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The Distance Squared Function on a Planar Curve

Given a regular unit speed curve $ \gamma(t)$ in $ {\bf R}^2$ , for a fixed point $ p$ in $ {\bf R}^2$ , define,

$\displaystyle F(t, p) = (\gamma(t) - p) \cdot (\gamma(t) - p),$ (21)

We are interested in finding the points on the curve which have either local minimal or local maximal distances to $ p$ . That is, for a fixed point $ p$ , we are trying to find those $ t$ s, such that $ F_p'(t) =0$ and either $ F_p''(t) > 0$ or $ F_p''(t) < 0$ .

The singular set of $ F$ ,

$\displaystyle F'(t, p) = 2(\gamma(t) - p) \cdot T = 0,$ (22)

is a surface in $ {\bf R}^3$ , since the Jacobian,

$\displaystyle 2(F''  -T_x  -T_y),$ (23)

has rank 1 ($ T_x$ and $ T_y$ , the two components of $ T$ , can not be both zero).

Equation (22) simply means

$\displaystyle p = \gamma(t) + \lambda N(t),  $   for some$\displaystyle   \lambda.$ (24)

and the surface is just the spread out of all the curve normals, lifted along $ t$ -axis by the parameter value $ t$ .

The fold (or silhouette generator viewed from $ t$ -axis) of this surface is defined by the following equation, in addition to equation (22),

$\displaystyle F''(t, p) = 2( 1 + (\gamma(t) - p) * \kappa N ) = 0,$ (25)

which is regular, since the Jacobain of the map $ (F', F''):{\bf R^3} \rightarrow {\bf R^2}$ ,

$\displaystyle \left( \begin{array}{ccc} 0 & -2T_x & -2T_y F''' & -2\kappa N_x & -2\kappa N_y \end{array} \right)$ (26)

has a non-singular submatrix consisting of the last two columns if $ \kappa \neq 0$ , or a non-sinuglar submatrix consisting of the first column and either the seond or the third column ($ T_x$ and $ T_y$ can not be both zero) if $ F''' \neq 0$ .

When projecting back to $ {\bf R}^2$ , we have the envelope of the normals of curve, which is just the evolute of the curve, because from equation 25, we have,

$\displaystyle \kappa \neq 0,$    and $\displaystyle p = \gamma(t) + 1/\kappa N(t).$ (27)

The evolute is therefore smooth so long as $ F''' \neq 0$ (See Section 3). Otherwise, we have the regression point,

$\displaystyle F'''(t, p) = 2 * \kappa' (\gamma(t) - p) * N = 0,$ (28)

which is the curvature center of some vertex on the curve. And the evolute has an ordinary cusp there if $ F'''' \neq 0 $ , because
  1. $ F_{p_0}(t)$ ($ p_0$ is the curvature center of the vertex) has $ A_3$ singularity at the corresponding $ t$ .
  2. It can be proved that $ F(t,x)$ in equation (21) is the (p)versal unfolding of $ F_{p_0}(t)$ .
  3. By Section 6.1, the evolute has an ordinary cusp there.

Now back to our minimal or maximal distance problem, we have the following local result,

  1. For a fixed point $ p$ , the local minimal or maximal distance point has to be the curve point, say $ \gamma(t_0)$ , with the normal passing through $ p$ . See equation (24).
  2. $ \gamma(t_0)$ is a local minimal distance point if $ F'' > 0$ , i.e., $ \kappa = 0$ , or $ p$ is closer to it than the curvature center.
  3. $ \gamma(t_0)$ is a local maximal distance point if $ F'' < 0$ , i.e., $ p$ is farther to it than the curvature center.
  4. if $ \gamma(t_0)$ is not a vertex, and $ p$ is its curvature center, i.e. $ F'' = 0$ , but $ F''' \neq 0$ , then $ \gamma(t_0)$ is neither a local minimal distance point nor a local maximal one.
  5. $ \gamma(t_0)$ is a local minimal distance point, if it is a vertex of maximal curvatur, and $ p$ is its curvature center. i.e. $ F'' = 0$ , and $ F''' \neq 0$ , but $ F'''' > 0$ .
  6. $ \gamma(t_0)$ is a local maximal distance point, if it is a vertex of minimal curvatur, and $ p$ is its curvature center. i.e. $ F'' = 0$ , and $ F''' \neq 0$ , but $ F'''' < 0$ .

In the neighborhood of curvature center of vertex of maximal curvature, we have the following multi-local result(Figure 3),

  1. For any point $ p$ , outside the folded area, there is only one minimal distance point. ($ A_1$ -singularity)
  2. For any point $ p$ , inside the folded area, there are two minimal distance points and one maximal point. (all $ A_1$ -singularity)
  3. For any point $ p$ , on the fold but not the cusp, there is one minimal distance point.($ A_1$ -singularity, and there is another curve point of $ A_2$ singularity).
  4. For $ p$ on the cusp, there is one minimal distance point. ($ A_3$ singularity).
  5. The evolute divides the $ {\bf R}^2$ control space into 2 regions, with 1 and 3 critical points respectively. When moving the point $ p$ from the 1 critical point (which is minimal) region into the 3 critical point region, the change of minimal point on the curve is gradual. If we keep moving point $ p$ all the way through the 3 critical point region and crossing out into the other side of the 1 critical point region, there is a sudden change of minimal point on the curve, which is called catastrophe.

Figure: Evolute of the parabola $ \gamma (t) = (t, t^2)$ . It has an ordinary cusp, which is also the local structure of any versal 2-unfolding of $ A_2$ singularity
\begin{figure*}\begin{center}
\epsfig{file=demoEvolute.eps, scale=.3}
\end{center}
\end{figure*}


next up previous
Next: The Gravitational Catastrophe Machine Up: Introduction of Singularity Theory Previous: Local Structure of Silhouettes
Xianming Chen 2006-10-02