Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry? - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-19T07:10:04Z http://mathoverflow.net/feeds/question/79167 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79167/is-there-a-similar-formula-in-spherical-and-hyperbolic-geometry-as-euclidean-geom Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry? van abel 2011-10-26T15:14:14Z 2011-10-26T22:42:55Z <ol> <li><p>In an Euclidean plane, we know that the area of a triangle is determined by the length of base and the height, i.e., $$S_{\Delta}=\frac{1}{2}a.h,$$ where $a$ is the length of base and the $h$ is the height. Is there a similar formula in Spherical and hyperbolic spaces?</p></li> <li><p>In Euclidean plane, we know that the cosine law says that (suppose $\alpha$, $\beta$, $\gamma$ are the angles, and $a, b, c$ are the lengths opposite $a, b, c$ respectively.) $$\cos\gamma=\frac{a^2+b^2-c^2}{2ab}.$$ Then is there a analogue in spherical and hyperbolic geometry? I have noted that the First and Second Cosine law are not so clearly relevant with this formula.</p></li> </ol> http://mathoverflow.net/questions/79167/is-there-a-similar-formula-in-spherical-and-hyperbolic-geometry-as-euclidean-geom/79169#79169 Answer by Tony Huynh for Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry? Tony Huynh 2011-10-26T15:37:39Z 2011-10-26T22:42:55Z <p>For hyperbolic geometry, you can find the answers to your questions on <a href="http://en.wikipedia.org/wiki/Hyperbolic_triangle" rel="nofollow">Wikipedia</a>. An interesting thing is that there is an absolute upper bound on the area of a hyperbolic triangle, even though lengths are unbounded. </p> <p>For spherical geometry, you can find the area of a spherical triangle <a href="http://mathworld.wolfram.com/SphericalTriangle.html" rel="nofollow">here</a> and corresponding trigonometric formulas <a href="http://mathworld.wolfram.com/SphericalTrigonometry.html" rel="nofollow">here</a>.</p>