In mathematics, a Cayley–Klein metric is a
metric on the complement of a fixed
quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is d ...
in a
projective space which is defined using a
cross-ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, t ...
. The construction originated with
Arthur Cayley
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics.
As a child, Cayley enjoyed solving complex maths problems ...
's essay "On the theory of distance"
[Cayley (1859), p 82, §§209 to 229] where he calls the quadric the absolute. The construction was developed in further detail by
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
in papers in 1871 and 1873, and subsequent books and papers. The Cayley–Klein metrics are a unifying idea in geometry since the method is used to provide metrics in
hyperbolic geometry,
elliptic geometry, and
Euclidean geometry. The field of
non-Euclidean geometry rests largely on the footing provided by Cayley–Klein metrics.
Foundations
The
algebra of throws by
Karl von Staudt (1847) is an approach to geometry that is independent of
metric. The idea was to use the relation of
projective harmonic conjugates and
cross-ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, t ...
s as fundamental to the measure on a line. Another important insight was the
Laguerre formula by
Edmond Laguerre (1853), who showed that the Euclidean angle between two lines can be expressed as the
logarithm of a cross-ratio. Eventually, Cayley (1859) formulated relations to express distance in terms of a projective metric, and related them to general quadrics or
conic
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a speci ...
s serving as the ''absolute'' of the geometry. Klein (1871, 1873) removed the last remnants of metric concepts from von Staudt's work and combined it with Cayley's theory, in order to base Cayley's new metric on logarithm and the cross-ratio as a number generated by the geometric arrangement of four points. This procedure is necessary to avoid a circular definition of distance if cross-ratio is merely a double ratio of previously defined distances. In particular, he showed that non-Euclidean geometries can be based on the Cayley–Klein metric.
[Campo & Papadopoulos (2014)]
Cayley–Klein geometry is the study of the
group of motions that leave the Cayley–Klein metric
invariant. It depends upon the selection of a quadric or conic that becomes the ''absolute'' of the space. This group is obtained as the
collineation
In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is thus ...
s for which the absolute is
stable. Indeed, cross-ratio is invariant under any collineation, and the stable absolute enables the metric comparison, which will be equality. For example, the
unit circle is the absolute of the
Poincaré disk model and the
Beltrami–Klein model in
hyperbolic geometry. Similarly, the
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
is the absolute of the
Poincaré half-plane model.
The extent of Cayley–Klein geometry was summarized by Horst and Rolf Struve in 2004:
:There are three absolutes in the real projective line, seven in the real projective plane, and 18 in real projective space. All classical non-euclidean projective spaces as hyperbolic, elliptic, Galilean and Minkowskian and their duals can be defined this way.
Cayley-Klein
Voronoi diagrams are affine diagrams with linear
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperp ...
bisectors.
Cross ratio and distance
Suppose that ''Q'' is a fixed quadric in projective space that becomes the ''absolute'' of that geometry. If ''a'' and ''b'' are 2 points then the line through ''a'' and ''b'' intersects the quadric ''Q'' in two further points ''p'' and ''q''. The Cayley–Klein distance ''d''(''a'',''b'') from ''a'' to ''b'' is proportional to the logarithm of the
cross-ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, t ...
:
for some fixed constant ''C''.
When ''C'' is real, it represents the hyperbolic distance of
hyperbolic geometry, when imaginary it relates to
elliptic geometry. The absolute can also be expressed in terms of arbitrary quadrics or
conic
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a speci ...
s having the form in
homogeneous coordinates:
(where ''α'',''β''=1,2,3 relates to the plane and ''α'',''β''=1,2,3,4 to space), thus:
The corresponding hyperbolic distance is (with ''C''=1/2 for simplification):
or in elliptic geometry (with ''C'' = ''i''/2 for simplification)
Normal forms of the absolute
Any
quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is d ...
(or surface of second order) with real coefficients of the form
can be transformed into normal or canonical forms in terms of sums of squares, while the difference in the number of positive and negative signs doesn't change under a real homogeneous transformation of determinant ≠ 0 by
Sylvester's law of inertia, with the following classification ("zero-part" means real equation of the quadric, but no real points):
- Proper surfaces of second order.
- . Zero-part surface.
-
. Oval surface.
- Ellipsoid
- Elliptic
paraboloid
In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry.
Every plane ...
- Two-sheet hyperboloid
-
. Ring surface.
- One-sheet hyperboloid
- Hyperbolic paraboloid
- Conic surfaces of second order.
-
. Zero-part cone.
- Zero-part
cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines conn ...
- Zero-part cylinder
-
. Ordinary cone.
- Cone
- Elliptic cylinder
- Parabolic cylinder
- Hyperbolic cylinder
- Plane pairs.
-
. Conjugate imaginary plane pairs.
- Mutually intersecting imaginary planes.
- Parallel imaginary planes.
-
. Real plane pairs.
- Mutually intersecting planes.
- Parallel planes.
- One plane is finite, the other one infinitely distant, thus not existent from the affine point of view.
- Double counting planes.
-
.
- Double counting finite plane.
- Double counting infinitely distant plane, not existent in affine geometry.
The
collineation
In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is thus ...
s leaving invariant these forms can be related to
linear fractional transformation
In mathematics, a linear fractional transformation is, roughly speaking, a transformation of the form
:z \mapsto \frac ,
which has an inverse. The precise definition depends on the nature of , and . In other words, a linear fractional transfo ...
s or
Möbius transformations. Such forms and their transformations can now be applied to several kinds of spaces, which can be unified by using a parameter ''ε'' (where ''ε''=0 for Euclidean geometry, ''ε''=1 for elliptic geometry, ''ε''=−1 for hyperbolic geometry), so that the equation in the plane becomes
and in space
. For instance, the absolute for the Euclidean plane can now be represented by
.
The elliptic plane or space is related to zero-part surfaces in homogeneous coordinates:
or using inhomogeneous coordinates