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1
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Properties of Conflict Sets in the Plane

100%
Siersma D.
Banach Center Publications
|
1999
|
tom 50
|
nr 1
267-276
EN
This paper studies the smoothness and the curvature of conflict sets of the distance function in the plane. Conflict sets are also well known as 'bisectors'. We prove smoothness in the case of two convex sets and give a formula for the curvature. We generalize moreover to weighted distance functions, the so-called Johnson-Mehl model.
2
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Hypersurfaces with singular locus a plane curve and transversal type $A_1$

88%
Siersma D.
Banach Center Publications
|
1988
|
tom 20
|
nr 1
397-410
3
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Remarks on minimal round functions

64%
Khimshiashvili G., Siersma D.
Banach Center Publications
|
2003
|
tom 62
|
nr 1
159-172
EN
We describe the structure of minimal round functions on compact closed surfaces and three-dimensional manifolds. The minimal possible number of critical loops is determined and typical non-equisingular round function germs are interpreted in the spirit of isolated line singularities. We also discuss a version of Lusternik-Schnirelmann theory suitable for round functions.
4
Content available remote

Critical configurations of planar robot arms

51%
Khimshiashvili G., Panina G., Siersma D., Zhukova A.
Open Mathematics
|
2013
|
tom 11
|
nr 3
519-529
EN
It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, P can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or robot arms. We introduce the notion of the oriented area for an open polygonal chain, prove that critical points are exactly the cyclic configurations with antipodal endpoints and derive a formula for the Morse index of a critical configuration.
5
Content available remote

Curvatures of conflict surfaces in Euclidean 3-space

51%
Sotomayor J., Siersma D., Garcia R.
Banach Center Publications
|
1999
|
tom 50
|
nr 1
277-285
EN
This article extends to three dimensions results on the curvature of the conflict curve for pairs of convex sets of the plane, established by Siersma [3]. In the present case a conflict surface arises, equidistant from the given convex sets. The Gaussian, mean curvatures and the location of umbilic points on the conflict surface are determined here. Initial results on the Darbouxian type of umbilic points on conflict surfaces are also established. The results are expressed in terms of the principal directions and on the curvatures of the borders of the given convex sets.
6
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Configuration spaces and limits of voronoi diagrams

51%
Lindenbergh R., van der Kallen W., Siersma D.
Banach Center Publications
|
2003
|
tom 62
|
nr 1
183-195
EN
The Voronoi diagram of n distinct generating points divides the plane into cells, each of which consists of points most close to one particular generator. After introducing 'limit Voronoi diagrams' by analyzing diagrams of moving and coinciding points, we define compactifications of the configuration space of n distinct, labeled points. On elements of these compactifications we define Voronoi diagrams.
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