Puneet Varma

Lambert quadrilateral

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Lambert quadrilateral

In geometry, a Lambert quadrilateral, named after Johann Heinrich Lambert, is a quadrilateral in which three of its angles are right angles. Historically, the fourth angle of a Lambert quadrilateral was of considerable interest since if it could be shown to be a right angle, then the Euclidean parallel postulate could be proved as a theorem. It is now known that the type of the fourth angle depends upon the geometry in which the quadrilateral exists. In hyperbolic geometry the fourth angle is acute, in Euclidean geometry it is a right angle and in elliptic geometry it is an obtuse angle.

A Lambert quadrilateral can be constructed from a Saccheri quadrilateral by joining the midpoints of the base and summit of the Saccheri quadrilateral. This line segment is perpendicular to both the base and summit and so either half of the Saccheri quadrilateral is a Lambert quadrilateral.

Lambert quadrilateral in hyperbolic geometry

In hyperbolic geometry a Lambert quadrilateral AOBF where the angles F A O , A O B , O B F are right, and F is opposite O , A F B is an acute angle , and the curvature = -1 the following relations hold:

sinh A F = sinh O B cosh B F

tanh A F = cosh O A tanh O B

sinh B F = sinh O A cosh A F

tanh B F = cosh O B tanh O A

cosh O F = cosh O A cosh A F

cosh O F = cosh O B cosh B F


sin A F B = cosh O B cosh A F = cosh O A cosh B F

cos A F B = sinh O A sinh O B = tanh A F tanh B F

cot A F B = tanh O A sinh A F = tanh O B sinh B F

sin A O F = sinh A F sinh O F

cos A O F = tanh O A tanh O F

tan A O F = tanh A F sinh O A

Where tanh , cosh , sinh are hyperbolic functions

References

Lambert quadrilateral Wikipedia


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