Quasi-Newtonian methods for modeling of plan curve

The paper is devoted to the methods of geometric modeling of plane curves given in the natural parameterization. The paper considers numerical modeling methods that make it possible to find the equation of curvature of the desired curve for different cases of the input data. The unknown curvature distribution coefficients of the required curve are determined by solving a system of nonlinear integral equations. Various numerical methods are considered to solve this nonlinear system. The results of computer implementation of the proposed methods for modeling two curvilinear contours with different initial data are presented. For the first curve, the input data are the coordinates of three points, the angles of inclination of the tangents at the extreme points and the linear law of curvature distribution. The second example considers an S-shaped curve with a quadratic law of curvature distributi.

A note on the stability of pencils of plane curves

We investigate the problem of classifying pencils of plane curves of degree d up to projective equivalence. We obtain explicit stability criteria in terms of the log canonical threshold by relating the stability of a pencil to the stability of the curves lying on it.

Geometric Distance Fields of Plane Curves

This paper introduces a geometric generalization of signed distance fields for plane curves. We propose to store simplified geometric proxies to the curve at every sample. These proxies are constructed based on the differential geometric quantities of the represented curve and are used for queries such as closest point and distance calculations. We investigate the theoretical approximation order of these constructs and provide empirical comparisons between geometric and algebraic distance fields of higher order. We apply our results to font representation and rendering.

A New Class of Plane Curves with Arc Length Parametrization and Its Application to Linear Analysis of Curved Beams

The objective of this paper is to define one class of plane curves with arc-length parametrization. To accomplish this, we constructed a novel class of special polynomials and special functions. These functions form a basis of L2(R) space and some of their interesting properties are discussed. The developed curves are used for the linear static analysis of curved Bernoulli–Euler beam. Due to the parametrization with arc length, the exact analytical solution can be obtained. These closed-form solutions serve as the benchmark results for the development of numerical procedures. One such example is provided in this paper.

Mathematical Modeling of the Route of Logging Roads

The implementation of tasks related to the development of the transportation network as a whole and logging roads as an integral part of it requires scientifically based theoretical studies of the patterns of formation of spatial curves when combining elements of the plan and the longitudinal profile, since the rational laying of the route for many years determines its most important transport and operational characteristics (speed, traffic safety, traffic capacity). Consideration of the visual perception of the road by the driver will improve the quality of design decisions, which will allow to avoid emergencies in the future after setting the route into service. On the other hand, a decrease in speed before seemingly sharp turns of the road affects the efficiency of logging road transport. Therefore, the view of the road ahead should strongly orient the driver, i.e. be visually clear and clearly changing, ensuring the constancy or smooth reduction of the traffic flow mode. At the same time, the need for a successful spatial solution of the road increases. In the designs of logging roads, straight lines, transition curves, described in recent years most often according to the clotoid, and circular curves are found as elements of the route plan. It is found that the road view in perspective correctly orients the driver of the car, i.e. it is visually clear, provided that the lines describing the edges of the roadway and the edges of the roadway in the perspective image are curved in the same direction as in the road plan. The purpose of the work is to determine a set of quantitative indicators (curvature, radius of the curve in the plan, maximum curvature, maximum rate of change of curvature) for optimization of the visual smoothness and clarity of the central projections of elementary spatial and plane curves. The performed studies allow us to fully characterize the visual smoothness and clarity of the central projections of elementary spatial and plane curves. The above algorithm makes it possible to compile a computer program to determine the mentioned indicators. The indicators determined by this algorithm allow us to evaluate both the visual smoothness and clarity of curves on logging roads.