By Hans Hagen

This choice of rules and effects on subject matters of curve and floor layout is meant for learn within the educational atmosphere in addition to for functional use in commercial purposes. major emphasis is on minimum power splines and geometric spline curves, and on options past tensor product surfaces.

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**Example text**

For generic umbilics this index is ±1/2, this implies that the direction field is not orientable on a neighborhood of such points. As illustrated on Fig. 1, if one fixes an orientation of the field at a point on a circuit around an umbilic, propagating this orientation by continuity along the circuit gives the reverse orientation after one turn. In other words, there is no non vanishing continuous vector field inducing the direction field around the umbilic. The index can also be computed with the Monge cubic, this computation is point wise as opposed to the previous one, but need third order coefficients (hence it is likely to be less stable in practice).

14, the blue principal direction field is drawn —from which one infers that the normal is pointing outward so that the two principal curvatures are negative. The two elliptic ridges are closed curves without turning point. The four Lemon umbilics are the black dots, and they are linked by four separatrices —the yellow and green curves. The separatrices, which are curvature lines, are also ridges in that case. 162]). Notice also that the lines of curvatures which are not separatrices are all cycles.

Only one non-singular ridge branch passes through the umbilic. Moreover passing through the umbilic, there is one curve along which k1 (resp. k2 ) is constant equal to k. Such an umbilic is either a lemon, a monstar or a star. The number of ridges is given by the number of real factors lines of the Monge cubic, but these lines are not the tangent directions to ridge lines going through the umbilic. However, these tangent directions can be computed from the Monge cubic cf. [HGY+ 99]. Definition.