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Observing a solid angle from various viewpoints. (English)
Yokohama Math. J. 54, No. 2, 155-160 (2008).
Summary: Let $AOB$ be a triangle in $\Bbb R^3$. When we look at this triangle from various viewpoints, the angle $\angle AOB$ changes its appearance, and its ‘visual size’ is not constant. In [{\it Y. Maeda} and {\it H. Maehara}, Lecture Notes in Computer Science. 2866. Berlin: Springer, 200‒203 (2003)], it is proved that the average visual size of $\angle AOB$ is equal to the true size of the angle when viewpoints are chosen at random on the surface of a sphere centered at $O$. In this paper, a simpler proof of this result is presented. Furthermore, we extend the result to the case of a solid angle in $\Bbb R^4$.
Classification: G75 K55