Geoscience Reference
In-Depth Information
The problem is interesting not only for Euclidean circles and spheres but also for
all unit balls derived from a norm. In this section we consider such hypersphere
location problems for different types of norms and different objective functions.
Note that circle location deals with finding a circle in
2 minimizing the
distances from its circumference to a set of points in the plane. For circle location,
more and stronger results are known than for general hypersphere location; it will
hence be treated separately where appropriate.
R
7.4.1
Applications
Hyperspheres and circles are mathematical objects which are well-known for
hundreds of years. The Rhind Mathematical Papyrus, written around 1650 BC by
Egyptian mathematicians, already contains a method for approximating the surface
area of a circle, see Robins and Shute ( 1987 ). The problem of fitting a circle or a
sphere to a set of data points has also been mentioned in the fourth century BC by
notes of Aristotle on the earth's sphericity, see Dicks ( 1985 ).
Also nowadays, the location of circles and spheres has applications in different
fields. The Euclidean version of the problem is of major interest in measurement
science, where it is used as a model for the out-of-roundness problem which occurs
in quality control and consists of deciding whether or not the roundness of a
manufactured part is in the normal range (see, e.g., Farago and Curtis 1994 ;Ventura
and Yeralan 1989 ; Yeralan and Ventura 1988 ). To this end, measurements are taken
along the boundary of the manufactured part. In order to evaluate the roundness of
the part, a circle is searched which fits the measurements. Mathematical models for
different variants of the out-of-roundness problem are studied for instance in Le and
Lee ( 1991 ), Swanson et al. ( 1995 ), and Sun ( 2009 ).
Circle and hypersphere location problems have also applications in other dis-
ciplines, e.g., in particle physics (Moura and Kitney 1992 ;Crawford 1983 )when
fitting a circular trajectory to a large number of electrically charged particles
within uniform magnetic fields, or in archeology where minmax circles are used to
estimate the diameter of an ancient shard (Chernov and Sapirstein 2008 ). In Suzuki
( 2005 ), the construction of ring roads is mentioned as an application. Many further
applications are collected in Nievergelt ( 2010 ). They include
￿
the analysis of the design and layout of structures in archeology,
￿
the analysis of megalithic monuments in history,
￿
the identification of the shape of planetary surfaces in astronomy,
￿
computer graphics and vision,
￿
calibration of microwave devices in electrical engineering,
￿
measurement of the efficiency of turbines in mechanical engineering,
￿
monitoring of deformations in structural engineering, or
￿
the identification of particles in accelerators in particle physics.
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