Surface u and v derivatives

Rhino Developer
1

Hi,

When we get first and second derivative on a nurbssurface (typically used for x and y planes of a surface) are they orthogonal to any curved surface?

Or they are not , because a surface is curved?

2

On a NURBS surface, like on any parametric surface, if the derivatives are well defined at a point, then:

  • The first derivatives are in the tangent plane of the surface at that point, and so are orthogonal to the normal to the surface at that point. The first derivatives are generally not orthogonal to each other.
  • The second derivatives are not generally orthogonal to the surface’s normal, unless the surface has no curvature in the direction of (an isocurve along) that parameter. They are generally not orthogonal to each other.

The Wikipedia page on Parametric Surfaces is a good source of information on this topic, and the Rhino’s “Essential Mathematics” guide takes a more graphical approach.

3

Thank you