2022 # 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 i…

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“…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]:…”

confidence: 99%

“…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]:…”

confidence: 99%

“…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]:…”

confidence: 99%

“…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:…”

confidence: 99%