The Theodorus Variation

The Spiral of Theodorus, also known as the "root snail" from its connection with square roots, can be constructed by hand from triangles made with from paper with scissors, ruler, and protractor. See the Video Abstract. Once the triangles are made, two different but similar spirals can be made. This paper proves some things about the second spiral; in particular that the open curve generated by the inner vertices monotonically approaches a circle, and that the vertices are ultimately equidistributed around that inner circle.

A note on the solvability of homogeneous Riemann boundary problem with infinity index

In this note we establish a necessary and sufficient condition for solvability of the homogeneous Riemann boundary problem with infinity index on a rectifiable open curve. The index of the problem we deal with considers the influence of the requirement of the solutions of the problem, the degree of non-smoothness of the curve at the endpoints as well as the behavior of the coefficient at these points.

Knot polynomials of open and closed curves

In this manuscript, we introduce a method to measure entanglement of curves in 3-space that extends the notion of knot and link polynomials to open curves. We define the bracket polynomial of curves in 3-space and show that it has real coefficients and is a continuous function of the curve coordinates. This is used to define the Jones polynomial in a way that it is applicable to both open and closed curves in 3-space. For open curves, the Jones polynomial has real coefficients and it is a continuous function of the curve coordinates and as the endpoints of the curve tend to coincide, the Jones polynomial of the open curve tends to that of the resulting knot. For closed curves, it is a topological invariant, as the classical Jones polynomial. We show how these measures attain a simpler expression for polygonal curves and provide a finite form for their computation in the case of polygonal curves of 3 and 4 edges.

THE TRAVELING SALESMAN PROBLEM FOR LINES AND RAYS IN THE PLANE

In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of n regions (neighborhoods) and we seek a shortest tour that visits each region. In the path variant, we seek a shortest path that visits each region. We present several linear-time approximation algorithms with improved ratios for these problems for two cases of neighborhoods that are (infinite) lines, and respectively, (half-infinite) rays. Along the way we derive a tight bound on the minimum perimeter of a rectangle enclosing an open curve of length L.

A New Open Curve Detection Algorithm for Extracting the Laser Lines on the Road

This paper proposes a new open curve detection algorithm for recognizing the laser rays on the road to verify the roughness of the road surface by parsing the curvature of the laser rays. The laser line in the image is first enhanced by the bright line enhance filtering to remove the clutters generated by the background. The enhanced line has the characteristics that its gray value is greater than that of background, so the established energy functional in open curve detection process has a gray value constraint to drive the evolution curve to the brightest place in image. The line on the road surface is an open curve throughout the whole image, so we fixed the two endpoints of the evolution curve to maintain it is an open curve in curve evolution. Moreover, we design an edge inhibit operator to restrain the influence of objects edges to ensure the accuracy of the test laser line. We compared our algorithm with the edge detection operators, and the comparison results showed that our method is more accurate and more robustness to the background.