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:
“The basis of good manners is self-reliance. Necessity is the law of all who are not self-possessed.”
—Ralph Waldo Emerson (18031882)
“A quality is something capable of being completely embodied. A law never can be embodied in its character as a law except by determining a habit. A quality is how something may or might have been. A law is how an endless future must continue to be.”
—Charles Sanders Peirce (18391914)
“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)