The medial axis of a plane domain is defined to be the set of the centers of the maximal inscribed disks. It is essentially the cut loci of the inward unit normal bundle of the boundary. We prove that if a plane domain has finite number of boundary curves each of which consists of finite number of real analytic pieces, then the medial axis is a connected geometric graph in R 2 with finitely many vertices and edges. And each edge is a real analytic curve which can be extended in the C 1 manner at the end vertices. We clarify the relation between the vertex degree and the local geometry of the domain. We also analyze various continuity and regularity results in detail, and show that the medial axis is a strong deformation retract of the domain which means in the practical sense that it retains all the topological informations of the domain. We also obtain parallel results for the medial axis transform.
An adapted orthonormal frame (f 1 (ξ), f 2 (ξ), f 3 (ξ)) on a space curve r(ξ), ξ ∈ [ 0, 1 ] comprises the curve tangent f 1 (ξ) = r ′ (ξ)/|r ′ (ξ)| and two unit vectors f 2 (ξ), f 3 (ξ) that span the normal plane. The variation of this frame is specified by its angular velocity Ω = Ω 1 f 1 +Ω 2 f 2 +Ω 3 f 3 , and the twist of the framed curve is the integral of the component Ω 1 with respect to arc length. A minimal twist frame (MTF) has the least possible twist value, subject to prescribed initial and final orientations f 2 (0), f 3 (0) and f 2 (1), f 3 (1) of the normal-plane vectors. Employing the Euler-Rodrigues frame (ERF) -a rational adapted frame defined on spatial Pythagorean-hodograph curves -as an intermediary, an exact expression for an MTF with Ω 1 = constant is derived. However, since this involves rather complicated transcendental terms, a construction of rational MTFs is proposed by the imposition of a rational rotation on the ERF normal-plane vectors. For spatial PH quintics, it is shown that rational MTFs compatible with the boundary conditions can be constructed, with only modest deviations of Ω 1 about the mean value, by a rational quartic normal-plane rotation of the ERF. If necessary, subdivision methods can be invoked to ensure that the rational MTF is free of inflections, or to more accurately approximate a constant Ω 1 . The procedure is summarized by an algorithm outline, and illustrated by a representative selection of computed examples.
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