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A . This yields: It is well-known that ra = R cos R dA ≤ y/2 −y/2 y/2 θra da = −y/2 a y dx R cos da = 2R sin dx R R 2R so dA/dx ≤ 2R sin y/(2R). For a stadium of length x, the area on the sphere is bounded by A(x, y) as desired. A (1/x − x) × x stadium can be inscribed within any x × 1/x paper rectangle. By Proposition 2, this stadium only occupies A(1/x − x, x) area on the sphere. The remaining paper only has an area of x2 − πx2 /4. Upper Bound 3. x × 1/x paper can wrap an R-sphere only if 4πR2 ≤ x2 − πx2 /4 + A(1/x − x, x).

Cicerone and G. Di Stefano 1. x is a point p interior to some edge e of P . Hence the cut [v, x] is either a b, b -pcut or a a, c -pcut. In both cases, the new angles at p are convex and then they do not need further cuts. So, one cut is sufficient to “remove” the concave vertex v from P . 2. x coincides with a vertex v of P , but v is not forbidden. Also in this case no further cuts are needed to “remove” the concave vertex v from P . This proves that the above approach uses µ(P ) octilinear cuts exactly.

Variations and generalizations of the problem are studied in [6,8–10,12–14,19,28,30–34,37–39]. In addition, Dubins’ characterization plays a fundamental role in establishing the existence as well as the optimality of curvature-constrained paths. Jacobs and Canny [23] showed that even in the presence of obstacles it suffices to restrict attention to paths of Dubins form between obstacle contacts and that if such a path exists then the shortest such path is well-defined. Fortune and Wilfong [20] give a super-exponential time algorithm for determining the existence of, but not actually constructing, such a path.

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