Lambert quadrilateral

In hyperbolic geometry, a Lambert quadrilateral is a quadrilateralMathworldPlanetmath with exactly three right anglesMathworldPlanetmathPlanetmath. Since the angle sum of a triangle in hyperbolic geometry is strictly less than π radians, the angle sum of a quadrilateral in hyperbolic geometry is strictly less than 2π radians. Thus, in any Lambert quadrilateral, the angle that is not a right angle must be acute.

The discovery of Lambert quadrilaterals is attributed to Johann Lambert.

Both pairs of opposite sides of a Lambert quadrilateral are disjointly parallel since, in both cases, they have a common perpendicularMathworldPlanetmathPlanetmathPlanetmath. Therefore, Lambert quadrilaterals are parallelogramsMathworldPlanetmath. Note also that Lambert quadrilaterals are right trapezoidsMathworldPlanetmath.

Below are some examples of Lambert quadrilaterals in various models. In each example, the Lambert quadrilateral is labelled as ABCD.

  • In each of these examples, blue lines indicate verification of right angles by using the poles, and green lines indicate verification of acute angles by using the poles. (Recall that most other models of hyperbolic geometry are angle preserving. Thus, verification of angle measures is unnecessary in those models.)

  • The Poincaré disc model:

  • ABCD...
Title Lambert quadrilateral
Canonical name LambertQuadrilateral
Date of creation 2013-03-22 17:08:04
Last modified on 2013-03-22 17:08:04
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 24
Author Wkbj79 (1863)
Entry type Definition
Classification msc 51M10
Classification msc 51-00
Synonym Lambert’s quadrilateral
Related topic RightTrapezoid