On the one method of a third-degree bezier type spline curve construction

TL;DR

This paper introduces a new method for constructing third-degree Bezier-type spline curves that are smooth, continuous, and lie within the convex hull of control points, using an approach based on earlier parabolic spline methods.

Contribution

It presents a novel construction method for third-degree Bezier-type spline curves with guaranteed continuity and uniqueness, extending previous parabolic spline techniques.

Findings

The constructed spline is inside the convex hull of control points.

The spline segments are tangent to the control polygon.

The method ensures first derivative continuity and uniqueness.

Abstract

A method is proposed for constructing a spline curve of the Bezier type, which is continuous along with its first derivative by a piecewise polynomial function. Conditions for its existence and uniqueness are given. The constructed curve lies inside the convex hull of the control points, and the segments of the broken line connecting the control points are tangent to the curve. To construct the curve, we use the approach proposed earlier for constructing a parabolic spline. The idea is to use additional points with unknown values of some function. Additional points are used as spline nodes, and the function values are determined from the condition of the first derivative continuity of a piecewise polynomial curve. In multiple interpolation nodes, the function takes the given values and the values of the first derivative, which are determined by the control points. Examples of constructing a spline curve are given.