How to Find a Bezier Curve in $\mathbf{E}^{3}$

Abstract

"How to find any $n^{th}$ order B\'{e}zier curve if we know its first, second, and third derivatives?" Hence we have examined the way to find the B\'{e}zier curve based on the control points with matrix form, while derivatives are given in $\mathbf{E}^{3}$. Further, we examined the control points of a cubic B\'{e}zier 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 $\left( n-1\right) ^{th}$ order, the $\left(n-2\right) ^{th}$ and the $\left( n-3\right) ^{th}$ Bezier curves in respective order. There is a lot of the number of B\'{e}zier curves with known the derivatives with control points. Hence to find a B\'{e}zier curve we have to choose any control point of any derivation\. In this study we have chosen two special points which are the initial point $P_{0}$ and the endpoint $P_{n}$.