resp. (like above) (the 7th parameter is necessary because you can simultaneously $f$ by a constant $k$ and multiply $g$ by $k$). The equations are no longer linear, but the constraints on derivatives are more easily expressed.

You can also turn to what is known under the name of "Bezier patches" (or "Hermite patches").

Jean Marie
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One approach is to use a bicubic interpolant, as described here. You have the necessary point and partial derivative info. However, you don't have values for the mixed second partial derivative $f_{xy}$ at the four corners. You can estimate these second derivative values, somehow, or just set them equal to zero. Setting them to zero gives you a so-called Ferguson patch, or F-patch. See these notes.

bubba
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