Smoothness and geometry of boundaries associated to skeletal structures I: sufficient conditions for smoothness

Damon, James

Annales de l'Institut Fourier, Tome 53 (2003) no. 6, pp. 1941-1985.

Nous introduisons une structure squelette $(M,U)$ dans ${ℝ}^{n+1}$, qui consiste en un ensemble stratifié de Whitney de dimension $n$ sur lequel est défini un “champ radial de vecteurs” multiformes $U$. C’est une extension de la notion du “Blum medial axis” d’une région dans ${ℝ}^{n+1}$ avec un bord lisse générique. Puis, pour de telles structures squelettes, on peut définir “un bord associé” $ℬ$. Nous introduisons des invariants géométriques du champ radial de vecteurs $U$ et un “flot radial” de $M$ à $ℬ$. Ils nous permettent d’obtenir des conditions numériques suffisantes pour que le bord soit lisse, et de déterminer sa géométrie. Nous établissons en même temps l’existence d’un voisinage tubulaire d’un tel ensemble stratifié de Whitney.

We introduce a skeletal structure $(M,U)$ in ${ℝ}^{n+1}$, which is an $n$- dimensional Whitney stratified set $M$ on which is defined a multivalued “radial vector field” $U$. This is an extension of notion of the Blum medial axis of a region in ${ℝ}^{n+1}$ with generic smooth boundary. For such a skeletal structure there is defined an “associated boundary” $ℬ$. We introduce geometric invariants of the radial vector field $U$ on $M$ and a “radial flow” from $M$ to $ℬ$. Together these allow us to provide sufficient numerical conditions for the smoothness of the boundary $ℬ$ as well as allowing us to determine its geometry. In the course of the proof, we establish the existence of a tubular neighborhood for such a Whitney stratified set.

Classification : 57N80, 58A35, 68U05, 53A07
Keywords: skeletal structures, Whitney stratified sets, Blum medial axis, shock set, radial shape operator, grassfire flow, radial flow
Mot clés : structure squelette, ensemble stratifié de Whitney, axe moyen de Blum, ensemble de choc, opérateur de forme, flot radial

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