In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, a Lambert quadrilateral (also known as Ibn al-Haytham–Lambert quadrilateral), is a
quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
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
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
''If a line segmen ...
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
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P ...
the fourth angle is
acute
Acute may refer to:
Science and technology
* Acute angle
** Acute triangle
** Acute, a leaf shape in the glossary of leaf morphology
* Acute (medicine), a disease that it is of short duration and of recent onset.
** Acute toxicity, the adverse eff ...
, in
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
it is a
right angle and in
elliptic geometry
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines ...
it is an
obtuse angle.
A Lambert quadrilateral can be constructed from a
Saccheri quadrilateral
A Saccheri quadrilateral (also known as a Khayyam–Saccheri quadrilateral) is a quadrilateral with two equal sides perpendicular to the base. It is named after Giovanni Gerolamo Saccheri, who used it extensively in his book ''Euclides ab omni na ...
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
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P ...
a Lambert quadrilateral ''AOBF'' where the
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...
s
are
right
Rights are legal, social, or ethical principles of freedom or entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal system, social convention, or ethical ...
, and ''F'' is opposite ''O'' ,
is an
acute angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ar ...
, and the
curvature = -1 the following relations hold:
Where
are
hyperbolic functions
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the un ...
Examples
See also
*
Non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean g ...
Notes
{{Reflist
References
*George E. Martin, ''The Foundations of Geometry and the Non-Euclidean Plane'', Springer-Verlag, 1975
*M. J. Greenberg, ''Euclidean and Non-Euclidean Geometries: Development and History'', 4th edition, W. H. Freeman, 2008.
Hyperbolic geometry
Types of quadrilaterals