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Curves and Surfaces

3D modelling software usually supports the creation and manipulation of various kinds of spline curves: Bézier curves, B-splines, and non-uniform rational B-splines or NURBS. These curves can be used for generating surfaces.1

paths in Metafont and MetaPost are Bézier curves. It would be possible to implement three-dimensional Bézier curves in 3DLDF, but unfortunately they are not projectively invariant:

Let c_0 represent a Bézier curve in three dimensions, P the control points of c_0, and t a projection transformation. Further, let Q represent the points generated from applying t to P, and c_1 the curve generated from Q. Finally, let R represent the points generated from applying t to all of the points on c_0, and c_2 the curve through R: c_1 \not\equiv c_2.

NURBS, on the other hand, are projectively invariant,2 so I will probably concentrate on implementing them. On the other hand, it would be nice to be able to implement Metafont's way of specifying paths using `curl', `tension', and `dir' in 3DLDF. This may prove to be difficult or impossible. I do not yet know whether Metafont's path creation algorithm can be generalized to three dimensions.3

Curves and surfaces are advanced topics, so it may be a while before I implement them in 3DLDF.

Footnotes

  1. Huw Jones, Computer Graphics through Key Mathematics, and David Salomon, Computer Graphics and Geometric Modeling, are my main sources of information about spline curves.

  2. Jones, Huw. Computer Graphics through Key Mathematics, p. 282.

  3. Knuth, Donald Ervin. The METAFONTbook, p. 130, and Hobby, John D. Smooth, Easy to Compute Interpolating Splines. Discrete and Computational Geometery 1(2).