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Difference between revisions of "Pentaspherical coordinates"

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A kind of homogeneous coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072010/p0720101.png" /> for a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072010/p0720102.png" /> in complex inversive space. The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072010/p0720103.png" />, not all zero, are connected by the relation
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A kind of homogeneous coordinates $x_0:x_1:x_2:x_3:x_4$ for a point $(x)$ in complex inversive space. The numbers $x_\nu$, not all zero, are connected by the relation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072010/p0720104.png" /></td> </tr></table>
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$$(x,x)\equiv x_0^2+x_1^2+x_2^2+x_3^2+x_4^2=0.$$
  
All points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072010/p0720105.png" /> which satisfy a linear equation
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All points $(x)$ which satisfy a linear equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072010/p0720106.png" /></td> </tr></table>
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$$(y,x)\equiv y_0x_0+y_1x_1+y_2x_2+y_3x_3+y_4x_4=0$$
  
are said to form a [[Sphere|sphere]], with coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072010/p0720107.png" />. Two spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072010/p0720108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072010/p0720109.png" /> are orthogonal if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072010/p07201010.png" />, tangent if
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are said to form a [[Sphere|sphere]], with coordinates $(y)$. Two spheres $(y)$ and $(z)$ are orthogonal if $(y,z)=0$, tangent if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072010/p07201011.png" /></td> </tr></table>
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$$(y,y)(z,z)-(y,z)^2=0.$$
  
If two spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072010/p07201012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072010/p07201013.png" /> intersect, the expression
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If two spheres $(y)$ and $(z)$ intersect, the expression
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072010/p07201014.png" /></td> </tr></table>
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$$\frac{(y,z)}{\sqrt{(y,y)}\sqrt{(z,z)}}$$
  
 
measures the cosine of their angle (or the hyperbolic cosine of their inverse distance).
 
measures the cosine of their angle (or the hyperbolic cosine of their inverse distance).
  
Setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072010/p07201015.png" />, one obtains the analogous [[Tetracyclic coordinates|tetracyclic coordinates]], which lead to circles instead of spheres.
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Setting $x_4=0$, one obtains the analogous [[Tetracyclic coordinates|tetracyclic coordinates]], which lead to circles instead of spheres.
  
Completely analogous constructions can be performed for spaces of higher dimensions, which give polyspherical coordinates. In the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072010/p07201016.png" />-dimensional case they are called hexaspherical coordinates. Polyspherical coordinates are used in conformal geometry in examining manifolds of figures.
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Completely analogous constructions can be performed for spaces of higher dimensions, which give polyspherical coordinates. In the $4$-dimensional case they are called hexaspherical coordinates. Polyspherical coordinates are used in conformal geometry in examining manifolds of figures.
  
 
====References====
 
====References====

Revision as of 16:26, 15 April 2014

A kind of homogeneous coordinates $x_0:x_1:x_2:x_3:x_4$ for a point $(x)$ in complex inversive space. The numbers $x_\nu$, not all zero, are connected by the relation

$$(x,x)\equiv x_0^2+x_1^2+x_2^2+x_3^2+x_4^2=0.$$

All points $(x)$ which satisfy a linear equation

$$(y,x)\equiv y_0x_0+y_1x_1+y_2x_2+y_3x_3+y_4x_4=0$$

are said to form a sphere, with coordinates $(y)$. Two spheres $(y)$ and $(z)$ are orthogonal if $(y,z)=0$, tangent if

$$(y,y)(z,z)-(y,z)^2=0.$$

If two spheres $(y)$ and $(z)$ intersect, the expression

$$\frac{(y,z)}{\sqrt{(y,y)}\sqrt{(z,z)}}$$

measures the cosine of their angle (or the hyperbolic cosine of their inverse distance).

Setting $x_4=0$, one obtains the analogous tetracyclic coordinates, which lead to circles instead of spheres.

Completely analogous constructions can be performed for spaces of higher dimensions, which give polyspherical coordinates. In the $4$-dimensional case they are called hexaspherical coordinates. Polyspherical coordinates are used in conformal geometry in examining manifolds of figures.

References

[1] F. Klein, "Vorlesungen über höhere Geometrie" , Springer (1926)
[2] G.V. Bushmanova, A.P. Norden, "Elements of conformal geometry" , Kazan' (1972) (In Russian)


Comments

References

[a1] M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French)
[a2] J.L. Coolidge, "A treatise on the circle and the sphere" , Chelsea, reprint (1971)
How to Cite This Entry:
Pentaspherical coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pentaspherical_coordinates&oldid=15120
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article