MODELING OF CHANNEL SURFACES AND SOLIDS IN THE POINT CALCULUS
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Abstract
The paper proposes and implements an approach to geometric modeling of channel surfaces in the point calculus by determining the movable plane of section perpendicular to the guiding line of the channel surface. For this purpose, by constructing a normal and a binormal to the tangent, a movable simplex of three-dimensional space accompanying the guiding curve, which is analogous to the Frenet trihedron in the pointwise calculus, is formed. The normal and binormal are defined by the tools of the point calculus using the metric operator of three points and the exit point from the plane, which are analogs, respectively, of scalar and vector products of vectors. Examples of modeling channel surfaces with algebraic planar and transcendental spatial curve are given. Examples of using an ellipse, a closed curve of "sinusoidal" type, and a closed bypass of the 1st order of smoothness are given as formants. In the part of developing a mathematical apparatus for constructing high-performance geometric solid modeling systems, the definition of channel solids having both constant and variable functionally controllable thicknesses is performed. Here, a geometric algorithm for modeling channel surfaces and solids that is described analytically by a sequence of point equations is implemented. For its computer implementation in the form of a computational algorithm, parametric equations obtained by means of a subordinate calculation are given in parallel with the point equations.
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