Describing relations of hyperbolic geometry,
Franz Taurinus showed in 1826[4] that the
spherical law of cosines can be related to spheres of imaginary radius, thus he arrived at the hyperbolic law of cosines in the form:[5]
Take a hyperbolic plane whose
Gaussian curvature is . Given a
hyperbolic triangle with angles and side lengths , , and , the following two rules hold. The first is an analogue of Euclidean law of cosines, expressing the length of one side in terms of the other two and the angle between the latter:
(1)
The second law has no Euclidean analogue, since it expresses the fact that lengths of sides of a hyperbolic triangle are determined by the interior angles:
Houzel indicates that the hyperbolic law of cosines implies the
angle of parallelism in the case of an ideal hyperbolic triangle:[11]
When that is when the vertex A is rejected to infinity and the sides BA and CA are "parallel", the first member equals 1; let us suppose in addition that so that and The angle at B takes a value β given by this angle was later called "angle of parallelism" and Lobachevsky noted it by "F(a)" or "Π(a)".
Hyperbolic law of Haversines
In cases where is small, and being solved for, the numerical precision of the standard form of the hyperbolic law of cosines will drop due to
rounding errors, for exactly the same reason it does in the
Spherical law of cosines. The hyperbolic version of the
law of haversines can prove useful in this case:
Relativistic velocity addition via hyperbolic law of cosines
Setting in (1), and by using hyperbolic identities in terms of the
hyperbolic tangent, the hyperbolic law of cosines can be written:
It turns out that this result corresponds to the hyperbolic law of cosines - by identifying with relativistic
rapidities the equations in (2) assume the form:[10][3]
Houzel, Christian (1992). "The Birth of Non-Euclidean Geometry". In Boi, L.; Flament, D.; Salanskis, J. M. (eds.). 1830–1930: A Century of Geometry: Epistemology, History and Mathematics.
Lecture Notes in Physics. Vol. 402.
Springer-Verlag. pp. 3–21.
ISBN3-540-55408-4.
Lobachevsky, N. (1898) [1830]. "Über die Anfangsgründe der Geometrie" [On the beginnings of geometry]. In Engel, F.; Stäckel, P. (eds.).
Zwei geometrische Abhandlungen [Two Geometric Treatises] (in German). Leipzig: Teubner. pp.
21–65.
Describing relations of hyperbolic geometry,
Franz Taurinus showed in 1826[4] that the
spherical law of cosines can be related to spheres of imaginary radius, thus he arrived at the hyperbolic law of cosines in the form:[5]
Take a hyperbolic plane whose
Gaussian curvature is . Given a
hyperbolic triangle with angles and side lengths , , and , the following two rules hold. The first is an analogue of Euclidean law of cosines, expressing the length of one side in terms of the other two and the angle between the latter:
(1)
The second law has no Euclidean analogue, since it expresses the fact that lengths of sides of a hyperbolic triangle are determined by the interior angles:
Houzel indicates that the hyperbolic law of cosines implies the
angle of parallelism in the case of an ideal hyperbolic triangle:[11]
When that is when the vertex A is rejected to infinity and the sides BA and CA are "parallel", the first member equals 1; let us suppose in addition that so that and The angle at B takes a value β given by this angle was later called "angle of parallelism" and Lobachevsky noted it by "F(a)" or "Π(a)".
Hyperbolic law of Haversines
In cases where is small, and being solved for, the numerical precision of the standard form of the hyperbolic law of cosines will drop due to
rounding errors, for exactly the same reason it does in the
Spherical law of cosines. The hyperbolic version of the
law of haversines can prove useful in this case:
Relativistic velocity addition via hyperbolic law of cosines
Setting in (1), and by using hyperbolic identities in terms of the
hyperbolic tangent, the hyperbolic law of cosines can be written:
It turns out that this result corresponds to the hyperbolic law of cosines - by identifying with relativistic
rapidities the equations in (2) assume the form:[10][3]
Houzel, Christian (1992). "The Birth of Non-Euclidean Geometry". In Boi, L.; Flament, D.; Salanskis, J. M. (eds.). 1830–1930: A Century of Geometry: Epistemology, History and Mathematics.
Lecture Notes in Physics. Vol. 402.
Springer-Verlag. pp. 3–21.
ISBN3-540-55408-4.
Lobachevsky, N. (1898) [1830]. "Über die Anfangsgründe der Geometrie" [On the beginnings of geometry]. In Engel, F.; Stäckel, P. (eds.).
Zwei geometrische Abhandlungen [Two Geometric Treatises] (in German). Leipzig: Teubner. pp.
21–65.