Minimum-Width Double-Strip and Parallelogram Annulus

In this paper, we study the problem of computing a minimum-width double-strip or parallelogram annulus that encloses a given set of n points in the plane. A double-strip is a closed region in the plane whose boundary consists of four parallel lines and a parallelogram annulus is a closed region between two edge-parallel parallelograms. We present several first algorithms for these problems. Among them are O(n^2) and O(n^3 log n)-time algorithms that compute a minimum-width double-strip and parallelogram annulus, respectively, when their orientations can be freely chosen

L_1 Geodesic Farthest Neighbors in a Simple Polygon and Related Problems

In this paper, we investigate the L_1 geodesic farthest neighbors in a simple polygon P, and address several fundamental problems related to farthest neighbors. Given a subset S subseteq P, an L_1 geodesic farthest neighbor of p in P from S is one that maximizes the length of L_1 shortest path from p in P. Our list of problems include: computing the diameter, radius, center, farthest-neighbor Voronoi diagram, and two-center of S under the L_1 geodesic distance. We show that all these problems can be solved in linear or near-linear time based on our new observations on farthest neighbors and extreme points. Among them, the key observation shows that there are at most four extreme points of any compact subset S subseteq P with respect to the L_1 geodesic distance after removing redundancy

EVALUATION OF CALF MUSCULAR FUNCTION DURING THE RECOVERY PHASE AFTER THE REPAIR OF AN ACHILLES TENDON RUPTURE

During the recovery phase after the repair of an Achilles tendon rupture the calf muscular function determines the recovery time of the Achilles tendon. Tensiomyography(TMG) is a non-invasive diagnostic method for skeletal muscular contractile properties based on displacement of the muscle belly and contraction time through electrical stimulation. The purpose of this study is to evaluate the recovery process of calf muscular function in relation to Achilles tendon repair. TMG can be used to evaluate the calf muscular function of the recovery phase of an Achilles tendon surgery and can also be used to indirectly evaluate the function of Achilles tendon and ipsilateral or bilateral muscular symmetries. It would be especially helpful for rehabilitation of weakened muscles, for operational methods and in use as one of the criteria of return to play

Empty Squares in Arbitrary Orientation Among Points

This paper studies empty squares in arbitrary orientation among a set $P$ of $n$ points in the plane. We prove that the number of empty squares with four contact pairs is between $\Omega(n)$ and $O(n^2)$, and that these bounds are tight, provided $P$ is in a certain general position. A contact pair of a square is a pair of a point $p\in P$ and a side $\ell$ of the square with $p\in \ell$. The upper bound $O(n^2)$ also applies to the number of empty squares with four contact points, while we construct a point set among which there is no square of four contact points. These combinatorial results are based on new observations on the $L_\infty$ Voronoi diagram with the axes rotated and its close connection to empty squares in arbitrary orientation. We then present an algorithm that maintains a combinatorial structure of the $L_\infty$ Voronoi diagram of $P$, while the axes of the plane continuously rotates by $90$ degrees, and simultaneously reports all empty squares with four contact pairs among $P$ in an output-sensitive way within $O(s\log n)$ time and $O(n)$ space, where $s$ denotes the number of reported squares. Several new algorithmic results are also obtained: a largest empty square among $P$ and a square annulus of minimum width or minimum area that encloses $P$ over all orientations can be computed in worst-case $O(n^2 \log n)$ time.Comment: 39 pages, 11 figure