Overview
This study examines the geometry of the principal curvature surface that is associated with a Whitney umbrella. This specific surface is generated by extending the Whitney umbrella along its normal direction by its bounded principal curvature. The research establishes that this construction possesses a natural geometric interpretation.
Approach
The investigation proceeded by first analyzing the intersection of the principal curvature surface with the normal plane. Subsequently, the focal conic was deduced. The study then focused on the geometry of the principal curvature surface along its exceptional set. This involved a detailed examination of its geodesic and normal curvatures, as well as its Gaussian and mean curvatures. Finally, the research aimed to determine all possible generic numbers of zeros for these curvature functions.
Findings
- The intersection of the principal curvature surface with the normal plane was proven to coincide with the pedal curve of the curvature parabola.
- The focal conic associated with this construction was deduced to be the inversion of the aforementioned pedal curve.
- The study investigated the geodesic and normal curvatures of the principal curvature surface along its exceptional set.
- The Gaussian and mean curvatures of the principal curvature surface along the exceptional set were also examined.
- All possible generic numbers of zeros for the geodesic, normal, Gaussian, and mean curvature functions were determined.