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# Kuen's Surface, Aqua Twilight Eco Canvas

## Kuen's Surface, Aqua Twilight Eco Canvas

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On a surface in Euclidean three-space, straightest paths, those of locally-shortest length, are called "geodesics." Kuen's surface has "Gaussian curvature" equal to $$-1$$ (away from curves where the surface pinches to a sharp edge). Concretely, for every geodesic triangle on the surface, the sum of the interior angles (in radians) is equal to $$\pi$$ minus the area of the triangle. On a surface in Euclidean three-space, "straightest" paths, those of locally-shortest length, are called "geodesics." Geodesic triangles on a surface generally do not have total interior angle equal to a straight angle: The "angular defect," the sum of the interior angles (in radians) minus $$\pi$$, is generally non-zero. For triangles with short sides the angular defect is nearly proportional to the area. If we fix a point $$p$$ and consider geodesic triangles containing $$p$$, the limiting constant of proportionality as the sides approach $$0$$ is the "Gaussian curvature" at $$p$$. Kuen's surface has Gaussian curvature equal to $$-1$$ at each point (except for curves where the surface pinches to a sharp edge), and is consequently a model for small pieces of the hyperbolic plane.