Constructive Geometry of some Algebraic Curves Represented by Splines in CAD

Abstract

The problem of representing segments of some algebraic curves (curves of the second order and cubic parabolas) is solved using spline curves of CAD systems – Bezier curves and rational splines. Algorithms are being developed for constructing curves, mainly cubic parabolas, according to given conditions in the representation by Bezier curves of 3rd order. The concepts of special pencils of cubic parabolas of the first and second kind are introduced, their properties and application to the construction of cubic parabolas are shown. A way to representing cubic splines in a CAD system is proposed. The representation of arcs of conic curves, as well as arcs of Bezier curves using rational splines is shown. The developed algorithms were used to create a model of the canal surface.