Point Tools of Geometric Modeling, Invariant Relating to Parallel Projection

Abstract

The purpose of this paper is to familiarize experts in geometric and computer modeling with specific tools for point calculus; demonstrate the possibilities of point calculus as a mathematical apparatus for modeling of multidimensional space’s geometric objects. In the paper with specific examples have been described the basic constructive tools for point calculus, having invariant properties relating to parallel projection. These tools are used to model geometric objects, including: affine ratio of three points of a straight line, intersection of two straight lines, intersection of a straight line with a plane, parallel translation and tangent to a curve. The theoretical foundations of point tools for geometric modeling, invariant relating to parallel projection, have been presented. For example, instead of traditional determination for straight lines intersection point by composing and solving a system of equations in coordinate form, zeroing of a moving triangle’s area is used. This approach allows to define geometric objects in multidimensional spaces keeping the symbolic representation of point equation, as well as to perform its coordinate-wise calculation at the last stage of modeling, which allows to significantly reduce computing resources in the process of solving the problems related to engineering geometry and computer graphics.
 The local results of the research presented in this paper, which served as examples for the use of point calculation constructive tools, are: definition of the cubic Bezier curve as a curve of one relation in point and coordinate form; determination of excessive parameterization of the plane and bypass arcs based on it; determination of the tangent to the spatial curve by differentiation the original curve with respect to a current parameter, followed by parallel transfer of the obtained segment to the tangency point; the general point equation for the torso surface has been obtained on account of its definition as a geometric place of tangents to its cusp edge, and examples for the construction of torso surfaces based on the cubic Bezier curve and a transcendental space curve have been presented.

Visualization of Geometric Models of Faceted Solids in Point Calculus

“…Next, we determine the value of the  parameter, at which the volume of the SBCD tetrahedron is equal to zero. In accordance with the point calculus S  theorem [14], we obtain:…”

“…To determine the , , u v w and t parameters it is necessary to perform a coordinate-bycoordinate calculation [14]. For three-dimensional space we can formulate the following equation system:…”

“…But in most cases, the task is reduced to determining the function of intersection of the rendered object with the ray and determining the signed distance function [11][12][13]. Based on the fact that geometric solids in the form of a three-parameter set of points in three-dimensional space are parametrized in point calculus [14][15], to determine the distance function, it is necessary to solve the problem of intersection of geometric solids with projecting rays in point calculus.…”