MODELING AND COMPUTER IMPLEMENTATION VECTOR-PARAMETRIC CURVE ON THE GIVEN TWO POINTS AND THE FIRST, SECOND AND THIRD DERIVATIVES IN THEM

Abstract

The paper considers the construction of a vector-parametric curve of the seventh degree for given two points and the first, second and third derivatives in them. This allows you to set the curvature and torsion at the boundaries of the curve. On the basis of this curve it is possible to construct a strip of a surface on the set two boundary curves and the set first, second and third derivatives in them.

You can also build a portion of the surface at a given four points and the first, second and third derivatives in them in two directions, along and across the portion of the surface. A portion of the surface at four points allows you to build a smooth surface with a third order of smoothness on a given list of points in three-dimensional space. In addition, the task of the first and second derivatives makes it possible to set the curvature as well as the third derivative of torsion. In this case, you can first specify the first derivatives, and the second will be determined by the formula of curvature. The third derivatives will be determined by the formula of torsion. These strips and portions of surfaces are advantageous to use in the design of surfaces of machines and units operating in a moving environment (surfaces of aircraft, cars, ships), in which it is important to specify the law of curvature along the surface. The law of curvature is very important in this case, because the fracture of the curvature on the surface mimics the turbulent disruption of the flow of the moving medium, which increases the resistance of the unit to the moving medium. Increasing resistance to the moving medium causes a decrease in speed. And when used in aircraft construction, the failure of the moving environment can lead to a peak and a plane crash. The torsion task provides the design of gas pipelines with a given torsion.

Keywords: surface strip, surface portions, curve of the seventh degree for given two points and the first, second and third derivatives in them, curvature, torsion.