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The branch of geometry in which properties of figures are studied that are invariant under conformal transformations (cf. [[Conformal transformation|Conformal transformation]]). The main invariant in conformal geometry is the angle between two directions.
 
The branch of geometry in which properties of figures are studied that are invariant under conformal transformations (cf. [[Conformal transformation|Conformal transformation]]). The main invariant in conformal geometry is the angle between two directions.
  
Conformal geometry is the geometry defined in Euclidean space extended by a single (ideal) point at infinity having as corresponding fundamental group of transformations the group of point transformations taking spheres into spheres. This space is called the conformal space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c0247701.png" />, and its fundamental group is called the group of conformal transformations. In conformal space, a plane is a sphere passing through the point at infinity.
+
Conformal geometry is the geometry defined in Euclidean space extended by a single (ideal) point at infinity having as corresponding fundamental group of transformations the group of point transformations taking spheres into spheres. This space is called the conformal space $  M _ {n} $,  
 +
and its fundamental group is called the group of conformal transformations. In conformal space, a plane is a sphere passing through the point at infinity.
  
This definition of conformal geometry is valid for Euclidean spaces of arbitrary dimension; in the two-dimensional case one speaks about circles instead of spheres. For dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c0247702.png" />, the transformations taking spheres into spheres account for all angle-preserving transformations (Liouville's theorem). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c0247705.png" />, the group of transformations preserving angles is larger; however, even in this case the name conformal geometry is kept for geometries having as fundamental group the group of point transformations taking circles into circles.
+
This definition of conformal geometry is valid for Euclidean spaces of arbitrary dimension; in the two-dimensional case one speaks about circles instead of spheres. For dimension $  n \geq  3 $,  
 +
the transformations taking spheres into spheres account for all angle-preserving transformations (Liouville's theorem). For $  n = 2 $,  
 +
the group of transformations preserving angles is larger; however, even in this case the name conformal geometry is kept for geometries having as fundamental group the group of point transformations taking circles into circles.
  
 
Each transformation in the fundamental group of a conformal geometry decomposes into a finite number of Euclidean motions, similarity transformations and inversions.
 
Each transformation in the fundamental group of a conformal geometry decomposes into a finite number of Euclidean motions, similarity transformations and inversions.
  
The fundamental group of the conformal geometry of the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c0247706.png" /> is isomorphic to a subgroup of the projective group, namely the subgroup of projective transformations of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c0247707.png" />-dimensional projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c0247708.png" /> taking an oval surface of the second order (an elliptic quadric) into itself, that is, the group of hyperbolic motions of the three-dimensional space. This enables one to apply to conformal geometry a similar analytic apparatus as is used in non-Euclidean geometries.
+
The fundamental group of the conformal geometry of the plane $  M _ {2} $
 +
is isomorphic to a subgroup of the projective group, namely the subgroup of projective transformations of the $  3 $-
 +
dimensional projective space $  P _ {3} $
 +
taking an oval surface of the second order (an elliptic quadric) into itself, that is, the group of hyperbolic motions of the three-dimensional space. This enables one to apply to conformal geometry a similar analytic apparatus as is used in non-Euclidean geometries.
  
Every point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c0247709.png" /> is determined by four homogeneous coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477011.png" />, or by the pseudo-vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477012.png" /> with these coordinates. Let
+
Every point of $  P _ {3} $
 +
is determined by four homogeneous coordinates $  x _ {i} $,  
 +
$  i = 1 \dots 4 $,  
 +
or by the pseudo-vector $  \mathbf x $
 +
with these coordinates. Let
  
<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/c/c024/c024770/c02477013.png" /></td> </tr></table>
+
$$
 +
( \mathbf x \mathbf y )  = \
 +
x _ {1} y _ {1} + x _ {2} y _ {2} + x _ {3} y _ {3} - x _ {4} y _ {4}  $$
  
be a form in two vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477014.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477015.png" /> be the elliptic quadric in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477016.png" /> defined by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477017.png" />, or by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477018.png" />. One has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477019.png" /> for the points outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477021.png" /> for those inside. By using the absolute <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477022.png" />, the [[Stereographic projection|stereographic projection]] can be performed, taking points on and outside the absolute into the conformal plane and its set of circles. The coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477024.png" />, of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477025.png" /> are called the [[Tetracyclic coordinates|tetracyclic coordinates]] of the points and the circles on the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477026.png" />. Since under stereographic projection points on the absolute are taking to points in the plane, while points outside the absolute are taking to circles in the plane, the group of hyperbolic motions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477027.png" /> with absolute <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477028.png" /> corresponds to the group of transformations of the plane under which points are taken to points and circles to circles, that is, the fundamental group of the conformal geometry of the plane. This group is given analytically by the formulas
+
be a form in two vectors $  \mathbf x , \mathbf y $,  
 +
and let $  K $
 +
be the elliptic quadric in $  P _ {3} $
 +
defined by the equation $  x _ {1}  ^ {2} + x _ {2}  ^ {2} + x _ {3}  ^ {2} - x _ {4}  ^ {2} = 0 $,  
 +
or by $  ( \mathbf x \mathbf x ) = 0 $.  
 +
One has $  ( \mathbf x \mathbf x ) > 0 $
 +
for the points outside $  K $
 +
and $  ( \mathbf x \mathbf x ) < 0 $
 +
for those inside. By using the absolute $  K $,  
 +
the [[Stereographic projection|stereographic projection]] can be performed, taking points on and outside the absolute into the conformal plane and its set of circles. The coordinates $  x _ {i} $,  
 +
$  i = 1 \dots 4 $,  
 +
of points of $  P _ {3} $
 +
are called the [[Tetracyclic coordinates|tetracyclic coordinates]] of the points and the circles on the plane $  M _ {2} $.  
 +
Since under stereographic projection points on the absolute are taking to points in the plane, while points outside the absolute are taking to circles in the plane, the group of hyperbolic motions in $  P _ {3} $
 +
with absolute $  K $
 +
corresponds to the group of transformations of the plane under which points are taken to points and circles to circles, that is, the fundamental group of the conformal geometry of the plane. This group is given analytically by the formulas
  
<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/c/c024/c024770/c02477029.png" /></td> </tr></table>
+
$$
 +
x _ {k}  ^ {*}  = \
 +
\sum _ { l= } 1 ^ { 4 }
 +
p _ {k}  ^ {l} x _ {l} ,\ \
 +
k = 1 \dots 4 ; \ \
 +
\mathop{\rm det}  \| p _ {k}  ^ {l} \|  \neq  0 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477031.png" /> are coordinates of a point before and after transformation, with the restriction that the expression
+
where $  x _ {i} $
 +
and $  x _ {i}  ^ {*} $
 +
are coordinates of a point before and after transformation, with the restriction that 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/c/c024/c024770/c02477032.png" /></td> </tr></table>
+
$$
 +
( \mathbf x \mathbf x )  = \
 +
x _ {1}  ^ {2} + x _ {2}  ^ {2} + x _ {3}  ^ {2} -
 +
x _ {4}  ^ {2}
 +
$$
  
 
differs from
 
differs from
  
<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/c/c024/c024770/c02477033.png" /></td> </tr></table>
+
$$
 +
( \mathbf x  ^ {*} \mathbf x  ^ {*} )  = \
 +
( x _ {1}  ^ {*} )  ^ {2} +
 +
( x _ {2}  ^ {*} )  ^ {2} +
 +
( x _ {3}  ^ {*} )  ^ {2} -
 +
( x _ {4}  ^ {*} )  ^ {2}
 +
$$
  
 
only by a factor. By setting
 
only by a factor. By setting
  
<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/c/c024/c024770/c02477034.png" /></td> </tr></table>
+
$$
 +
e  ^ {ij}  = p _ {1}  ^ {i}
 +
p _ {1}  ^ {j} + p _ {2}  ^ {i} p _ {2}  ^ {j} + p _ {3}  ^ {i} p _ {3}  ^ {j} -
 +
p _ {4}  ^ {i} p _ {4}  ^ {j} ,
 +
$$
  
 
the conditions of preservation of the quadratic form can be written as
 
the conditions of preservation of the quadratic form can be written as
  
<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/c/c024/c024770/c02477035.png" /></td> </tr></table>
+
$$
 +
- e  ^ {44}  = e  ^ {11}  = \
 +
e  ^ {22}  = e  ^ {33}  = 1 ; \ \
 +
e  ^ {ij}  = 0 \  \textrm{ if } \
 +
i \neq j .
 +
$$
  
 
Under conformal transformations, the point at infinity can be taken to any other point, therefore a circle can be taken to a line and vice versa. If it is required that the point at infinity be taken to itself, i.e. that lines be taken to lines, then the group of such transformations is the group of similarity transformations ([[Homothety|homothety]] and Euclidean motion).
 
Under conformal transformations, the point at infinity can be taken to any other point, therefore a circle can be taken to a line and vice versa. If it is required that the point at infinity be taken to itself, i.e. that lines be taken to lines, then the group of such transformations is the group of similarity transformations ([[Homothety|homothety]] and Euclidean motion).
  
The similarity subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477036.png" /> corresponds to the subgroup of hyperbolic motions leaving some given point of the absolute fixed.
+
The similarity subgroup in $  P _ {3} $
 +
corresponds to the subgroup of hyperbolic motions leaving some given point of the absolute fixed.
 +
 
 +
Another important class of conformal transformations consists of the inversions (cf. [[Inversion|Inversion]]). An inversion in  $  P _ {3} $
 +
corresponds to a polar homology, that is, a hyperbolic motion under which each pair of corresponding points  $  p $
 +
and  $  p  ^ {*} $
 +
lies on a line passing through some fixed point  $  C $
 +
outside the absolute and such that for the cross ratio the condition  $  ( p :  p  ^ {*} :  C :  N ) = - 1 $
 +
is satisfied, where  $  N $
 +
is the point of intersection of the above line with the plane that is polar to  $  C $
 +
with respect to the absolute. Just as each hyperbolic motion can be obtained as composition of a finite number of polar homologies, any conformal transformation can be obtained as composition of a finite number of inversions.
  
Another important class of conformal transformations consists of the inversions (cf. [[Inversion|Inversion]]). An inversion in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477037.png" /> corresponds to a polar homology, that is, a hyperbolic motion under which each pair of corresponding points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477039.png" /> lies on a line passing through some fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477040.png" /> outside the absolute and such that for the cross ratio the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477041.png" /> is satisfied, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477042.png" /> is the point of intersection of the above line with the plane that is polar to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477043.png" /> with respect to the absolute. Just as each hyperbolic motion can be obtained as composition of a finite number of polar homologies, any conformal transformation can be obtained as composition of a finite number of inversions.
+
The main invariant in conformal geometry on the plane is the angle  $  \phi $
 +
between two circles. It is expressed by the formula
  
The main invariant in conformal geometry on the plane is the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477044.png" /> between two circles. It is expressed by the formula
+
$$
 +
\cos  ^ {2}  \phi  = \
  
<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/c/c024/c024770/c02477045.png" /></td> </tr></table>
+
\frac{( \mathbf x \mathbf y )  ^ {2} }{( \mathbf x \mathbf x )
 +
( \mathbf y \mathbf y ) }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477047.png" /> are the vectors corresponding to the two circles in tetracyclic coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477050.png" />. In the hyperbolic geometry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477051.png" />, the angle between two circles in the plane is equal to the non-Euclidean distance between the points in space corresponding to the circles. The invariance of angles follows from that of distances. The condition of orthogonality of two circles is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477052.png" />, and the condition of tangency is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477053.png" />. If one of the circles reduces to a point, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477054.png" />, then one obtains the condition of incidence of the point and the circle, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477055.png" />.
+
where $  \mathbf x $
 +
and $  \mathbf y $
 +
are the vectors corresponding to the two circles in tetracyclic coordinates $  x _ {i} $
 +
and $  y _ {i} $,  
 +
$  i = 1 \dots 4 $.  
 +
In the hyperbolic geometry of $  P _ {3} $,  
 +
the angle between two circles in the plane is equal to the non-Euclidean distance between the points in space corresponding to the circles. The invariance of angles follows from that of distances. The condition of orthogonality of two circles is $  ( \mathbf x \mathbf y ) = 0 $,  
 +
and the condition of tangency is $  ( \mathbf x\mathbf x ) ( \mathbf y\mathbf y ) - ( \mathbf x\mathbf y )  ^ {2} = 0 $.  
 +
If one of the circles reduces to a point, $  ( \mathbf x \mathbf x ) = 0 $,  
 +
then one obtains the condition of incidence of the point and the circle, $  ( \mathbf x \mathbf y ) = 0 $.
  
The simplest figure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477056.png" /> is a pencil of circles. It is defined by an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477057.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477059.png" /> are fixed circles of the pencil. In dependence on the sign of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477060.png" />, the pencils become: a) elliptic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477061.png" />; b) hyperbolic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477062.png" />; or c) parabolic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477063.png" /> (see Fig. a).
+
The simplest figure in $  M _ {2} $
 +
is a pencil of circles. It is defined by an equation $  \mathbf t = \alpha \mathbf p + \beta \mathbf q $,  
 +
where $  \mathbf p $
 +
and $  \mathbf q $
 +
are fixed circles of the pencil. In dependence on the sign of $  \Delta = ( \mathbf p \mathbf p ) ( \mathbf q \mathbf q ) - ( \mathbf p \mathbf q )  ^ {2} $,  
 +
the pencils become: a) elliptic $  ( \Delta > 0 ) $;  
 +
b) hyperbolic $  ( \Delta < 0 ) $;  
 +
or c) parabolic $  ( \Delta = 0 ) $(
 +
see Fig. a).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024770a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024770a.gif" />
Line 51: Line 149:
 
Figure: c024770a
 
Figure: c024770a
  
In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477064.png" /> straight lines correspond to pencils of circles. An elliptic pencil corresponds to a straight line not intersecting the absolute, a hyperbolic pencil — to a straight line intersecting the absolute, and a parabolic pencil — to a straight line tangent to the absolute. Since each line of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477065.png" /> has a conjugate, each pencil in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477066.png" /> has a conjugate pencil.
+
In $  P _ {3} $
 +
straight lines correspond to pencils of circles. An elliptic pencil corresponds to a straight line not intersecting the absolute, a hyperbolic pencil — to a straight line intersecting the absolute, and a parabolic pencil — to a straight line tangent to the absolute. Since each line of $  P _ {3} $
 +
has a conjugate, each pencil in $  M _ {2} $
 +
has a conjugate pencil.
  
 
The transformations belonging to the fundamental group of the conformal geometry of the plane are given by the fractional-linear functions of a complex variable.
 
The transformations belonging to the fundamental group of the conformal geometry of the plane are given by the fractional-linear functions of a complex variable.
  
In the conformal geometry of the three-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477067.png" /> the main objects are points and spheres. They are defined by [[Pentaspherical coordinates|pentaspherical coordinates]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477069.png" />, or by a pseudo-vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477070.png" /> in the five-dimensional space. The angle between spheres is defined by the same formula as the angle between circles in the plane.
+
In the conformal geometry of the three-dimensional space $  M _ {3} $
 +
the main objects are points and spheres. They are defined by [[Pentaspherical coordinates|pentaspherical coordinates]] $  x _ {i} $,  
 +
$  i = 1 \dots 5 $,  
 +
or by a pseudo-vector $  \mathbf x $
 +
in the five-dimensional space. The angle between spheres is defined by the same formula as the angle between circles in the plane.
  
The simplest figures in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477071.png" /> are: pencils of spheres, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477072.png" />, two-parameter bundles, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477073.png" />, and three-parameter bundles, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477074.png" />, of spheres.
+
The simplest figures in $  M _ {3} $
 +
are: pencils of spheres, $  \mathbf w = \alpha \mathbf y + \beta \mathbf z $,  
 +
two-parameter bundles, $  \mathbf w = \alpha \mathbf x + \beta \mathbf y + \gamma \mathbf z $,  
 +
and three-parameter bundles, $  \mathbf w = \alpha \mathbf x + \beta \mathbf y + \gamma \mathbf z + \delta \mathbf t $,  
 +
of spheres.
  
A circle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477075.png" /> is defined by an elliptic pencil of spheres, that is, by a formula
+
A circle in $  M _ {3} $
 +
is defined by an elliptic pencil of spheres, that is, by a formula
  
<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/c/c024/c024770/c02477076.png" /></td> </tr></table>
+
$$
 +
\mathbf x  = \sum _ { i= } 1 ^ { 2 }
 +
\alpha  ^ {i} \mathbf x _ {i}  $$
  
 
under the extra condition
 
under the extra condition
  
<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/c/c024/c024770/c02477077.png" /></td> </tr></table>
+
$$
 +
( \mathbf x _ {1} \mathbf x _ {1} )
 +
( \mathbf x _ {2} \mathbf x _ {2} ) -
 +
( \mathbf x _ {1} \mathbf x _ {2} )
 +
^ {2}  > 0 .
 +
$$
  
The angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477078.png" /> between circles determined by the spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477080.png" /> and the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477081.png" /> is defined by the formula
+
The angle $  \theta $
 +
between circles determined by the spheres $  \mathbf x _ {1} $,  
 +
$  \mathbf x _ {2} $
 +
and the sphere $  \mathbf y $
 +
is defined by the formula
  
<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/c/c024/c024770/c02477082.png" /></td> </tr></table>
+
$$
 +
\cos  ^ {2}  \theta  = \
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477083.png" /> are the cofactors of the elements of the determinant formed from the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477085.png" />. The pair of circles
+
\frac{A ^ {\alpha \beta }
 +
( \mathbf x _  \alpha  \mathbf y )
 +
( \mathbf x _  \beta  \mathbf y ) }{( \mathbf y \mathbf y ) }
 +
,
 +
$$
  
<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/c/c024/c024770/c02477086.png" /></td> </tr></table>
+
where  $  A ^ {\alpha \beta } $
 +
are the cofactors of the elements of the determinant formed from the  $  A _ {\alpha \beta }  = ( \mathbf x _  \alpha  \mathbf x _  \beta  ) $,
 +
$  \alpha , \beta = 1 , 2 $.
 +
The pair of circles
 +
 
 +
$$
 +
\mathbf x  = \sum _ { i= } 1 ^ { 2 }
 +
\alpha  ^ {i} \mathbf x _ {i} \ \
 +
\textrm{ and } \  \widetilde{\mathbf x}  = \
 +
\sum _ { i= } 1 ^ { 2 }
 +
\beta  ^ {i} \widetilde{\mathbf x}  _ {i}  $$
  
 
has the absolute invariants
 
has the absolute invariants
  
<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/c/c024/c024770/c02477087.png" /></td> </tr></table>
+
$$
 +
=
 +
\frac{s  ^ {2} }{A \widetilde{A}  }
 +
\  \textrm{ and } \ \
 +
=
 +
\frac{1}{2}
 +
A  ^ {ij}
 +
\widetilde{A}  ^ {kl} S _ {ik} S _ {jl} ,
 +
$$
  
 
where
 
where
  
<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/c/c024/c024770/c02477088.png" /></td> </tr></table>
+
$$
 +
A _ {ij}  = ( \mathbf x _ {i} \mathbf x _ {j} ) ,\  \widetilde{A}  _ {ij}  = ( \widetilde{\mathbf x}  _ {i} \widetilde{\mathbf x}  _ {j} ) ,\ \
 +
=   \mathop{\rm det}  \| A _ {ij} \| ,
 +
$$
  
<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/c/c024/c024770/c02477089.png" /></td> </tr></table>
+
$$
 +
\widetilde{A}  =   \mathop{\rm det}  \| \widetilde{A}  _ {ij} \| ,\  S _ {ij}  = ( \mathbf x _ {i} \widetilde{\mathbf x}  _ {j} ) .
 +
$$
  
For each pair of circles one can choose from the components of their pencils two principal spheres. The latter are defined by the property that for the pencils in terms of these spheres the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477090.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477092.png" /> are satisfied. The pencils themselves are defined in terms of these spheres by
+
For each pair of circles one can choose from the components of their pencils two principal spheres. The latter are defined by the property that for the pencils in terms of these spheres the conditions $  A _ {11} = A _ {22} = \widetilde{A}  _ {11} = \widetilde{A}  _ {22} = 1 $,  
 +
$  A _ {12} = \widetilde{A}  _ {12} = 0 $,  
 +
$  S _ {12} = S _ {21} = 0 $
 +
are satisfied. The pencils themselves are defined in terms of these spheres by
  
<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/c/c024/c024770/c02477093.png" /></td> </tr></table>
+
$$
 +
\mathbf x  = \mathbf x _ {1} \
 +
\cos  \phi + \mathbf x _ {2} \
 +
\sin  \phi ,\  \widetilde{\mathbf x}
 +
= \widetilde{\mathbf x}  _ {1}  \cos \
 +
\phi + \widetilde{\mathbf x}  _ {2}  \sin  \phi ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477094.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477095.png" />) is the angle between the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477096.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477097.png" />) and the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477098.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c02477099.png" />). The angles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c024770100.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c024770101.png" /> at which the principal spheres of the first circle intersect those of the second circle are called the principal angles of the circles (they are the same as the angles at which the principal spheres of the second circle intersect those of the first circle). The invariants of a pair of circles are expressed in terms of the principal angles as follows:
+
where $  \phi $(
 +
$  \phi _ {1} $)  
 +
is the angle between the sphere $  \mathbf x _ {1} $(
 +
$  \widetilde{\mathbf x}  _ {2} $)  
 +
and the sphere $  \widetilde{\mathbf x}  _ {1} $(
 +
$  \widetilde{\mathbf x}  _ {2} $).  
 +
The angles $  \theta _ {1} $
 +
and $  \theta _ {2} $
 +
at which the principal spheres of the first circle intersect those of the second circle are called the principal angles of the circles (they are the same as the angles at which the principal spheres of the second circle intersect those of the first circle). The invariants of a pair of circles are expressed in terms of the principal angles as follows:
  
<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/c/c024/c024770/c024770102.png" /></td> </tr></table>
+
$$
 +
= \cos  ^ {2} \
 +
\theta _ {1}  \cos
 +
^ {2}  \theta _ {2} ,\ \
 +
=
 +
\frac{1}{2}
  
The principal angles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c024770103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c024770104.png" /> define the extremal values of the angles that the spheres of the first circle form with those of the other. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c024770105.png" />, then for all spheres of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c024770106.png" />, and such a pair of circles is called isogonal. The mutual position of the two circles can be characterized by means of the invariants of the pair: a) linked <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c024770107.png" />; b) disjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c024770108.png" />; or c) intersecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c024770109.png" />; and the condition of linear independence of the spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c024770110.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c024770111.png" /> (see Fig. b).
+
( \cos  ^ {2}  \theta _ {1} + \cos  ^ {2} \
 +
\theta _ {2} ) .
 +
$$
 +
 
 +
The principal angles $  \theta _ {1} $
 +
and $  \theta _ {2} $
 +
define the extremal values of the angles that the spheres of the first circle form with those of the other. If $  \theta _ {1} = \theta _ {2} $,  
 +
then for all spheres of the pair $  \theta = \theta _ {1} = \theta _ {2} $,  
 +
and such a pair of circles is called isogonal. The mutual position of the two circles can be characterized by means of the invariants of the pair: a) linked $  ( 1 - 2 h + k > 0 ) $;  
 +
b) disjoint $  ( 1 - 2 h + k < 0 ) $;  
 +
or c) intersecting $  ( 1 - 2 h + k = 0 ) $;  
 +
and the condition of linear independence of the spheres $  \mathbf x _ {i} $
 +
and $  \widetilde{\mathbf x}  _ {i} $(
 +
see Fig. b).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024770b.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024770b.gif" />
Line 99: Line 283:
 
Figure: c024770b
 
Figure: c024770b
  
A necessary and sufficient condition for isogonality of a pair of circles is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024770/c024770112.png" />.
+
A necessary and sufficient condition for isogonality of a pair of circles is $  h  ^ {2} - k = 0 $.
  
 
The use of methods of mathematical analysis in conformal geometry leads to the creation of [[Conformal-differential geometry|conformal-differential geometry]]. The geometry of a space with a [[Conformal connection|conformal connection]] is constructed on the basis of conformal geometry, and this geometry is related to conformal geometry in the same way as Riemannian geometry is related to Euclidean geometry. The following terminology is also customary for conformal geometry: the geometry of inverse radii, circular geometry, inversion geometry, as well as Möbius geometry (named after A. Möbius who first studied the geometry of circular transformations).
 
The use of methods of mathematical analysis in conformal geometry leads to the creation of [[Conformal-differential geometry|conformal-differential geometry]]. The geometry of a space with a [[Conformal connection|conformal connection]] is constructed on the basis of conformal geometry, and this geometry is related to conformal geometry in the same way as Riemannian geometry is related to Euclidean geometry. The following terminology is also customary for conformal geometry: the geometry of inverse radii, circular geometry, inversion geometry, as well as Möbius geometry (named after A. Möbius who first studied the geometry of circular transformations).
Line 105: Line 289:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Klein,  "Vorlesungen über höhere Geometrie" , Springer  (1926)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Blaschke,  "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einstein's Relativitätstheorie" , '''3. Differentialgeometrie der Kreisen und Kugeln''' , Springer  (1929)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.V. Bushmanova,  A.P. Norden,  "Elements of conformal geometry" , Kazan'  (1972)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Klein,  "Vorlesungen über höhere Geometrie" , Springer  (1926)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Blaschke,  "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einstein's Relativitätstheorie" , '''3. Differentialgeometrie der Kreisen und Kugeln''' , Springer  (1929)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.V. Bushmanova,  A.P. Norden,  "Elements of conformal geometry" , Kazan'  (1972)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 17:46, 4 June 2020


The branch of geometry in which properties of figures are studied that are invariant under conformal transformations (cf. Conformal transformation). The main invariant in conformal geometry is the angle between two directions.

Conformal geometry is the geometry defined in Euclidean space extended by a single (ideal) point at infinity having as corresponding fundamental group of transformations the group of point transformations taking spheres into spheres. This space is called the conformal space $ M _ {n} $, and its fundamental group is called the group of conformal transformations. In conformal space, a plane is a sphere passing through the point at infinity.

This definition of conformal geometry is valid for Euclidean spaces of arbitrary dimension; in the two-dimensional case one speaks about circles instead of spheres. For dimension $ n \geq 3 $, the transformations taking spheres into spheres account for all angle-preserving transformations (Liouville's theorem). For $ n = 2 $, the group of transformations preserving angles is larger; however, even in this case the name conformal geometry is kept for geometries having as fundamental group the group of point transformations taking circles into circles.

Each transformation in the fundamental group of a conformal geometry decomposes into a finite number of Euclidean motions, similarity transformations and inversions.

The fundamental group of the conformal geometry of the plane $ M _ {2} $ is isomorphic to a subgroup of the projective group, namely the subgroup of projective transformations of the $ 3 $- dimensional projective space $ P _ {3} $ taking an oval surface of the second order (an elliptic quadric) into itself, that is, the group of hyperbolic motions of the three-dimensional space. This enables one to apply to conformal geometry a similar analytic apparatus as is used in non-Euclidean geometries.

Every point of $ P _ {3} $ is determined by four homogeneous coordinates $ x _ {i} $, $ i = 1 \dots 4 $, or by the pseudo-vector $ \mathbf x $ with these coordinates. Let

$$ ( \mathbf x \mathbf y ) = \ x _ {1} y _ {1} + x _ {2} y _ {2} + x _ {3} y _ {3} - x _ {4} y _ {4} $$

be a form in two vectors $ \mathbf x , \mathbf y $, and let $ K $ be the elliptic quadric in $ P _ {3} $ defined by the equation $ x _ {1} ^ {2} + x _ {2} ^ {2} + x _ {3} ^ {2} - x _ {4} ^ {2} = 0 $, or by $ ( \mathbf x \mathbf x ) = 0 $. One has $ ( \mathbf x \mathbf x ) > 0 $ for the points outside $ K $ and $ ( \mathbf x \mathbf x ) < 0 $ for those inside. By using the absolute $ K $, the stereographic projection can be performed, taking points on and outside the absolute into the conformal plane and its set of circles. The coordinates $ x _ {i} $, $ i = 1 \dots 4 $, of points of $ P _ {3} $ are called the tetracyclic coordinates of the points and the circles on the plane $ M _ {2} $. Since under stereographic projection points on the absolute are taking to points in the plane, while points outside the absolute are taking to circles in the plane, the group of hyperbolic motions in $ P _ {3} $ with absolute $ K $ corresponds to the group of transformations of the plane under which points are taken to points and circles to circles, that is, the fundamental group of the conformal geometry of the plane. This group is given analytically by the formulas

$$ x _ {k} ^ {*} = \ \sum _ { l= } 1 ^ { 4 } p _ {k} ^ {l} x _ {l} ,\ \ k = 1 \dots 4 ; \ \ \mathop{\rm det} \| p _ {k} ^ {l} \| \neq 0 , $$

where $ x _ {i} $ and $ x _ {i} ^ {*} $ are coordinates of a point before and after transformation, with the restriction that the expression

$$ ( \mathbf x \mathbf x ) = \ x _ {1} ^ {2} + x _ {2} ^ {2} + x _ {3} ^ {2} - x _ {4} ^ {2} $$

differs from

$$ ( \mathbf x ^ {*} \mathbf x ^ {*} ) = \ ( x _ {1} ^ {*} ) ^ {2} + ( x _ {2} ^ {*} ) ^ {2} + ( x _ {3} ^ {*} ) ^ {2} - ( x _ {4} ^ {*} ) ^ {2} $$

only by a factor. By setting

$$ e ^ {ij} = p _ {1} ^ {i} p _ {1} ^ {j} + p _ {2} ^ {i} p _ {2} ^ {j} + p _ {3} ^ {i} p _ {3} ^ {j} - p _ {4} ^ {i} p _ {4} ^ {j} , $$

the conditions of preservation of the quadratic form can be written as

$$ - e ^ {44} = e ^ {11} = \ e ^ {22} = e ^ {33} = 1 ; \ \ e ^ {ij} = 0 \ \textrm{ if } \ i \neq j . $$

Under conformal transformations, the point at infinity can be taken to any other point, therefore a circle can be taken to a line and vice versa. If it is required that the point at infinity be taken to itself, i.e. that lines be taken to lines, then the group of such transformations is the group of similarity transformations (homothety and Euclidean motion).

The similarity subgroup in $ P _ {3} $ corresponds to the subgroup of hyperbolic motions leaving some given point of the absolute fixed.

Another important class of conformal transformations consists of the inversions (cf. Inversion). An inversion in $ P _ {3} $ corresponds to a polar homology, that is, a hyperbolic motion under which each pair of corresponding points $ p $ and $ p ^ {*} $ lies on a line passing through some fixed point $ C $ outside the absolute and such that for the cross ratio the condition $ ( p : p ^ {*} : C : N ) = - 1 $ is satisfied, where $ N $ is the point of intersection of the above line with the plane that is polar to $ C $ with respect to the absolute. Just as each hyperbolic motion can be obtained as composition of a finite number of polar homologies, any conformal transformation can be obtained as composition of a finite number of inversions.

The main invariant in conformal geometry on the plane is the angle $ \phi $ between two circles. It is expressed by the formula

$$ \cos ^ {2} \phi = \ \frac{( \mathbf x \mathbf y ) ^ {2} }{( \mathbf x \mathbf x ) ( \mathbf y \mathbf y ) } , $$

where $ \mathbf x $ and $ \mathbf y $ are the vectors corresponding to the two circles in tetracyclic coordinates $ x _ {i} $ and $ y _ {i} $, $ i = 1 \dots 4 $. In the hyperbolic geometry of $ P _ {3} $, the angle between two circles in the plane is equal to the non-Euclidean distance between the points in space corresponding to the circles. The invariance of angles follows from that of distances. The condition of orthogonality of two circles is $ ( \mathbf x \mathbf y ) = 0 $, and the condition of tangency is $ ( \mathbf x\mathbf x ) ( \mathbf y\mathbf y ) - ( \mathbf x\mathbf y ) ^ {2} = 0 $. If one of the circles reduces to a point, $ ( \mathbf x \mathbf x ) = 0 $, then one obtains the condition of incidence of the point and the circle, $ ( \mathbf x \mathbf y ) = 0 $.

The simplest figure in $ M _ {2} $ is a pencil of circles. It is defined by an equation $ \mathbf t = \alpha \mathbf p + \beta \mathbf q $, where $ \mathbf p $ and $ \mathbf q $ are fixed circles of the pencil. In dependence on the sign of $ \Delta = ( \mathbf p \mathbf p ) ( \mathbf q \mathbf q ) - ( \mathbf p \mathbf q ) ^ {2} $, the pencils become: a) elliptic $ ( \Delta > 0 ) $; b) hyperbolic $ ( \Delta < 0 ) $; or c) parabolic $ ( \Delta = 0 ) $( see Fig. a).

Figure: c024770a

In $ P _ {3} $ straight lines correspond to pencils of circles. An elliptic pencil corresponds to a straight line not intersecting the absolute, a hyperbolic pencil — to a straight line intersecting the absolute, and a parabolic pencil — to a straight line tangent to the absolute. Since each line of $ P _ {3} $ has a conjugate, each pencil in $ M _ {2} $ has a conjugate pencil.

The transformations belonging to the fundamental group of the conformal geometry of the plane are given by the fractional-linear functions of a complex variable.

In the conformal geometry of the three-dimensional space $ M _ {3} $ the main objects are points and spheres. They are defined by pentaspherical coordinates $ x _ {i} $, $ i = 1 \dots 5 $, or by a pseudo-vector $ \mathbf x $ in the five-dimensional space. The angle between spheres is defined by the same formula as the angle between circles in the plane.

The simplest figures in $ M _ {3} $ are: pencils of spheres, $ \mathbf w = \alpha \mathbf y + \beta \mathbf z $, two-parameter bundles, $ \mathbf w = \alpha \mathbf x + \beta \mathbf y + \gamma \mathbf z $, and three-parameter bundles, $ \mathbf w = \alpha \mathbf x + \beta \mathbf y + \gamma \mathbf z + \delta \mathbf t $, of spheres.

A circle in $ M _ {3} $ is defined by an elliptic pencil of spheres, that is, by a formula

$$ \mathbf x = \sum _ { i= } 1 ^ { 2 } \alpha ^ {i} \mathbf x _ {i} $$

under the extra condition

$$ ( \mathbf x _ {1} \mathbf x _ {1} ) ( \mathbf x _ {2} \mathbf x _ {2} ) - ( \mathbf x _ {1} \mathbf x _ {2} ) ^ {2} > 0 . $$

The angle $ \theta $ between circles determined by the spheres $ \mathbf x _ {1} $, $ \mathbf x _ {2} $ and the sphere $ \mathbf y $ is defined by the formula

$$ \cos ^ {2} \theta = \ \frac{A ^ {\alpha \beta } ( \mathbf x _ \alpha \mathbf y ) ( \mathbf x _ \beta \mathbf y ) }{( \mathbf y \mathbf y ) } , $$

where $ A ^ {\alpha \beta } $ are the cofactors of the elements of the determinant formed from the $ A _ {\alpha \beta } = ( \mathbf x _ \alpha \mathbf x _ \beta ) $, $ \alpha , \beta = 1 , 2 $. The pair of circles

$$ \mathbf x = \sum _ { i= } 1 ^ { 2 } \alpha ^ {i} \mathbf x _ {i} \ \ \textrm{ and } \ \widetilde{\mathbf x} = \ \sum _ { i= } 1 ^ { 2 } \beta ^ {i} \widetilde{\mathbf x} _ {i} $$

has the absolute invariants

$$ k = \frac{s ^ {2} }{A \widetilde{A} } \ \textrm{ and } \ \ h = \frac{1}{2} A ^ {ij} \widetilde{A} ^ {kl} S _ {ik} S _ {jl} , $$

where

$$ A _ {ij} = ( \mathbf x _ {i} \mathbf x _ {j} ) ,\ \widetilde{A} _ {ij} = ( \widetilde{\mathbf x} _ {i} \widetilde{\mathbf x} _ {j} ) ,\ \ A = \mathop{\rm det} \| A _ {ij} \| , $$

$$ \widetilde{A} = \mathop{\rm det} \| \widetilde{A} _ {ij} \| ,\ S _ {ij} = ( \mathbf x _ {i} \widetilde{\mathbf x} _ {j} ) . $$

For each pair of circles one can choose from the components of their pencils two principal spheres. The latter are defined by the property that for the pencils in terms of these spheres the conditions $ A _ {11} = A _ {22} = \widetilde{A} _ {11} = \widetilde{A} _ {22} = 1 $, $ A _ {12} = \widetilde{A} _ {12} = 0 $, $ S _ {12} = S _ {21} = 0 $ are satisfied. The pencils themselves are defined in terms of these spheres by

$$ \mathbf x = \mathbf x _ {1} \ \cos \phi + \mathbf x _ {2} \ \sin \phi ,\ \widetilde{\mathbf x} = \widetilde{\mathbf x} _ {1} \cos \ \phi + \widetilde{\mathbf x} _ {2} \sin \phi , $$

where $ \phi $( $ \phi _ {1} $) is the angle between the sphere $ \mathbf x _ {1} $( $ \widetilde{\mathbf x} _ {2} $) and the sphere $ \widetilde{\mathbf x} _ {1} $( $ \widetilde{\mathbf x} _ {2} $). The angles $ \theta _ {1} $ and $ \theta _ {2} $ at which the principal spheres of the first circle intersect those of the second circle are called the principal angles of the circles (they are the same as the angles at which the principal spheres of the second circle intersect those of the first circle). The invariants of a pair of circles are expressed in terms of the principal angles as follows:

$$ k = \cos ^ {2} \ \theta _ {1} \cos ^ {2} \theta _ {2} ,\ \ h = \frac{1}{2} ( \cos ^ {2} \theta _ {1} + \cos ^ {2} \ \theta _ {2} ) . $$

The principal angles $ \theta _ {1} $ and $ \theta _ {2} $ define the extremal values of the angles that the spheres of the first circle form with those of the other. If $ \theta _ {1} = \theta _ {2} $, then for all spheres of the pair $ \theta = \theta _ {1} = \theta _ {2} $, and such a pair of circles is called isogonal. The mutual position of the two circles can be characterized by means of the invariants of the pair: a) linked $ ( 1 - 2 h + k > 0 ) $; b) disjoint $ ( 1 - 2 h + k < 0 ) $; or c) intersecting $ ( 1 - 2 h + k = 0 ) $; and the condition of linear independence of the spheres $ \mathbf x _ {i} $ and $ \widetilde{\mathbf x} _ {i} $( see Fig. b).

Figure: c024770b

A necessary and sufficient condition for isogonality of a pair of circles is $ h ^ {2} - k = 0 $.

The use of methods of mathematical analysis in conformal geometry leads to the creation of conformal-differential geometry. The geometry of a space with a conformal connection is constructed on the basis of conformal geometry, and this geometry is related to conformal geometry in the same way as Riemannian geometry is related to Euclidean geometry. The following terminology is also customary for conformal geometry: the geometry of inverse radii, circular geometry, inversion geometry, as well as Möbius geometry (named after A. Möbius who first studied the geometry of circular transformations).

References

[1] F. Klein, "Vorlesungen über höhere Geometrie" , Springer (1926)
[2] W. Blaschke, "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einstein's Relativitätstheorie" , 3. Differentialgeometrie der Kreisen und Kugeln , Springer (1929)
[3] G.V. Bushmanova, A.P. Norden, "Elements of conformal geometry" , Kazan' (1972) (In Russian)

Comments

An exhausting treatment of Möbius geometry in dimension 2 is given in [a1].

References

[a1] H. Schwerdtfeger, "Geometry of complex numbers" , Dover, reprint (1979)
How to Cite This Entry:
Conformal geometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conformal_geometry&oldid=16231
This article was adapted from an original article by G.V. Bushmanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article