Difference between revisions of "Circular symmetrization"
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| + | A geometrical transformation of an open (closed) set $ G $ | ||
| + | in the plane, relative to a ray $ \lambda $ | ||
| + | emanating from a point $ P $, | ||
| + | onto a set $ G ^ {*} $ | ||
| + | in the same plane defined as follows: 1) the intersection of $ G ^ {*} $ | ||
| + | with some circle with centre at $ P $ | ||
| + | is either empty or is the entire circle, depending on whether the intersection of $ G $ | ||
| + | with the same circle is empty or the entire circle, respectively; and 2) if the intersection of $ G $ | ||
| + | with a circle with centre at $ P $ | ||
| + | has angular Lebesgue measure $ \Phi $, | ||
| + | then the intersection of $ G ^ {*} $ | ||
| + | with the same circle is an open (closed) arc intersecting $ \lambda $, | ||
| + | symmetric about $ \lambda $ | ||
| + | and visible from $ P $ | ||
| + | at angle $ \Phi $. | ||
The above definition carries over in a natural way to the three-dimensional case (symmetrization relative to a half-plane). See also [[Symmetrization|Symmetrization]]. | The above definition carries over in a natural way to the three-dimensional case (symmetrization relative to a half-plane). See also [[Symmetrization|Symmetrization]]. | ||
Latest revision as of 16:44, 4 June 2020
A geometrical transformation of an open (closed) set $ G $
in the plane, relative to a ray $ \lambda $
emanating from a point $ P $,
onto a set $ G ^ {*} $
in the same plane defined as follows: 1) the intersection of $ G ^ {*} $
with some circle with centre at $ P $
is either empty or is the entire circle, depending on whether the intersection of $ G $
with the same circle is empty or the entire circle, respectively; and 2) if the intersection of $ G $
with a circle with centre at $ P $
has angular Lebesgue measure $ \Phi $,
then the intersection of $ G ^ {*} $
with the same circle is an open (closed) arc intersecting $ \lambda $,
symmetric about $ \lambda $
and visible from $ P $
at angle $ \Phi $.
The above definition carries over in a natural way to the three-dimensional case (symmetrization relative to a half-plane). See also Symmetrization.
References
| [1] | G. Pólya, G. Szegö, "Isoperimetric inequalities in mathematical physics" , Princeton Univ. Press (1951) |
| [2] | W.K. Hayman, "Multivalent functions" , Cambridge Univ. Press (1958) |
| [3] | J.A. Jenkins, "Univalent functions and conformal mapping" , Springer (1958) |
How to Cite This Entry:
Circular symmetrization. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Circular_symmetrization&oldid=13034
Circular symmetrization. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Circular_symmetrization&oldid=13034
This article was adapted from an original article by I.P. Mityuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article