8. References 6. Concluding Remarks 7. Acknowledgments

Maximum intersection of spherical polygons and work-piece orientation for 4-and 5-axis machining, " of convex objects in two and three dimensions , " Accessibility analysis for the automatic inspec-29-fixed. Furthermore, only one point will have an integral x-coordinate and thus achieve the maximum aperture angle by the construction of Q. The adversary will ensure that this point is fixed only after n-2 chains are fixed. Therefore, Ω(n log(m/n)) queries must be made before the x-coordinate of the point with the maximum aperture angle is fixed. On each floating chain, space is reserved for the ((m/n)j + k)th point, for 1 ≤ k < m/n, at both j-1/2 + k/(4m/n) and j+ 1/4 + k/(4m/n). The adversary responds to a query for x i as follows: if x i is fixed, it reports (x i ,-1/x i). Otherwise x i is in a floating chain and i = ((m/n)j + k) for some integer 1 ≤ k < m/n. If this is the last floating chain, then the adversary fixes x i = j and fixes all other floating points in that chain to their reserved spots before or after x i as needed to preserve their order. If x i is not in the last floating chain, then the adversary fixes either all points of the floating chain before and including x i to their reserved spots less than j or all points including x i and after to their spots greater than j, which ever causes fewer points to be fixed. These actions force all but one floating chain to be fixed before the algorithm discovers which point is fixed to (j,-1/j) for some integer j and, thereby, finds out which aperture angle achieves the maximum value of π/2. Since the adversary ensures that Ω(log(m/n)) steps are required to fix each chain, this gives the bound of Ω(n log(m/n)) queries. Q.E.D. Notice that all computations can be performed in rational arithmetic with numerators and denominators bounded by small polynomials in n and m. Thus, the adversary can operate within the standard unit-cost RAM model of computation with word length logarithmic in n + m. In this paper we considered the problem of computing the aperture angle of a camera that is allowed to travel in a convex region in the plane and is required to maintain some other convex region within its field of view at all times. We …