Approximate predicates can be used to reduce the number of comparisons made by expensive, complex predicates. For example, to check if a point is within a region (expensive predicate) we can first check if the point is within a bounding rectangle (approximate predicate). In general, approximate predicates may have false positive and false negative errors. In this paper we study the problem of s...

In this paper, we introduce a new approach for fitting of a bounding rectangle to closed regions. In this approach the coordinates of the vertices are computed directly using a closed-form solution. This approach is based on simple coordinate geometry and uses the boundary points of regions. Using a least-square approach we determine the directions of major and minor axes of the object, which g...

We study when an arrangement of axis-aligned rectangles can be transformed into an arrangement of axisaligned squares in R while preserving its structure. We found a counterexample to the conjecture of J. Klawitter, M. Nöllenburg and T. Ueckerdt whether all arrangements without crossing and side-piercing can be squared. Our counterexample also works in a more general case when we only need to p...

In wireless communication networks, optimal use of the directional antenna is very important. The directional antenna coverage (DAC) problem is to cover all clients with the smallest number of directional antennas. In this paper, we consider the variable-size rectangle covering (VSRC) problem, which is a transformation of the DAC problem. There are n points above the base line y = 0 of the two-...

In a witness rectangle graph (WRG) on vertex point set P with respect to witness point set W in the plane, two points x, y in P are adjacent whenever the open rectangle with x and y as opposite corners contains at least one witness point in W . WRGs are representative of a much larger family of witness proximity graphs introduced in two previous papers. We study graph-theoretic properties of WR...

We prove that any simple polytope (and some non-simple polytopes) in R admits an inscribed regular octahedron.

We present a scheme to construct C1 continuous toolpath for 5-axis pocketing of high speed machining. A continuous toolpath synthesis is proposed, that forms out circular segments of maximally inscribed sizes and linear segments connecting these circular segments. Extending earlier work that employed the Medial Axis Transform to compute these maximally inscribed circles for 2D planar pockets, t...

For the optimal approximation of convex bodies by inscribed or circumscribed polytopes there are precise asymptotic results with respect to different notions of distance. In this paper we want to derive some results on optimal approximation without restricting the polytopes to be inscribed or circumscribed. Let Pn and P(n) denote the set of polytopes with at most n vertices and n facets, respec...

Finding wirelength optimal packings of rectangles is a well known problem in VLSI design. We propose a branch and bound algorithm for this problem that is based on the rectangle packing algorithm of Moffitt and Pollack. It makes use of a very efficient implementation of an incremental network simplex algorithm. Our algorithm allows for the first time to find optimum solutions of three instances...

Every year a large number of bridges flooding in the river just when they most need them there destroy all born. Carried out. In this study, the four fulcrum forms a semi-circular, trapezoidal, rectangular edges rounded rectangle 90 degrees at 70 degrees in Sagittarius with four different discharges depth of constant scouring around the abutment in pure water were examined. For material of flum...