Law of Cosines in Non-Euclidean Geometry
A version of the law of cosines also holds in non-Euclidean geometry. In spherical geometry, a triangle is defined by three points u, v, and w on the unit sphere, and the arcs of great circles connecting those points. If these great circles make angles A, B, and C with opposite sides a, b, c then the spherical law of cosines asserts that each of the following relationships hold:
In hyperbolic geometry, a pair of equations are collectively known as the hyperbolic law of cosines. The first is
where sinh and cosh are the hyperbolic sine and cosine, and the second is
Like in Euclidean geometry, one can use the law of cosines to determine the angles A, B, C from the knowledge of the sides a, b, c. However, unlike Euclidean geometry, the reverse is also possible in each of the models of non-Euclidean geometry: the angles A, B, C determine the sides a, b, c.
Read more about this topic: Law Of Cosines
Famous quotes containing the words law of, law and/or geometry:
“According to the law of nature it is only fair that no one should become richer through damages and injuries suffered by another.”
—Marcus Tullius Cicero (10643 B.C.)
“I had often stood on the banks of the Concord, watching the lapse of the current, an emblem of all progress, following the same law with the system, with time, and all that is made ... and at last I resolved to launch myself on its bosom and float whither it would bear me.”
—Henry David Thoreau (18171862)
“The geometry of landscape and situation seems to create its own systems of time, the sense of a dynamic element which is cinematising the events of the canvas, translating a posture or ceremony into dynamic terms. The greatest movie of the 20th century is the Mona Lisa, just as the greatest novel is Grays Anatomy.”
—J.G. (James Graham)