Numerical hyperinterpolation over spherical triangles

Sommerville and Davies classified the spherical triangles that can tile the sphere in an edge-to-edge fashion. Relaxing this condition yields other triangles, which tile the sphere but have some tiles intersecting in partial edges. This paper shows that no right triangles in a certain subfamily can tile the sphere, although multilayered tilings are possible.

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The Ambiguous Cases in the Solution of Spherical Triangles

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500033046 ◽

1904 ◽

Vol 23 ◽

pp. 55

Author(s):

T. J. I'A Bromwich

Keyword(s):

Spherical Triangles

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Areas, Angles, and Polyhedra

Heavenly Mathematics ◽

10.23943/princeton/9780691175997.003.0007 ◽

2017 ◽

Author(s):

Glen Van Brummelen

Keyword(s):

Eighteenth Century ◽

Unit Sphere ◽

Regular Polyhedron ◽

Spherical Triangle ◽

French Mathematician ◽

Regular Polyhedra ◽

Leonhard Euler ◽

Spherical Triangles

This chapter explains how to find the area of an angle or polyhedron. It begins with a discussion of how to determine the area of a spherical triangle or polygon. The formula for the area of a spherical triangle is named after Albert Girard, a French mathematician who developed a theorem on the areas of spherical triangles, found in his Invention nouvelle. The chapter goes on to consider Euler's polyhedral formula, named after the eighteenth-century mathematician Leonhard Euler, and the geometry of a regular polyhedron. Finally, it describes an approach to finding the proportion of the volume of the unit sphere that the various regular polyhedra occupy.

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