Namespaces
Variants
Actions

Difference between revisions of "Tetracyclic coordinates"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
 +
{{TEX|done}}
 
''of a point in the plane''
 
''of a point in the plane''
  
A kind of homogeneous coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092480/t0924801.png" /> for a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092480/t0924802.png" /> in the complex inversive plane. The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092480/t0924803.png" />, not all zero, are connected by the relation
+
A kind of homogeneous coordinates $x_0:x_1:x_2:x_3$ for a point $(x)$ in the complex inversive plane. 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/t/t092/t092480/t0924804.png" /></td> </tr></table>
+
$$(xx)\equiv x_0^2+x_1^2+x_2^2+x_3^2=0.$$
  
All points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092480/t0924805.png" /> which satisfy a linear equation
+
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/t/t092/t092480/t0924806.png" /></td> </tr></table>
+
$$(yx)\equiv y_0x_0+y_1x_1+y_2x_2+y_3x_3=0$$
  
are said to form a circle with  "coordinates"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092480/t0924807.png" />. Two circles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092480/t0924808.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092480/t0924809.png" /> are orthogonal if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092480/t09248010.png" />, tangent if
+
are said to form a circle with  "coordinates"  $(y)$. Two circles $(y)$ and $(z)$ are orthogonal if $(yz)=0$, 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/t/t092/t092480/t09248011.png" /></td> </tr></table>
+
$$(yy)(zz)-(yz)^2=0.$$
  
If two circles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092480/t09248012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092480/t09248013.png" /> intersect, the expression
+
If two circles $(y)$ 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/t/t092/t092480/t09248014.png" /></td> </tr></table>
+
$$\frac{(yz)}{\sqrt{(yy)}\sqrt{(zz)}}$$
  
 
measures the cosine of their angle (or the hyperbolic cosine of their inversive distance).
 
measures the cosine of their angle (or the hyperbolic cosine of their inversive distance).
  
In three dimensions, with an extra coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092480/t09248015.png" />, one obtains the analogous [[Pentaspherical coordinates|pentaspherical coordinates]], which lead to spheres instead of circles.
+
In three dimensions, with an extra coordinate $x_4$, one obtains the analogous [[Pentaspherical coordinates|pentaspherical coordinates]], which lead to spheres instead of circles.
  
 
According to an alternative definition, involving only real numbers, tetracyclic coordinates of points and circles in the plane can be introduced using [[Stereographic projection|stereographic projection]]. Here the tetracyclic coordinates of a point in the plane are the homogeneous coordinates of the point on the sphere corresponding to it under stereographic projection. The tetracyclic coordinates of a circle in the plane are the homogeneous coordinates of the point in space that is the pole of the plane of the circle on the sphere which corresponds to the circle in the plane under stereographic projection with respect to that sphere.
 
According to an alternative definition, involving only real numbers, tetracyclic coordinates of points and circles in the plane can be introduced using [[Stereographic projection|stereographic projection]]. Here the tetracyclic coordinates of a point in the plane are the homogeneous coordinates of the point on the sphere corresponding to it under stereographic projection. The tetracyclic coordinates of a circle in the plane are the homogeneous coordinates of the point in space that is the pole of the plane of the circle on the sphere which corresponds to the circle in the plane under stereographic projection with respect to that sphere.
  
 
====Comments====
 
====Comments====
The inversive plane, also called conformal plane, is obtained by adding an ideal point  "∞"  at infinity to the plane. The name derives from the fact that with this point added, inversion in a circle becomes an everywhere well-defined  "automorphism"  of period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092480/t09248016.png" />. (Given a circle of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092480/t09248017.png" /> and centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092480/t09248018.png" />, two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092480/t09248019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092480/t09248020.png" /> correspond under inversion in this circle if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092480/t09248021.png" />.) In the inversive plane all lines pass through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092480/t09248022.png" /> and a line (of the original plane) is a circle with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092480/t09248023.png" />. Now all circles (lines) either touch or have two points of intersection.
+
The inversive plane, also called conformal plane, is obtained by adding an ideal point  "∞"  at infinity to the plane. The name derives from the fact that with this point added, inversion in a circle becomes an everywhere well-defined  "automorphism"  of period $2$. (Given a circle of radius $r$ and centre $O$, two points $P$ and $P'$ correspond under inversion in this circle if and only if $(OP)(OP')=r^2$.) In the inversive plane all lines pass through $\infty$ and a line (of the original plane) is a circle with centre at $\infty$. Now all circles (lines) either touch or have two points of intersection.
  
Inversive space (conformal space) is obtained by adding an ideal point at infinity to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092480/t09248024.png" />-space (making inversion in a sphere everywhere well-defined of period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092480/t09248025.png" />).
+
Inversive space (conformal space) is obtained by adding an ideal point at infinity to $3$-space (making inversion in a sphere everywhere well-defined of period $2$).
  
 
Any angle inverts into an equal angle, whence the terms  "conformal space"  and  "conformal plane"  for inversive space and inversive plane.
 
Any angle inverts into an equal angle, whence the terms  "conformal space"  and  "conformal plane"  for inversive space and inversive plane.

Latest revision as of 16:16, 15 April 2014

of a point in the plane

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

$$(xx)\equiv x_0^2+x_1^2+x_2^2+x_3^2=0.$$

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

$$(yx)\equiv y_0x_0+y_1x_1+y_2x_2+y_3x_3=0$$

are said to form a circle with "coordinates" $(y)$. Two circles $(y)$ and $(z)$ are orthogonal if $(yz)=0$, tangent if

$$(yy)(zz)-(yz)^2=0.$$

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

$$\frac{(yz)}{\sqrt{(yy)}\sqrt{(zz)}}$$

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

In three dimensions, with an extra coordinate $x_4$, one obtains the analogous pentaspherical coordinates, which lead to spheres instead of circles.

According to an alternative definition, involving only real numbers, tetracyclic coordinates of points and circles in the plane can be introduced using stereographic projection. Here the tetracyclic coordinates of a point in the plane are the homogeneous coordinates of the point on the sphere corresponding to it under stereographic projection. The tetracyclic coordinates of a circle in the plane are the homogeneous coordinates of the point in space that is the pole of the plane of the circle on the sphere which corresponds to the circle in the plane under stereographic projection with respect to that sphere.

Comments

The inversive plane, also called conformal plane, is obtained by adding an ideal point "∞" at infinity to the plane. The name derives from the fact that with this point added, inversion in a circle becomes an everywhere well-defined "automorphism" of period $2$. (Given a circle of radius $r$ and centre $O$, two points $P$ and $P'$ correspond under inversion in this circle if and only if $(OP)(OP')=r^2$.) In the inversive plane all lines pass through $\infty$ and a line (of the original plane) is a circle with centre at $\infty$. Now all circles (lines) either touch or have two points of intersection.

Inversive space (conformal space) is obtained by adding an ideal point at infinity to $3$-space (making inversion in a sphere everywhere well-defined of period $2$).

Any angle inverts into an equal angle, whence the terms "conformal space" and "conformal plane" for inversive space and inversive plane.

References

[a1] W.K. Clifford, "On the powers of spheres" , Mathematical Papers , Macmillan (1882) pp. 332–336
[a2] G. Darboux, "Sur les rélations entre les groupes de points, de circles et de sphères dans le plan et dans l'espace" Ann. Ecole Norm. Sup. , 1 (1872) pp. 323–392
[a3] F. Klein, "Vorlesungen über höhere Geometrie" , Chelsea, reprint (1949)
[a4] R. Lachlan, "On systems of circles and spheres" Philos. Trans. Royal Soc. London A , 177 (1886) pp. 481–625
[a5] J.L. Coolidge, "A treatise on the circle and the sphere" , Clarendon Press (1916)
[a6] F.G. Woods, "Higher geometry" , Ginn (1922)
[a7] J.B. Wilker, "Inversive geometry" C. Davis (ed.) B. Grünbaum (ed.) F.A. Sherk (ed.) , The geometric vein (Coxeter-Festschrift) , Springer (1980) pp. 374–442
[a8] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1969) pp. §6.4, §6.8
How to Cite This Entry:
Tetracyclic coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tetracyclic_coordinates&oldid=31750