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Area of a Triangle in the Poincaré Disk
Area of a Triangle in the Poincaré Disk
This Demonstration shows a triangle formed by three geodesics in the Poincaré disk. At each vertex, tangent vectors to the two intersecting geodesics are shown. The angle in radians between each pair of vectors is displayed and labeled as , or per the Gauss-Bonnet formula for the area of a hyperbolic triangle, which is shown above the disk. The boundary of the disk is dashed to indicate that the boundary is not part of the Poincaré disk. Below the disk, the area is computed approximately. Notice that if all three vertices are "infinite" vertices (vertices on the boundary circle) the area of the triangle is .
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