introduce an apparatus that completely describes all the relevant properties of a curve. After that, we move on to surfaces and their properties. We shall study the first and second fundamental forms, geodesics, and curvature – an apparatus that completely describes a surface. We shall formulate Gauss’s Theorema Egregium (Remarkable Theorem) that allows the concept of curvature to be generalize...

In this paper we study k-noncrossing RNA structures with minimum arc-length 4 and at most k − 1 mutually crossing bonds. Let T [4] k (n) denote the number of k-noncrossing RNA structures with arc-length ≥ 4 over n vertices. We prove (a) a functional equation for the generating function P n≥0 T [4] k (n)z n and (b) derive for k ≤ 9 the asymptotic formula T [4] k (n) ∼ c k n −((k−1) 2 +(k−1)/2) γ...

In order to establish titanium welding technology TIG arc weldability and MIG arc weldability were surveyed. For TIG arc weldability influence of arc length and thickness to proper region of welding condition became obvious. For MIG arc weldability a new method to defend wandering phenomenon was developed.

Curvature Scale Space (CSS) representations have been shown to be very useful for recognition of noisy curves of arbitrary shapes at unknown orientations and scales [10,14]. This paper contains a number of important results on the convergence properties of CSS representations and on the evolution and arc length evolution of planar curves [6,12]. The processes which convolve arc length parametri...

We propose a method to detect circular arcs, including the determinations of their centers, radii, lengths, and midpoints. The center, radius of a circular arc can easily be determined using circular Hough transform. But, the determinations of the arc midpoint and length are more complicated. Theoretical analysis of the centroid of the near peak Hough transform data is presented. Then, the arc ...

Most scenarios for accelerating muons require recirculating acceleration. A racetrack shape for the accelerator requires particles with lower energy in early passes to traverse almost the same length of arc as particles with the highest energy. This extra arc length may lead to excess decays and excess cost. Changing the geometry to a “dogbone” shape, where there is a single linac and the beam ...