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Problem 332

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Spherical triangles

A spherical triangle is a figure formed on the surface of a sphere by three great circular arcs intersecting pairwise in three vertices.

Let C(r) be the sphere with the centre (0,0,0) and radius r.
Let Z(r) be the set of points on the surface of C(r) with integer coordinates.
Let T(r) be the set of spherical triangles with vertices in Z(r). Degenerate spherical triangles, formed by three points on the same great arc, are not included in T(r).
Let A(r) be the area of the smallest spherical triangle in T(r).

For example A(14) is 3.294040 rounded to six decimal places.

Find $\sum_{r=1}^{50}$ A(r). Give your answer rounded to six decimal places.

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球面三角形

球面三角形是在球面上由三条大圆圆弧两两相交所构成的图形。

记C(r)为中心在(0,0,0)，半径为r的球。
记Z(r)为C(r)表面坐标为整数的点构成的集合。
记T(r)为顶点均在Z(r)中的球面三角形构成的集合。T(r)中不包含三个顶点在同一大圆圆弧上的退化球面三角形。
记A(r)为T(r)中最小的球面三角形的面积。

举例来说，A(14)四舍五入到六位小数为3.294040。

求$\sum_{r=1}^{50}$ A(r)。将你的答案四舍五入到六位小数。

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