Abstract

Combining methods from analysis and (convex) geometry, we derive geometric properties of generalizations of the following four classical types of bifocal curves: ellipses, hyperbolas, Cassini curves, and Apollonius circles. It is natural to consider them together, since the points of one such curve have (in the same sequence) constant sum, constant difference, constant product, and constant ratio of distances to the two given foci. Our generalization of this classical concept is two-fold: first we extend the used distance functions from the Euclidean and the (in the literature only partially investigated) normed case to gauges, where the unit ball no longer has to be symmetric with respect to the origin. And second we study, in the cases where it makes sense (ellipses and Cassini curves), even multifocal curves. Most of our investigations are even extended to higher dimensions. We also survey known results about these classes of curves which are close to our purpose (namely, referring to the normed case, to multifocality, and to higher dimensions), since no comparable survey exists and these results are widespread and not systematized in the literature.

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Keywords

  • Apollonius circle
  • bifocal curve
  • Birkhoff orthogonality
  • multifocal Cassini curve
  • convex distance function
  • multifocal ellipse
  • gauge
  • generalized hyperbola
  • generalized Minkowski space
  • multifocal curve
  • n-lemniscate
  • sublevel set
  • Voronoi cell

Mathematics Subject Classification

14H45, 14H50, 46B20, 46C15, 49J53, 51M04, 52A10, 52A21, 53A04, 53A05

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Suggested citation

T. Jahn, H. Martini, C. Richter. “Bi- and Multifocal Curves and Surfaces for Gauges.” Journal of Convex Analysis 23 (2016), No. 3, 733–774. https://doi.org/10.68381/jca23029

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