Research of algorithm for calculating the vector-parametric bispline based on polynomial of the fourth degree

Authors

  • Александр Михайлович Ковтун Danube Institute of National University «Odessa Maritime Academy», 9, Fanagoriyskaja Str., Izmail, Odessa Region, 68600-Ukraine, Ukraine https://orcid.org/0000-0002-6531-2561

DOI:

https://doi.org/10.15587/2312-8372.2016.80457

Keywords:

segment of three points and two first derivatives, vector-parametric spline of fourth degree

Abstract

In the course of the audit process of the vector-parametric spline of fourth degree on the basis of a segment of three points and two first derivatives at the end points is easy to see that it cannot be set to the same number of boundary conditions at both ends, as for polynomials of third and fifth degree, because a polynomial of the fourth degree is «unbalanced».

New method is proposed to eliminate these disadvantages in the design of fourth degree splines and bisplines (vector parametric surfaces) based on them.

It is proposed to consider the next variant of polynomial of the fourth degree for bispline design: the endpoints, derivatives in them and another middle point are given.

Based on the proposed functions of the polynomial:

Vector parametric spline of fourth degree on the basis of a segment of three points and two first derivatives is noted:

Based on the segment of the fourth degree for the portions of the surface recorded this equation is noted:

 To specify a portion it must have not only first derivatives but also the mixed derivatives at the nodal points.

Based on these formulas, it became possible to write a test program for visualization of bispline (vector parametric surface) fourth degree in the language Auto Lisp in AutoCAD, spline of fourth degree showed good «custom» properties, the surface is adequate to the input data, subjectively nice-looking.

The paper shows the ability of the splines of the fourth degree to give biplane. Due to the peculiarities of their structure (the ability to give an additional medial condition) the proposed curve has an additional possibility of a more correct and adequate to the task of specifying the conditions. The achieved effect (a new polynomial) gives a method the right to life for designing smooth curves and surfaces.

Author Biography

Александр Михайлович Ковтун, Danube Institute of National University «Odessa Maritime Academy», 9, Fanagoriyskaja Str., Izmail, Odessa Region, 68600-Ukraine

Candidate of Technical Sciences, Associate Professor

Department of Technical Subjects 

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Published

2016-09-29

How to Cite

Ковтун, А. М. (2016). Research of algorithm for calculating the vector-parametric bispline based on polynomial of the fourth degree. Technology Audit and Production Reserves, 5(1(31), 22–26. https://doi.org/10.15587/2312-8372.2016.80457

Issue

Vol. 5 No. 1(31) (2016): Mathematical modeling. Mechanical engineering and machine building. Electrical engineering and industrial electronics

Section

Mathematical Modeling: Original Research

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Copyright (c) 2016 Александр Михайлович Ковтун

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