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

James Damon^{[1]}

- [1] University of North Carolina, Department of Mathematics, Chapel Hill NC 27599 (USA)

Annales de l’institut Fourier (2003)

- Volume: 53, Issue: 6, page 1941-1985
- ISSN: 0373-0956

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topDamon, James. "Smoothness and geometry of boundaries associated to skeletal structures I: sufficient conditions for smoothness." Annales de l’institut Fourier 53.6 (2003): 1941-1985. <http://eudml.org/doc/116090>.

@article{Damon2003,

abstract = {We introduce a skeletal structure $(M,U)$ in $\{\mathbb \{R\}\}^\{n+1\}$, 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 $\{\mathbb \{R\}\}^\{n+1\}$ with generic smooth boundary. For such a skeletal structure there is defined an
“associated boundary” $\{\mathcal \{B\}\}$. We introduce geometric invariants of the radial vector
field $U$ on $M$ and a “radial flow” from $M$ to $\{\mathcal \{B\}\}$. Together these allow us to
provide sufficient numerical conditions for the smoothness of the boundary $\{\mathcal \{B\}\}$ 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.},

affiliation = {University of North Carolina, Department of Mathematics, Chapel Hill NC 27599 (USA)},

author = {Damon, James},

journal = {Annales de l’institut Fourier},

keywords = {skeletal structures; Whitney stratified sets; Blum medial axis; shock set; radial shape operator; grassfire flow; radial flow; Whitney stratified set; skeletal strucuture; shape operator; tubular neighborhood},

language = {eng},

number = {6},

pages = {1941-1985},

publisher = {Association des Annales de l'Institut Fourier},

title = {Smoothness and geometry of boundaries associated to skeletal structures I: sufficient conditions for smoothness},

url = {http://eudml.org/doc/116090},

volume = {53},

year = {2003},

}

TY - JOUR

AU - Damon, James

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

JO - Annales de l’institut Fourier

PY - 2003

PB - Association des Annales de l'Institut Fourier

VL - 53

IS - 6

SP - 1941

EP - 1985

AB - We introduce a skeletal structure $(M,U)$ in ${\mathbb {R}}^{n+1}$, 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 ${\mathbb {R}}^{n+1}$ with generic smooth boundary. For such a skeletal structure there is defined an
“associated boundary” ${\mathcal {B}}$. We introduce geometric invariants of the radial vector
field $U$ on $M$ and a “radial flow” from $M$ to ${\mathcal {B}}$. Together these allow us to
provide sufficient numerical conditions for the smoothness of the boundary ${\mathcal {B}}$ 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.

LA - eng

KW - skeletal structures; Whitney stratified sets; Blum medial axis; shock set; radial shape operator; grassfire flow; radial flow; Whitney stratified set; skeletal strucuture; shape operator; tubular neighborhood

UR - http://eudml.org/doc/116090

ER -

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