Minkowski Symmetry Sets of Plane Curves

Abstract

We study in this paper the Minkowski Symmetry Set (MSS) of a closed smooth curve γ in the Minkowski plane. We answer the following question which is analogous to one concerning curves in Euclidean plane that was treated in [7]: given a point p on γ, does there exist a bi-tangent pseudo-circle that is tangent to γ at both p and at some other point q on γ? The answer is yes, but as pseudo-circles with non-zero radii have two branches (connected components) it is possible to refine the above question to the following one: given a point p on γ, does there exist a branch of a pseudo-circle that is tangent to γ at both p and at some other point q on γ? This question is motivated by the quest in [12] to define the Minkowski Blum medial axis, a counterpart of the Blum medial axis of curves in the Euclidean plane ([3]).

Minkowski medial axes and shocks of plane curves

“…It follows from Theorem 3.2 in [9] that the family f is always a versal unfolding of its generic singularities, so the M SS is locally diffeomorphic to the bifurcation set of the models of such singularities (Corollary 3.3 in [9]). Thus the configuration of the M SS at the generic multi-local singularities of f c are as in Figure 4.…”

“…Theorem 4.1 of [9] states that for any point p of a closed smooth curve γ there exists another point q ∈ γ and a pseudo-circle that is tangent to γ at both p and q. From this, and the fact that it is possible to reconstruct the curve locally, it follows that it is possible to reconstruct any smooth closed curve from its Minkowski symmetry set.…”

“…The two sets are not equal because not all 1-branch bi-tangent pseudo-circles are maximal, see for example Theorem 4.5 of [9] states that for any point p on a spacelike or timelike curve γ without inflections there exists another point q on γ and a pseudo-circle that is tangent to γ at both p and q with both points being on a single branch of the pseudo-circle. From this it follows that any closed convex curve can be reconstructed from its 1-branch medial axis.…”

“…Proof: Consider a neighbourhood of an A 3 point on a spacelike M M A (which necessarily corresponds to a timelike piece of γ, see Theorem 5.2 in [9]). Orient the M M A so that its tangent line points towards the branch of the bi-tangent pseudo-circle that contains the tangent points.…”

“…If the M M A is timelike (the curve γ must be spacelike, Theorem 5.2 in [9]). Following the same arguments as above and using formula (2) at an A 3 -point, yields −r∂r/∂s > 0.…”