Geometric Modeling of the Contour Arcs Passing Through the Predefined Points

Abstract

Multidimensional interpolation is an important scientific task in demand of various science fields and technology. The geometrical theory of multidimensional interpolation has been further developed in terms of extending the shaping tools of geometrical modeling at the expense of the contour arcs passing through the predefined points. The proposed method is based on modification of Bezier’s curve with preservation of tangents in its initial and/or final points. Then the Bezier curve retains its geometrical properties of the arc, but additionally has the possibility to pass through several points set in advance. Examples are given of modifying a 5th-order Bezier curve arc into a curve arc passing through four predefined points and a 3rd-order Bezier curve arc passing through three predefined points and having a tangent in the initial point, which is proposed to be used as the onion dome surface forming arc. Such a method helps to reduce the piecewise character of composite curves when building curves. The introduction of such research results into computeraided design and solid-state modeling (CAD/CAM) systems will allow to expand their toolbox in terms of shaping surfaces and solids of technical shapes of various purposes.

Bionics in architecture and geometric modelling of thin shell surfaces

“…Taking into account structural requirements, the geometry of shells [7,10,12,13] is considered in detail, depending on different ways of their support on: walls (supporting rings); columns (columns); directly on foundations (solid or separate); supports, which are an organic transition between the shell and the foundation; combined (mixed) support on walls and cables; console support; "floating" foundations. The influence of overlapped span size on the choice of shell shape was studied.…”

“…Taking into account the functional series of peculiarities of the geometric surface of the dome [12,13], it is important to fulfill this relation b  H. С. Taking into account the design mathematical parameters of the surface, the following inequality corresponds to the above inequality: R  Hk.…”

“…In research works, a number of authors consider the problem of the stress-strain state of thin geometric shells under a vertical load arising under given rigid displacements of a number of cross sections [5][6][7][8]. Scientific papers [9][10][11][12] are devoted to the problems of nonlinear deformation of the stress-strain state of thin-walled building structures of complex geometric shape.…”

“…In the present study, an attempt is made to analyze different predetermined requirements to the designed shells and their geometrical interpretation is given. For example, based on the calculation requirements, the dependence of the calculation complexity on the geometry of the shell is shown, and the static features of shells with parabolic, hyperbolic and elliptical points are given [10][11][12]. Studies of the shells delineated by the transfer surface have been carried out.…”