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Abstract
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The equidistant set of two nonempty subsets
and
in the
Euclidean plane is the set of all points that have the same distance from
and
. Since the
classical conics can be also given in this way, equidistant sets can be considered as one of their
generalizations:
and
are called the focal sets. The points of an equidistant set are difficult to determine in
general because there are no simple formulas to compute the distance between a point
and a set. As a simplification of the general problem, we are going to investigate
equidistant sets with finite focal sets. The main result is the characterization of the
equidistant points in terms of computable constants and parametrization. The
process is presented by a Maple algorithm. Its motivation is a kind of continuity
property of equidistant sets. Therefore we can approximate the equidistant points of
and
with the equidistant
points of finite subsets
and
.
Such an approximation can be applied to the computer simulation, as some examples
show in the last section.
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Keywords
generalized conics, equidistant sets
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Mathematical Subject Classification 2010
Primary: 51M04
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Milestones
Received: 21 August 2016
Revised: 26 January 2017
Accepted: 4 February 2017
Published: 17 September 2017
Communicated by Michael Dorff
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