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In this study we have examined, involute of the cubic Bezier curve based on the control points with matrix form in E3. Frenet vector fields and also curvatures of involute of the cubic Bezier curve are examined based on the Frenet apparatus of the first cubic Bezier curve in E3.

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In this study, we have examined how to find any 5 th order Bézier curve with its known first, second and third derivatives, which are the 4 th order, the cubic and the quadratic Bézier curves, respectively, based on the control points of given the derivatives. Also we give an example to find the 5 th order Bézier curve with the given derivatives.

How to find any n th order B ézier curve if we know its first, second, and third derivatives?" Hence we have examined the way to find the B ézier curve based on the control points with matrix form, while derivatives are given in E 3 . Further, we examined the control points of a cubic B ézier curve with given derivatives as an example. In this study first we have examined how to find any n th order Bezier curve with known its first, second and third derivatives, which are inherently, the (n − 1) th order, the (n − 2) th and the (n − 3) th Bezier curves in respective order. There is a lot of the number of B ézier curves with known the derivatives with control points. Hence to find a B ézier curve we have to choose any control point of any derivation İn this study we have chosen two special points which are the initial point P 0 and the endpoint P n .