Generalized Focal Surfaces: (S. Hahmann)

The generalized focal surfaces are a surface interrogation method. The idea of generalized focal surfaces is quite related to hedgehog diagrams. In stead of drawing surface normals proportional to a function value, only the point on the surface normal proportional to the function is drawn. The loci of all these points is the generalized focal surface. This method was introduced by Hagen and Hahmann (1992), and is based on the concept of focal surfaces which are known from line geometry. The focal surfaces are the loci of all focal points of an special line congruence, the normal congruence.
Generalized focal surfaces consists of variational surface offstes:

F(u,w) = X(u,w) + c f(kappa_max, kappa_min) N(u,w)

where kappa_max, kappa_min are the principal curvatures of the given surface X and f is a real valued function.

Different offset functions can be used to interrogate and visualize surfaces with respect to the following criteria:
    convexity test
    detection of flat points
    detection of surface irregularities
    visualization of curvature behaviour
    visualization of technical smoothness
    visualization of $C^2$- and $C^3$-discontinuities
    test of technical aspects

Examples:

Convexity test
the intersection of the surface with its focal surface indicate the line of zero Gaussian curvature


variable offset factor:
f = kappa_max * kappa_min

vrml2.0

C2-continuity test on hair dryer surface
The gaps in the focal surface (right) indicate the curvature discontinuity.

variable offset factor:
f = kappa_max^2 + kappa_min^2

continuity test
left: C2-discont. surface
right: C3-discont. surface

C-1 (left) and C0-continuous (right) focal surfaces.

References:

    Return to Stefanie Hahmann's homepage.