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:
“No. I am not the law in your mind,
the grandfather of watchfulness.
I am the law of your members,
the kindred of blackness and impulse.
See. Your hand shakes.
It is not palsy or booze.”
—Anne Sexton (1928–1974)
“Unless we maintain correctional institutions of such character that they create respect for law and government instead of breeding resentment and a desire for revenge, we are meeting lawlessness with stupidity and making a travesty of justice.”
—Mary B. Harris (1874–1957)
“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 Gray’s Anatomy.”
—J.G. (James Graham)