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I have worked on triangle mesh for a while. The mesh is reconstructed from range images captured via one 3D scanner. Currently I am very interested in using parametric surfaces to represent one mesh. It says from one book that Parametric surfaces are defined by a vector-valued parameterization function that maps a 2D parameter domain to the surface. However, the parameter domain is always written as (u, v) instead of (x,y). In my view, parameter domain(x,y) is very straightforward.

In computer-aided geometric design where NURBS surface is popularly employed, the NURBS surface is represented by parameter domain (u,v), is it difficult to find the parameter domain(u,v) from cartesian coordinates of points?

Jogging Song
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    What are you looking for is either [Interpolation](http://en.wikipedia.org/wiki/Interpolation) or [Approximation](http://en.wikipedia.org/wiki/Approximation) of the mesh region into NURBS surface. The problem is well-known, there're plenty of info on the subject in the web. Check out these readings: [this](http://mustard.postech.ac.kr/data/surface_reconstruction_parameterization.pdf) and [this](http://gul.sourceforge.net/viewdog-manual/node25.html) – Andrey.Dankevich Apr 19 '14 at 05:22
  • Thanks, Andrey. I am learning NURBS surfaces on my own. I have several questions. From the link http://en.wikipedia.org/wiki/B%C3%A9zier_surface about Bézier surface, it seems that there is no the concept of knot vector. Is the statement right?. For the NURBS surface, how is the factor weights in the formula is calculated in practical applications? – Jogging Song Apr 24 '14 at 08:01
  • Right, Bézier curves don't have knots. Sorry, I guess I don't understand your question about weights. Weights are not calculated, they're properties of control point. Usually, you weights == 1, unless you need to create e.g. a conic shape. – Andrey.Dankevich May 05 '14 at 09:48

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