Simplexes in spaces of constant curvature

✍ Scribed by B. V. Dekster; J. B. Wilker

• Publisher: Springer
• Year: 1991
• Tongue: English
• Weight: 543 KB
• Volume: 38
• Category: Article
• ISSN: 0046-5755
• DOI: 10.1007/bf00147732

✦ Synopsis

In hyperbolic, Euclidean and spherical n-space, we determine, for each positive number l, the largest interval of the form ),.(/) ~< l u ~< l which guarantees the existence of an nsimplex Pl P2 "'" P,+t with edge-lengths pipj=lu. (In spherical geometry of curvature 1 the interval is empty unless 1 ~< 2 arcsin ~.)

The assertion that these intervals are as large as possible is justified because each of them allows certain degenerate simplexes. We determine explicitly all of these critical configurations.