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Transcript
Area of Hyperbolic Triangles Heldine Aguiluz Emily Ki Trinity Shen The Hyperbolic Geometry Song LYRICS: QuickTime™ and a decompressor are needed to see this picture. Hyperbolic triangles Which sort of look like trees Are always curved strangely So that they're less than 180 degrees Not a regular line on a regular plane Shaped like a potato chip on a subway train Not a regular line on a regular plane A complicated challenge for the human brain Background Check Triangle: Polygon with 3 corners/vertices & 3 sides/edges Euclidean geometry Sides/edges are straight lines Right, obtuse, & acute triangles angles add up to 180° Area = ½ bh b a f (x)dx QuickTime™ and a decompressor are needed to see this picture. Hyperbolic Triangles In hyperbolic geometry, a hyperbolic triangle is a figure in the hyperbolic plane, consisting of three sides and three angles. The 3 sides are made of geodesics and the three angles sum up to less than 180° Hyperbolic Area If D is a region in H ,2 hyperbolic area is defined as 1 Areahyp D D y 2 dxdy 0 2 Area of Hyperbolic Triangle In hyperbolic geometry, a hyperbolic quadrilateral has angle sum less than 2π, therefore cannot have four right angles. Instead, we use triangles as basic figures. The Gauss-Bonnet Formula If the hyperbolic triangle ABC has angles α, β,γ, then its area is Areahyp (ABC) 0 2 Areahyp(ABC) AC (AB BC) Areahyp(ABC) AC (AB BC) How to Have Hyperbolic Fun! http://www.geometrygames.org/HyperbolicGames/
