Convex Quality of Alignment Lines and Geometric Design of Arches (Part I)

The design of arches in the production of accurate topographic maps such as road geometric design, short-range photogrammetry and medical simulations is based on arch type and in accordance with design goals. What is connected with establishing proper interconnection between design points is very important. The algorithm of linkage between design points, if it is based on the shortest path, leads to a combination of broken lines and, if it is necessary that it have curvature upward or downward, and to what extent, are among the matters that have always been of particular interest to the cartograph engineers, and proportional to users expectation, the Spline, Fitcurve, and TIN methods have been recommended. Each technique has a share in alignment with the user's desire to simulate the surface features. This paper examines the advantages of marking based on nodal spacing for B-Spline curves and provides formulas based on nodal spacing for common B-Spline operations such as node’s degree and derivative. Using a node spacing-based marking, Spline introduces a multi-degree curve that is similar to the B-Spline type and consists of polynomials or with a number of degrees. MD-Splines are a generalization of the B-Spline curve, in the sense that if the curved sections in a MD-Spline have the same degree, MD-Spline will be reduced to a B-Spline curve. This section deals with MD-Splines of grades 1, 2, 3, and also degrees 1 and n. MD-Splines has local support, follows the body, the convex structure and the reduction property of the variation, and is at least of (Cn-1)th degree, in which n is smaller than the degrees of the two parts of the adjacent curve.