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Math Help - Hyperbolic Area

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    Junior Member hercules's Avatar
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    Hyperbolic Area

    Find the hyperbolic area of the euclidean rectangle with vertices (0,1) , (0,3),(5,3),(5,1).


    the area of a hyperbolic triangle is equal to its Defect. Now any hyperbolic polygon can be divided into hyperbolic triangles to get the area. But with this euclidean rectangle two sides are not geodesics. Will an inversion work? Is there another method to find the Hyperbolic area of a euclidean figure?

    Thank You.
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    Quote Originally Posted by hercules View Post
    Find the hyperbolic area of the euclidean rectangle with vertices (0,1) , (0,3),(5,3),(5,1).


    the area of a hyperbolic triangle is equal to its Defect. Now any hyperbolic polygon can be divided into hyperbolic triangles to get the area. But with this euclidean rectangle two sides are not geodesics. Will an inversion work? Is there another method to find the Hyperbolic area of a euclidean figure?

    Thank You.
    Draw a diagnol, which in this case is a curved geodesic. You end up with 2 hyperbolic triangles. Add up the areas.
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    Junior Member hercules's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    Draw a diagnol, which in this case is a curved geodesic. You end up with 2 hyperbolic triangles. Add up the areas.
    That was my original thought but the bottom and top sides of the rectangle are not geodesics. Is that still considered a hyperbolic triangle?
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    Junior Member hercules's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    Draw a diagnol, which in this case is a curved geodesic. You end up with 2 hyperbolic triangles. Add up the areas.

    I'm still trying to figure this out. Drawing the diagonal doesn't work because of the reason i mentioned above. All lines of the hyperbolic triangle must be geodesics. A possible route to the solution is using integrals (double at that). I'm horrible with those. Going to review all of calculus this summer then definitely i will be a great helper on this forum......But i need help now .
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    Quote Originally Posted by hercules View Post
    I'm still trying to figure this out. Drawing the diagonal doesn't work because of the reason i mentioned above. All lines of the hyperbolic triangle must be geodesics. A possible route to the solution is using integrals (double at that). I'm horrible with those. Going to review all of calculus this summer then definitely i will be a great helper on this forum......But i need help now .
    I misunderstood your question. I see that you have an Euclidean triangle, not a hyberbolic one.

    Let R = \{ (a,b)\in\mathbb{R}^2| 0\leq a\leq 5\mbox{ and }1\leq b\leq 3\}.

    Then a(R) = \iint_R \frac{dA}{y^2} = \int_0^5 \int_1^3 \frac{1}{y^2} ~ dy~dx
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