Transcript Hyperbolic Triangles

Area of Hyperbolic Triangles
Heldine Aguiluz
Emily Ki
Trinity Shen
The Hyperbolic Geometry Song
LYRICS:
QuickTime™ and a
decompressor
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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/