Generalized Circumcenter: minimizing the range of distances from a point to the vertices of a polygon












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It is well known that the circumcenter of a polygon exists if and only if the polygon is cyclic.



I would like to extend the definition of a circumcenter for noncyclic polygons.



Let us define $c(A)$ as the range of the lengths of the distances from $A$ to the vertices of the polygon; that is, the longest minus shortest distance from $A$ to the vertices of the polygon.

 The range is chosen as a simple measure of spread.




If there exists an $A_0$ such that $0 leq c(A_0) < c(A)$ for all $A$ not equal to $A_0$, and $A_0$ is not equivalently at infinity, then this $A_0$ is defined to be the generalized circumcenter of the polygon.




Note that this generalization follows from the fact that the distances from the circumcenter to the vertices of a cyclic polygon are equal to each other.




Does the generalized circumcenter exist for all n-gons?







geometry statistics euclidean-geometry







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  • $begingroup$
    What does $A_0$ ` is not equivalently at infinity` mean?
    $endgroup$
    – sds
    Jan 2 at 21:17












  • $begingroup$
    If we define $A_0$ arbitrarily far away then we can make $c(A_0)$ as close to zero as possible, since the range of the lengths of the distances from $A_0$ to the vertices of the polygon would increase as $A_0$ tends to infinity. This is not what we want, so I have excluded it in the definition.
    $endgroup$
    – pacosta
    Jan 5 at 0:21
















1












$begingroup$


It is well known that the circumcenter of a polygon exists if and only if the polygon is cyclic.



I would like to extend the definition of a circumcenter for noncyclic polygons.



Let us define $c(A)$ as the range of the lengths of the distances from $A$ to the vertices of the polygon; that is, the longest minus shortest distance from $A$ to the vertices of the polygon.

 The range is chosen as a simple measure of spread.




If there exists an $A_0$ such that $0 leq c(A_0) < c(A)$ for all $A$ not equal to $A_0$, and $A_0$ is not equivalently at infinity, then this $A_0$ is defined to be the generalized circumcenter of the polygon.




Note that this generalization follows from the fact that the distances from the circumcenter to the vertices of a cyclic polygon are equal to each other.




Does the generalized circumcenter exist for all n-gons?







geometry statistics euclidean-geometry







share|cite|improve this question











$endgroup$












  • $begingroup$
    What does $A_0$ ` is not equivalently at infinity` mean?
    $endgroup$
    – sds
    Jan 2 at 21:17












  • $begingroup$
    If we define $A_0$ arbitrarily far away then we can make $c(A_0)$ as close to zero as possible, since the range of the lengths of the distances from $A_0$ to the vertices of the polygon would increase as $A_0$ tends to infinity. This is not what we want, so I have excluded it in the definition.
    $endgroup$
    – pacosta
    Jan 5 at 0:21














1












1








1


0



$begingroup$


It is well known that the circumcenter of a polygon exists if and only if the polygon is cyclic.



I would like to extend the definition of a circumcenter for noncyclic polygons.



Let us define $c(A)$ as the range of the lengths of the distances from $A$ to the vertices of the polygon; that is, the longest minus shortest distance from $A$ to the vertices of the polygon.

 The range is chosen as a simple measure of spread.




If there exists an $A_0$ such that $0 leq c(A_0) < c(A)$ for all $A$ not equal to $A_0$, and $A_0$ is not equivalently at infinity, then this $A_0$ is defined to be the generalized circumcenter of the polygon.




Note that this generalization follows from the fact that the distances from the circumcenter to the vertices of a cyclic polygon are equal to each other.




Does the generalized circumcenter exist for all n-gons?







geometry statistics euclidean-geometry







share|cite|improve this question











$endgroup$




It is well known that the circumcenter of a polygon exists if and only if the polygon is cyclic.



I would like to extend the definition of a circumcenter for noncyclic polygons.



Let us define $c(A)$ as the range of the lengths of the distances from $A$ to the vertices of the polygon; that is, the longest minus shortest distance from $A$ to the vertices of the polygon.

 The range is chosen as a simple measure of spread.




If there exists an $A_0$ such that $0 leq c(A_0) < c(A)$ for all $A$ not equal to $A_0$, and $A_0$ is not equivalently at infinity, then this $A_0$ is defined to be the generalized circumcenter of the polygon.




Note that this generalization follows from the fact that the distances from the circumcenter to the vertices of a cyclic polygon are equal to each other.




Does the generalized circumcenter exist for all n-gons?







geometry statistics euclidean-geometry




geometry statistics euclidean-geometry






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 2 at 23:01









Blue

49.3k870157




49.3k870157










asked Jan 2 at 21:08









pacostapacosta

1535




1535












  • $begingroup$
    What does $A_0$ ` is not equivalently at infinity` mean?
    $endgroup$
    – sds
    Jan 2 at 21:17












  • $begingroup$
    If we define $A_0$ arbitrarily far away then we can make $c(A_0)$ as close to zero as possible, since the range of the lengths of the distances from $A_0$ to the vertices of the polygon would increase as $A_0$ tends to infinity. This is not what we want, so I have excluded it in the definition.
    $endgroup$
    – pacosta
    Jan 5 at 0:21


















  • $begingroup$
    What does $A_0$ ` is not equivalently at infinity` mean?
    $endgroup$
    – sds
    Jan 2 at 21:17












  • $begingroup$
    If we define $A_0$ arbitrarily far away then we can make $c(A_0)$ as close to zero as possible, since the range of the lengths of the distances from $A_0$ to the vertices of the polygon would increase as $A_0$ tends to infinity. This is not what we want, so I have excluded it in the definition.
    $endgroup$
    – pacosta
    Jan 5 at 0:21
















$begingroup$
What does $A_0$ ` is not equivalently at infinity` mean?
$endgroup$
– sds
Jan 2 at 21:17






$begingroup$
What does $A_0$ ` is not equivalently at infinity` mean?
$endgroup$
– sds
Jan 2 at 21:17














$begingroup$
If we define $A_0$ arbitrarily far away then we can make $c(A_0)$ as close to zero as possible, since the range of the lengths of the distances from $A_0$ to the vertices of the polygon would increase as $A_0$ tends to infinity. This is not what we want, so I have excluded it in the definition.
$endgroup$
– pacosta
Jan 5 at 0:21




$begingroup$
If we define $A_0$ arbitrarily far away then we can make $c(A_0)$ as close to zero as possible, since the range of the lengths of the distances from $A_0$ to the vertices of the polygon would increase as $A_0$ tends to infinity. This is not what we want, so I have excluded it in the definition.
$endgroup$
– pacosta
Jan 5 at 0:21










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$begingroup$

Assuming by "range" you mean the maximum distance from $A_0$ to the vertices of the polygon, your "generalized circumcenter" is actually the center of the minimum covering circle



It is indeed unique: if there were two such points and circles with the same radius all the points then they would overlap and the a circle centered on that overlap would have a strictly smaller radius



It is not really a circumcenter. For example with an obtuse-angled triangle, the minimum covering circle is smaller than the circumcircle, the center of the minimum covering circle is the midpoint of the longest edge, and the minimum covering circle only passes through two of the vertices






share|cite|improve this answer









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    1 Answer
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    $begingroup$

    Assuming by "range" you mean the maximum distance from $A_0$ to the vertices of the polygon, your "generalized circumcenter" is actually the center of the minimum covering circle



    It is indeed unique: if there were two such points and circles with the same radius all the points then they would overlap and the a circle centered on that overlap would have a strictly smaller radius



    It is not really a circumcenter. For example with an obtuse-angled triangle, the minimum covering circle is smaller than the circumcircle, the center of the minimum covering circle is the midpoint of the longest edge, and the minimum covering circle only passes through two of the vertices






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      Assuming by "range" you mean the maximum distance from $A_0$ to the vertices of the polygon, your "generalized circumcenter" is actually the center of the minimum covering circle



      It is indeed unique: if there were two such points and circles with the same radius all the points then they would overlap and the a circle centered on that overlap would have a strictly smaller radius



      It is not really a circumcenter. For example with an obtuse-angled triangle, the minimum covering circle is smaller than the circumcircle, the center of the minimum covering circle is the midpoint of the longest edge, and the minimum covering circle only passes through two of the vertices






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        Assuming by "range" you mean the maximum distance from $A_0$ to the vertices of the polygon, your "generalized circumcenter" is actually the center of the minimum covering circle



        It is indeed unique: if there were two such points and circles with the same radius all the points then they would overlap and the a circle centered on that overlap would have a strictly smaller radius



        It is not really a circumcenter. For example with an obtuse-angled triangle, the minimum covering circle is smaller than the circumcircle, the center of the minimum covering circle is the midpoint of the longest edge, and the minimum covering circle only passes through two of the vertices






        share|cite|improve this answer









        $endgroup$



        Assuming by "range" you mean the maximum distance from $A_0$ to the vertices of the polygon, your "generalized circumcenter" is actually the center of the minimum covering circle



        It is indeed unique: if there were two such points and circles with the same radius all the points then they would overlap and the a circle centered on that overlap would have a strictly smaller radius



        It is not really a circumcenter. For example with an obtuse-angled triangle, the minimum covering circle is smaller than the circumcircle, the center of the minimum covering circle is the midpoint of the longest edge, and the minimum covering circle only passes through two of the vertices







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 2 at 22:51









        HenryHenry

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        101k482169






























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