8 research outputs found
Level sets of functions and symmetry sets of smooth surface sections
We prove that the level sets of a real C^s function of two variables near a
non-degenerate critical point are of class C^[s/2] and apply this to the study
of planar sections of surfaces close to the singular section by the tangent
plane at hyperbolic points or elliptic points, and in particular at umbilic
points.
We also analyse the cases coming from degenerate critical points,
corresponding to elliptic cusps of Gauss on a surface, where the
differentiability is now reduced to C^[s/4].
However in all our applications to symmetry sets of families of plane curves,
we assume the C^infty smoothness.Comment: 15 pages, Latex, 6 grouped figures. The final version will appear in
Mathematics of Surfaces. Lecture Notes in Computer Science (2005
A path algebra approach to the computation of exact rational parametrisations for CAD/CAM applications
New minimal bounds for the derivatives of rational Bézier paths and rational rectangular Bézier surfaces
A Functional Equation Approach to the Computation of the Parameter Symmetries of Spline Paths
Level sets of functions and symmetry sets of surface sections
We prove that the level sets of a real C s function of two variables near a non-degenerate critical point are of class C [s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at an elliptic or hyperbolic point, and in particular at an umbilic point. We go on to use the results to study symmetry sets of the planar sections. We also analyse one of the cases coming from a degenerate critical point, corresponding to an elliptic cusp of Gauss on a surface, where the differentiability is reduced to C [s/4]. However in all our applications we assume C  ∞  smoothness