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copied!<p>If all you have is a single value of p, such that p is between the minimum and maximum values at the four corners of the square, then no, it is not possible in general to find a SINGLE solution (s,t) such that the bilinear interpolant will give you that value. </p> <p>In general, there will be an infinite number of solutions (s,t) inside the square. They will lie along a curved (hyperbolic) path through the square. </p> <p>If your function is a vector valued one, so you have two known values at some unknown point in the square? Given known values of two parameters at each corner of the square, then a solution MAY exist, but there is no assurance of that. Remember that we can view this as two separate, independent problems. The solution to each of them will lie along a hyperbolic contour line through the square. If the pair of contours cross inside the square, then a solution exists. If they do not cross, then no solution exists.</p> <p>You also ask if a simple formula exists to solve the problem. Sorry, but not really that I see. As I said, the curves are hyperbolic.</p> <p>One solution is to switch to a different method of interpolation. So instead of bilinear, break the square into a pair of triangles. Within each triangle, we can now use truly linear interpolation. So now we can solve the linear system of 2 equations in 2 unknowns within each triangle. There may be one solution in each triangle, except for a rare degenerate case where the corresponding piecewise linear contour lines happen to be co-incident.</p>

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