Hello,

Bilinear interpolation generates visible crosses, see image from wikipedia to see what I mean, I mean the horizontal and vertical structures and sharp edges:

Bicubic interpolation looks much rounder:

Imagine now that the colors are types of world (tundra, desert, grass, ...): the rounder shapes are *highly* preferable over the ugly "crosses" of bilinear interpolation.

However, bicubic interpolation has two disadvantages (the second being the most problematic for me):

1.) It requires 16 points, rather than only 4 points. It would be highly preferable for me to only have to use the 4 corners of a square zone, not corner points of neighboring zones as well

2.) Values can overshoot, that is, they can become lower or higher than the 4 corners of this zone. I really don't want this, "conditions" in a zone should be bounded by its corners for convenience reasons (predictable range of values in a zone).

So, the question is:

Does there exist a way of 2D interpolating that only uses 4 corner points and only returns values in that range, but, looks "rounder" than bilinear interpolation?

Thanks!

Awesome! That is a great idea and works great, thanks!

EDIT: never mind the other comment I posted here, Smoothstep does look visually better than http://dinodini.wordpress.com/2010/04/05/normalized-tunable-sigmoid-functions/ :)

I personally prefer Bezier (it doesn't overshoot, hits each control point exactly, and look absolutely gorgeous).

How does that work? The cubic bezier on Wikipedia does not hit each control point, only the outer two - but then it's also 2D intead of 1D:

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

Can this be calculated in a way similar to bicubic interpolation? I couldn't find anything.

THanks a lot!

I personally prefer Bezier (it doesn't overshoot, hits each control point exactly, and look absolutely gorgeous).

How does that work? The cubic bezier on Wikipedia does not hit each control point, only the outer two - but then it's also 2D intead of 1D:

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

Can this be calculated in a way similar to bicubic interpolation? I couldn't find anything.

Look no further than Wikipedia: http://en.wikipedia.org/wiki/Bezier_patch

Basically, you do Bézier interpolation between Bézier curves along the additional parameter dimension.

However, a really good looking surface is going to need data beyond the 4 vertex heights of one rectangular grid cell, because it needs good looking normals at grid vertices, and figuring them out from other grid vertices is better than assigning fixed normals (all up, like in "smoothstep" C1 alternatives to C0 linear interpolation) or random normals (which would show patterns). Storing a precomputed normal (or even a normal and curvature) for each vertex might be more convenient than looking up adjacent vertices.