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

Abstract

There are many ways to approximate cosine curve. In this study we have examined the way how the cosine curve can be written as any order Bezier curve. As a result using the Maclaurin series we have examined cosine curve as the 4th and the 6th order Bezier curve based on the control points with matrix form in E2. We give the control points of the 4th and the 6th order Bezier curve based on the coefficients. Also we give the coefficients based on the the control points of the 4th and the 6th order Bezier curve too.

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

“…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].…”

A Modelling on the Exponential Curves as $Cubic$, $5^{th}$ and $7^{th}$ B\'{e}zier Curve in Plane

“…Particularly, the 5 th order Bezier curve and its derivatives were studied by matrices in [4]. Besides, it has been investigated approximation methods in matrix form for Helix, sin waves and cosin curves by different order Bézier curves in [5][6][7]. The curve is also subjected to the differential geometry.…”