  We geometrically analyze the problem of estimating parameters related to the
shape and size of a two-dimensional target object on the plane by using
randomly distributed distance sensors whose locations are unknown. Based on the
analysis using geometric probability, we discuss the observability of these
parameters: which parameters we can estimate and what conditions are required
to estimate them. For a convex target object, its size and perimeter length are
observable, and other parameters are not observable. For a general polygon
target object, convexity in addition to its size and perimeter length is
observable. Parameters related to a concave vertex can be observable when some
conditions are satisfied. We also propose a method for estimating the convexity
of a target object and the perimeter length of the target object.
