# Visualization of Gaussian curvature for NURBS surface.

Hello everyone,

I need to visualize Gaussian curvature of NURBS surface like a colored surface. What is a best way to do it?
Should I use evaluators for that? Can I evaluate not just point on the surface and normal vector, but also tangent vectors?

Surfing in Internet for many days with no results at all.

OpenGL is a low level rendering API, it does not have functionality to compute geometric properties like curvature and only handles simple primitives: triangles, lines, and points - not more complex surface representations like NURBS. There are the evaluators inside GLU that can be used to turn a NURBS surface into a triangle based representation - off hand I’m not sure if it has support to compute curvature.

Assuming you have found a way to compute curvature at the vertices of the triangle approximation of your NURBS surface, you can store that as a vertex attribute and pass it to your shaders just like any other attribute (position, normal, texture coordinates) - in your shader you can turn the curvature value into a color (through some formula or by sampling a look up texture). Alternatively (in particular if you are not using shaders) you can turn the curvature value into a color on the CPU and pass that color as (secondary) color attribute to OpenGL.

If you can get the control points of a Bezier patch (i.e. the arguments to glMap2()), you can convert those to the coefficients of the numerator and denominator polynomials, differentiate those with respect to u and v, then convert the coefficients of the derivatives back to the control points for a Bezier patch. Thus you can generate a Bezier patch for the tangents (and second derivatives if needed), and also for the normal (which is the cross product of the tangents).

For a polynomial Bezier patch (where the control points all have the same W), the derivatives have the same or lower degree as the original. E.g. a cubic patch has coefficients up to u^3v3. The derivative w.r.t. u has coefficients up to u^2v^3 while the derivative w.r.t. v has coefficients up to u^3v^2 The normal will have derivatives up to u^5*v^5.

However, NURBS surfaces use rational patches, and for a rational Bezier patch (where control points have different W values and so X/W etc is the ratio of polynomials), the derivatives have a higher degree than the original because of the quotient rule:
d(g/h)/dx = (hg’ - gh’)/(h^2).

For a cubic patch, h and g both contain terms up to u^3v^3. Differentiating w.r.t. u means that g’ and h’ will have terms up to u^2v^3 so both hg’ and gh’ will have terms up to u^5v^6 while h^2 has terms up to u^6v^6, so the resulting patch will have degree 6 (order 7, i.e. 7x7=49 control points). The normal and second derivative will have degree 12 (order 13), but OpenGL only requires support for a maximum order of 8.

Depending upon what you want to do, tessellation shaders (which require OpenGL 4) may be of use.

That’s probably simplest way for me. Thank you for help.