Quality of Geometric Education in Various Approaches to Teaching Methods

Vitovtov²

2021

Physical spline is a resilient element whose cross-sectional dimensions are very small compared to its axis’s length and radius of curvature. Such a resilient element, passing through given points, acquires a "nature-like" form, having a minimum energy of internal stresses, and, as a consequence, a minimum of average curvature. For example, a flexible metal ruler, previously used to construct smooth curves passing through given coplanar points, can be considered as a physical spline. The theoretical search for the equation of physical spline’s axis is a complex mathematical problem with no elementary solution. However, the form of a physical spline passing through given points can be obtained experimentally without much difficulty. In this paper polynomial and parametric methods for approximation of experimentally produced physical spline with large deflections are considered. As known, in the case of small deflections it is possible to obtain a good approximation to a real elastic line by a set of cubic polynomials ("cubic spline"). But as deflections increase, the polynomial model begins to differ markedly from the experimental physical spline, that limits the application of polynomial approximation. High precision approximation of an elastic line with large deflections is achieved by using a parameterized description based on Ferguson or Bézier curves. At the same time, not only the basic points, but also the tangents to the elastic line of the real physical spline should be given as boundary conditions. In such a case it has been shown that standard cubic Bézier curves have a significant computational advantage over Ferguson ones. Examples for modelling of physical splines with free and clamped ends have been considered. For a free spline an error of parametric approximation is equal to 0.4 %. For a spline with clamped ends an error of less than 1.5 % has been obtained. The calculations have been performed with SMath Studio computer graphics system.

Approximation of Physical Spline with Large Deflections

Vitovtov²

2021

Approximation of Physical Spline with Large Deflections

Vitovtov²

2021

Constructing a G2-Smooth Compound Curve Based on Cubic Bezier Segments

2021

The theory and practice of forming composite G2-smooth (two-continuously differentiable) curves, used in technical design since the mid-60s of the 20th century, is still not reflected in any way in the curriculum of technical universities or in Russian textbooks in engineering and computer graphics. Meanwhile, such curves are used in modeling a wide variety of geometric objects and physical processes. The article deals with the problem of constructing a composite G2-smooth curve passing through given points and touching at these points pre-specified straight lines. To solve the problem, cubic Bezier segments are used. The main problem in constructing a smooth compound curve is to ensure the continuity of curvature at the joints of the segments. The article shows that for parametrized cubic curves, this problem is reduced to solving a quadratic equation. A software module has been compiled that allows one to construct a plane G2-smooth curve passing through predetermined points and tangent at these points with predetermined straight lines. The shape of the curve (“completeness” of its segments) is adjusted by the user in the dialog mode of the program module. Solved the problem of constructing a cubic curve smoothly connecting unconnected Bezier segments. An algorithm for constructing a Bezier segment with given tangents and given curvature at its boundary points is proposed. Some properties of the cubic Bezier segment are considered. In particular, it was shown that for the case of parallel tangents, the curvature at the end of a segment is determined by the position of only one control point (Theorem 1). Cases are considered when the curvature at the ends of the Bezier segment is equal to zero (Theorem 2). An approximation of a three-point physical spline is performed using Bezier segments. The approximation error was less than 2%, which is comparable to the error in processing the experimental data. A method is proposed for modeling a spatial G2-smooth curve passing through points set in advance in space and touching at these points arbitrarily oriented lines in space. The article is of an educational nature and is intended for an in-depth study of the basics of computational geometry and computer graphics.