I am seeking clarification of the relationship between the focal locus and the cut locus of a curve $C$ in $\mathbb{R}^2$, and of a surface $S$ in $\mathbb{R}^3$. Essentially my question is,
Under what conditions is the focal locus and the cut locus identical, when do they differ, and when they differ, how do they differ.
For example, it believe the two coincide for a sphere (in any dimension): both are simply the center point of the sphere. It may be that these issues are primarily definitional rather than substantive. Let me offer the definitions with which I am working.
Cut Locus. Generally the cut locus is defined on a Riemannian manifold with respect to a point. But instead I want to define the cut locus of a set in $\mathbb{R}^n$. Let me follow the definition of Franz-Erich Wolter, who wrote his Ph.D. thesis on the topic:
"The cut locus $C_A$ of a closed set $A$ in the Euclidean space $E$ is defined as the closure of the set containing all points $p$ which have at least two shortest paths to $A$."
(This is quoted from reference (1) below.) This definition is in accord with that of the medial axis, extensively explored in computer science.
Focal locus. I am having more difficulty locating a widely accepted definition of the focal locus. Let me follow Thorpe:
"The focal locus of a plane curve $C$ is the locus of the centers of curvature and is often called the evolute of $C$." ... "The set of all focal points along all normal lines to an $n$-surface $S$ in $\mathbb{R}^{n+1}$ is called the focal locus of $S$." ... Let $\phi$ be a parametrized $n$-surface, and let $\beta$ be "a unit-speed parametrization of the line normal to Image $\phi$ at $\phi(p)$. A point $f$ is said to be a focal point of $\phi$ along $\beta$ if $f = \beta(s_0)$ where $s_0$ is such that the map ... $\phi(q) + s_0 N^\phi(q)$ is singular (not regular) at $p$" [where $N^\phi(q)$ is the normal at $q$].
(These quotes are from reference (2) below.)
One aspect of the focal locus that confuses me is that there
is a notion of focal surfaces, which derive from
"the reciprocal of the principal curvatures,"
as described in (3). Here there are two surfaces,
as opposed to one focal locus, as depicted in this intriguing figure:
It may be that there are references that would resolve my definitional confusions, in which case pointers would be much appreciated. Thanks!
References.
- Franz-Erich Wolter. "Cut Locus and Medial Axis in Global Shape Interrogation and Representation." MIT Ocean Engineering Design Laboratory Memorandum 92-2. December 1993. PDF.
- John A. Thorpe. Elementary Topics in Differential Geometry. Springer, 1979. Google books. Quote from p.137.
- Jingyi Yu, Xiaotian Yin, Xianfeng Gu, Leonard McMillan, and Steven Gortler. "Focal Surfaces of Discrete Geometry." 2007. Proceedings of the 5th Euro Graphics Symposium on Geometry Processing.