Let S be a finite set of n points in the plane in general position. A k-hole of S is a simple polygon with k vertices from S and no points of S in its interior. A simple polygon P is l-convex if no straight line intersects the interior of P in more than l connected components. Moreover, a point set S is l-convex if there exists an l-convex polygonalization of S. Considering a typical Erdős-Szek...

Let S = {l1, l2, l3, . . . , ln} be a set of n vertical line segments in the plane. Though not essential, to simplify proofs we assume that no two li’s are on the same vertical line. A convex polygon weakly intersects S if it contains a point of each line segment on its boundary or interior. In this paper, we propose an O(n log n) algorithm for the problem of finding a minimum area convex polyg...

We consider the motion planning problem for a point constrained to move along a smooth closed convex path of bounded curvature. The workspace of the moving point is bounded by a convex polygon with m vertices, containing an obstacle in a form of a simple polygon with n vertices. We present an O(m + n) time algorithm finding the path, going around the obstacle, whose curvature is the smallest po...

This paper considers reconngurations of polygons, where each polygon edge is a rigid link, no two of which can cross during the motion. We prove that one can reconngure any monotone polygon into a convex polygon; a polygon is monotone if any vertical line intersects the interior at a (possibly empty) interval. Our algorithm computes in O(n 2) time a sequence of O(n 2) moves, each of which rotat...

In this paper we give a parallel algorithm for finding the nearest-neighbor vertex of each vertex of a convex polygon. Our algoritb..z::J. runs in O(log n) time using O(njlogn) processors, in the parallel computation model CREW PRA.lvr (ConcurrentRead, Exclusive-Write Parallel RAM). This implies that the all nearest-neighbors problem for a convex polygon can be solved in O(n/p+logn) time using ...

In this paper, we consider the problem of computing the minimum area triangle that circumscribes a given n-sided convex polygon touching edge-to-edge. In other words, we compute the minimum area triangle that is the intersection of 3 half-planes out of n half-planes defined by a given convex polygon. Previously, O(n logn) time algorithms were known which are based on the technique for computing...

The problem of finding the convex hull of a planar set of points P, that is, finding the smallest convex region enclosing P, arises frequently in computer graphics. For example, to fit P into a square or a circle, it is necessary and sufficient that H(P), the convex hull of P, fits; and since it is usually the case that H(P) has many fewer points than P has, it is a simpler object to manipulate...

An n-gon is defined as a sequence P = (V0, . . . , Vn−1) of n points on the plane. An n-gon P is said to be convex if the boundary of the convex hull of the set {V0, . . . , Vn−1} of the vertices of P coincides with the union of the edges [V0, V1], . . . , [Vn−1, V0]; if at that no three vertices of P are collinear then P is called strictly convex. We prove that an n-gon P with n > 3 is strictl...

TheMinkowski sum of two setsA,B ∈ IR, denotedA⊕B, is defined as {a+ b | a ∈ A, b ∈ B}. We describe an efficient and robust implementation for the construction of Minkowski sums of polygons in IR using the convolution of the polygon boundaries. This method allows for faster computation of the sum of non-convex polygons in comparison to the widely-used methods for Minkowski-sum computation that d...

Given two points p, q and a sequence of n lines (n>1) in the plane, we want to find the shortest path of touring all the given lines that starts at p and ends at q. In this paper, we solve the problem by reducing it to the problem of finding the shortest path that tours all the segments in a convex polygon from p to q. We first introduce how to construct the convex polygon. Then, we propose the...