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,

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

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

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

$$2(F''  -T_x  -T_y),$$ (23)

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

Equation (22) simply means

$$p = \gamma(t) + \lambda N(t), \lambda.$$ (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),

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

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

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

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

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

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