Co-Euclidean space

dual Euclidean space

The space obtained from a Euclidean space by applying the duality principle for a projective space of the same dimension. It is denoted by $ \mathbf R _ {n} ^ {*} $, where $ n $ is the dimension of the space. The co-Euclidean space $ \mathbf R _ {n} ^ {*} $ is a space with a projective metric, defined in accordance with the general scheme of introducing projective metrics. If the projective metric of the Euclidean space $ \mathbf R _ {n} $ is defined by an absolute consisting of an $ ( n - 1 ) $- plane and an $ ( n - 2 ) $- imaginary quadric in this plane, then the projective metric of the co-Euclidean space $ \mathbf R _ {n} ^ {*} $ is defined by the dual absolute: a second-order imaginary cone, called the absolute cone, with as vertex the absolute point of the absolute.

The distance between two points in $ \mathbf R _ {n} ^ {*} $ is defined in accordance with the general scheme for the definition of the distance between points in a space with a projective metric, taking into account the dual character of this space with respect to $ \mathbf R _ {n} $. Let

$$ ( \mathbf u , \mathbf x ) + u _ {0} = 0 ,\ \ ( \mathbf v , \mathbf y ) + v _ {0} = 0 $$

be the normal equations for some planes in the Euclidean space $ \mathbf R _ {n} $, dual to $ \mathbf R _ {n} ^ {*} $, where

$$ ( \mathbf u , \mathbf u ) = 1 ,\ \ ( \mathbf v , \mathbf v ) = 1 , $$

and $ ( \mathbf u , \mathbf x ) $ is the scalar product of vectors in $ \mathbf R _ {n} $. One associates with these planes the points $ X ( x ^ {0} , \mathbf x ) $ and $ Y ( y ^ {0} , \mathbf y ) $ in $ \mathbf R _ {n} ^ {*} $ with coordinates

$$ x ^ {0} = \rho u _ {0} ,\ \ x ^ {i} = \rho u _ {i} ,\ \ y ^ {0} = \rho v _ {0} ,\ \ y ^ {i} = \rho v _ {i} ,\ \ \rho \in \mathbf R , $$

The coordinates of these points being normalized by the conditions

$$ ( \mathbf x , \mathbf x ) = \ \rho ^ {2} > 0 ,\ \ ( \mathbf y , \mathbf y ) = \ \rho ^ {2} > 0 $$

( $ x ^ {0} $ and $ y ^ {0} $ are the coordinates of the points $ X $ and $ Y $ in the improper plane at infinity). The distance $ \delta $ between $ X $ and $ Y $ is defined by the relation

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

in other words, it is expressed in terms of the angle between the planes dual to $ X $ and $ Y $. In accordance with the normalization of the vectors of the points $ X $ and $ Y $, this relation can be written as

$$ \cos \ \frac \delta \rho = \ \frac{1}{\rho ^ {2} } | ( \mathbf x , \mathbf y ) | . $$

The real number $ \rho $ is called the radius of curvature of the co-Euclidean space.

In the case when the points $ X , Y \in \mathbf R _ {n} ^ {*} $ correspond to parallel planes in the dual space $ \mathbf R _ {n} $, $ \delta = 0 $ and the distance between the points $ X $ and $ Y $ is defined as the Euclidean distance between these parallel planes.

The angle between two planes in $ \mathbf R _ {n} ^ {*} $ is defined as the normalized Euclidean distance between the corresponding two points in $ \mathbf R _ {n} $ according to the duality principle. This angle is also equal to the normalized distance between the points of the given planes in $ \mathbf R _ {n} ^ {*} $ that are the poles of the $ ( n - 2 ) $- plane of their intersection with respect to the quadrics cut out on these planes by the absolute cone. In this connection, one is always defining the angle between the planes that does not contain the absolute point. In particular, the angle between two straight lines in the co-Euclidean plane $ \mathbf R _ {2} ^ {*} $ is equal to the normalized distance between those two points of these lines that, together with the point of intersection of the given lines, harmonically divided the points of intersection of the lines with the absolute lines.

The motions of the co-Euclidean space $ \mathbf R _ {n} ^ {*} $ are defined as the transformations of this space induced by the motions of the corresponding dual space $ \mathbf R _ {n} $; thus, the motions of $ \mathbf R _ {n} ^ {*} $ are described by the orthogonal operators.

The geometry of the co-Euclidean plane $ \mathbf R _ {2} ^ {*} $ has properties dual to those of the plane $ \mathbf R _ {2} $. E.g., it follows from the invariance of the lengths of the sides of a triangle $ A B C $ under a motion of $ \mathbf R _ {2} $ that the angular excess $ \Delta = \widehat{A} + \widehat{B} - \widehat{C} $ is invariant under motions of $ \mathbf R _ {2} ^ {*} $ and is always positive. (Here and in what follows, it is supposed that the interior angle $ B $ of the triangle $ A B C $ in $ \mathbf R _ {2} ^ {*} $ contains the absolute point.) For the area $ S $ of a triangle in $ \mathbf R _ {2} ^ {*} $ one takes the quantity proportional to the angular excess, which is an additive function of the triangle: $ S = \rho ^ {2} \Delta $.

As a consequence of the dual character between sizes of the angles and lengths of the sides of a triangle in $ \mathbf R _ {2} ^ {*} $, there are the following trigonometric relations for a triangle $ A B C $:

$$ b = a + c , $$

$$ \widehat{A} {} ^ {2} = \widehat{B} {} ^ {2} + \widehat{C} {} ^ {2} - 2 \widehat{C} \widehat{B} \cos \frac{a} \rho , $$

$$ \frac{\sin a / \rho }{\widehat{A} } = \frac{\sin \ b / \rho }{\widehat{B} } = \frac{\sin c / \rho }{\widehat{C} } . $$

In the plane $ \mathbf R _ {2} ^ {*} $, the distance metric (on straight lines) is projective elliptic; the angle metric is parabolic. In the space $ \mathbf R _ {3} ^ {*} $, the projective distance metric (on straight lines) is elliptic; in planes, it is again elliptic; while in pencils of planes, it is parabolic.

The co-Euclidean space $ \mathbf R _ {n} ^ {*} $ is a limiting case both of the elliptic space and the Lobachevskii space: The projective metric of $ \mathbf R _ {n} ^ {*} $ can be obtained by limit transition from the projective metrics of the above spaces.

References

[1]	B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)

References

[a1]	D.M.Y. Sommerville, "Non-Euclidean geometry" , Dover, reprint (1958)