Focal Surface
Focal Surface is a derived surface, derived from another surface in the catalogue, immersed.
Open Focal Surface in the interactive viewer →
aperiodic def-derived derived immersed implemented tradition-classical
Properties
- Family
- derived
- Given by
- another surface in the catalogue
- Curvature
- no condition imposed
- Periodicity
- not periodic
- Ends
- 0
- Embedding
- immersed
- Fidelity
- exact
Definition
The surface of centres: both sheets x + N / kappa_i of principal curvature centres, evaluated ANALYTICALLY from exact source charts (never estimated from a mesh; discrete principal curvatures are noise precisely where the caustic is interesting). Singular at umbilics, where the sheets meet, and flaring to infinity at parabolic points, where the operator clips.
Sources
Derived from the Math Art generator registries and cross-referenced to the literature; not transcribed from any gallery's compilation.
- G. Monge, "Application de l'analyse a la geometrie" (1807) -- lines of curvature and the centres of curvature of a surface.
- A. Cayley, "On the centro-surface of an ellipsoid", Trans. Cambridge Phil. Soc. 12 (1873) -- the focal surface of the ellipsoid.
- L. P. Eisenhart, "A Treatise on the Differential Geometry of Curves and Surfaces" (1909), ch. on the surface of centres -- the classical treatment of the two sheets and their tangency to the normals.
- D. Hilbert and S. Cohn-Vossen, "Anschauliche Geometrie" (1932) -- centres of curvature and umbilics, read geometrically.
- I. R. Porteous, "Geometric Differentiation" (1994) -- the modern singularity-theory reading of focal sets, umbilics and ridges.
- R. Ferreol, "Encyclopedie des formes mathematiques remarquables" (mathcurve.com), "Surface focale (developpee d'une surface)".
- mathcurve, 'surface focale'.