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:
“All things are literally better, lovelier, and more beloved for the imperfections which have been divinely appointed, that the law of human life may be Effort, and the law of human judgment, Mercy.”
—John Ruskin (1819–1900)
“I believe it was a good job,
Despite this possible horror: that they might prefer the
Preservation of their law in all its sick dignity and their knives
To the continuation of their creed
And their lives.”
—Gwendolyn Brooks (b. 1917)
“... geometry became a symbol for human relations, except that it was better, because in geometry things never go bad. If certain things occur, if certain lines meet, an angle is born. You cannot fail. It’s not going to fail; it is eternal. I found in rules of mathematics a peace and a trust that I could not place in human beings. This sublimation was total and remained total. Thus, I’m able to avoid or manipulate or process pain.”
—Louise Bourgeois (b. 1911)