On the matrix representation of 5th order B\'{e}zier Curve and derivatives in E$^{3}$

Abstract

Using the matrix representation form, the first, second, third, fourth, and fifth derivatives of 5th order Bézier curves are examined based on the control points in E 3 . In addition to this, each derivative of 5th order Bézier curves is given by their control points. Further, a simple way has been given to find the control points of a Bézier curves and its derivatives by using matrix notations. An example has also been provided and the corresponding figures which are drawn by Geogebra v5 have been presented in the end. 2020 Mathematics Subject Classification. 53A04-53A05.

Analysis of Higher-Order Bézier Curves for Approximation of the Static Magnetic Properties of NO Electrical Steels

“…Another feature of Bézier curves is their matrix representation, which allows for a straightforward calculation of the polynomial coefficients for functions B x (t) and B y (t). For a Bézier curve of order n, the matrix equation is defined by Equation (11) [36]:…”

“…To obtain the B y (t) polynomial of the Bézier curve, the y coordinates of control points P i must be considered in Equation (11). The matrix M Béz,n is an (n + 1) × (n + 1)-sized coupling matrix between the variable t and the control point's coordinates P. By evaluating the matrix M Béz,n for Bézier curves of orders of up to 6 [36,37], the general form of matrix M Béz,n for curves of order n is determined by Equation ( 12) [38]:…”

“…A general rule is that the main counter diagonal elements of matrix M Béz,n always have a positive sign, and then the sign alternates between negative and positive for all other diagonals above the main counter diagonal. The formation of matrix M Béz,n is presented in [36,37].…”

How to approximate cosine curve with 4th and 6th order Bezier curve in plane?

“…For more detail see [8,15]. It is well known that Taylor series of a function is an infinite sum of the functions derivatives at a single point a, also a Maclaurin series is a taylor series where 0. a = For any function Taylor series expansion is:…”