SIMPLIFIED COMPUTATION OF TRADITIONAL FIRST DERIVATIVE VALUES OF POINT-BASED POLYNOMIAL SEGMENTS AT THEIR INITIAL POINTS

Abstract

In problems of geometric modeling of planar curves, the ability to efficiently compute derivatives—particularly the first derivative at the initial point of a segment—is of significant importance. This necessitates the development of formulas for computing the traditional (Newton–Leibniz) first derivative at point A₁ of a planar point-based polynomial segment, given the parameter value t₁ = 0. Expanded forms of second-, third-, fourth-, and fifth-degree point-based polynomials are presented, along with the corresponding expressions for computing the derivative at the initial point. For instance, for a second-degree polynomial, the derivative is calculated as twice the difference between the second and first points; for a third-degree polynomial, it involves a combination of vectors taking into account all three points, and so on. As the degree of the polynomial increases, the analytical expression becomes significantly more complex, making it difficult to derive a general formula for arbitrary n.

It is emphasized that the availability of formulas developed by the authors for determining the values of traditional first derivatives reduces the computational cost of finding projection centers when calculating differential projections to form a differential projection strip. Given these challenges, the creation of predefined analytical expressions for derivative calculation becomes practically essential for ensuring computational speed and accuracy. In the context of constructing a differential projection strip, such optimization is especially important, since these segments involve determining projection directions and centers, which require numerous derivatives. The obtained traditional derivatives serve as input values for further construction of compositional derivatives.

Compositional derivatives offer a number of advantages, the main one being their construction based on invariants of parallel projection. This ensures the stability of the characteristics of geometric objects regardless of their spatial position, which is especially valuable in transformations and unified model analysis. As a result, the use of compositional derivatives not only increases precision but also reduces the computational complexity of the developed software products.

Keywords: point-based polynomial, compositional derivative, traditional derivative, differential projection strip, formulas for derivative values at the initial point.