The Distance Squared Function on a Planar Curve

We are interested in finding the points on the curve which have either local minimal or local maximal distances to . That is, for a fixed point , we are trying to find those s, such that and either or .

The singular set of ,

is a surface in , since the Jacobian,

(23) |

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

Equation (22) simply means

for some | (24) |

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

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

which is regular, since the Jacobain of the map ,

(26) |

has a non-singular submatrix consisting of the last two columns if , or a non-sinuglar submatrix consisting of the first column and either the seond or the third column ( and can not be both zero) if .

When projecting back to , we have the envelope of the normals of curve, which is just the evolute of the curve, because from equation 25, we have,

The evolute is therefore smooth so long as (See Section 3). Otherwise, we have the regression point,

which is the curvature center of some vertex on the curve. And the evolute has an ordinary cusp there if , because

- ( is the curvature center of the vertex) has singularity at the corresponding .
- It can be proved that in equation (21) is the (p)versal unfolding of .
- 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,

- For a fixed point , the local minimal or maximal distance point has to be the curve point, say , with the normal passing through . See equation (24).
- is a local minimal distance point if , i.e., , or is closer to it than the curvature center.
- is a local maximal distance point if , i.e., is farther to it than the curvature center.
- if is not a vertex, and is its curvature center, i.e. , but , then is neither a local minimal distance point nor a local maximal one.
- is a local minimal distance point, if it is a vertex of maximal curvatur, and is its curvature center. i.e. , and , but .
- is a local maximal distance point, if it is a vertex of minimal curvatur, and is its curvature center. i.e. , and , but .

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

- For any point , outside the folded area, there is only one minimal distance point. ( -singularity)
- For any point , inside the folded area, there are two minimal distance points and one maximal point. (all -singularity)
- For any point , on the fold but not the cusp, there is one minimal distance point.( -singularity, and there is another curve point of singularity).
- For on the cusp, there is one minimal distance point. ( singularity).
- The evolute divides the control space into 2 regions, with 1 and 3 critical points respectively. When moving the point 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 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.