Visual distinguishability of polygons

Abstract

Considering equioptic curves of segments in the plane we prove that two polygons are distinguishable by their visual angles given on the boundary of an open domain that contains both polygons if the boundary either does not contain some exceptional types of arcs or is differentiable by order seven.

Can you see the bubbles in a foam?

“…In Section 3 we first calculate the directional derivatives of the masking function, which generalizes a number of earlier results on visual angles given in [6][7][8] and [10][11][12][13]. At the end, the Laplacian of the masking function is given for d-weighted masking functions (Theorem 3.5).…”

“…We reformulate our question above as a problem to find such sets S of sources and sets O of objects that O T → M T ,ω S is injective. There are numerous results of this kind for special multicurves in the literature: [6][7][8], [10][11][12][13], and also [2,10,16] can be listed 6 here. With these terms the author's original Conjecture in [10] takes the following more general form.…”

“…For convex curves this can be easily proved by the second formula of [10, Lemma 1] 13. If r j (s) or r i (s) is an endpoint of r j resp.…”

Finding Needles in a Haystack

“…2.2]. For a more direct proof one observes that M r J and r J are locally the sum of the functions ±φ j , hence one can assume without loss of generality that r J is a segment for which [7,Lem. A.1(2)] implies the harmonicity directly.…”

“…However it is proved in [6,8], that two polygons are always distinguishable from each other, so the question Are convex polygons distinguishable among convex domains? (1) emerged naturally in [4,Quest.…”