Given two sets of points
and
in the plane (called the
focal sets), the
equidistant set (or
midset) of
and
is the locus of points
equidistant from
and
.
This article studies envelope curves as realizations of focal sets.
We prove two results: First, given a closed convex focal set
that
lies within the convex region bounded by the graph of a concave-up function
, there is a
second focal set
(an envelope curve for a suitable family of circles) such that the graph of
lies in the
midset of
and
. Second, given
any function
with a continuous third derivative and bounded curvature, the envelope curves
and
associated to any family of circles of sufficiently small constant radius centered on the
graph of
will define a midset containing this graph.
