## Bezier Curves and Surfaces

Bezier Curves is the important topic of the Computer Graphics. Moreover, It is the Important subject of the Computer Science & Technological field.

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- It is developed by French engineer Pierre Bezier for the Renault automobile bodies.
- It has a number of properties and easy to implement so it is widely available in various CAD and graphics package.

**Bezier Curves **

- The Bezier curves section can fit with any number of control points.
- The number of control points and their relative position gives the degree of the Bezier polynomials.
- With the interpolation spline, Bezier curve can be specified with boundary condition or blending function.
- A most convenient method is to specify Bezier curve for blending function.
- Consider we are given n+1 control point position from p
_{0}to p_{n}where p_{k}= (x_{k}, y_{k}, z_{k}). - This is blended to gives position vector p(u) which gives a path of the approximate Bezier curve is:

Bezier curve is a polynomial of degree one less than the number of control points.

Below figure shows some possible Bezier Curves shapes by selecting various control point.

Fig.Â Example of 2D Bezier curves generated by the different number of control points.

Moreover, Efficient method for determining coordinate positions along a Bezier curve can set up using recursive calculation

For example, successive binomial coefficients can calculate as:

C(n, k) = (n-k+1)/k . C(n, k-1) Where n>=k

**Properties of Bezier curves **

- It always passes through first control point i.e. p(0) = p
_{0} - Moreover, It always passes through last control point i.e. p(1) = p
_{n} - Parametric first order derivatives of a Bezier curve at the endpoints can obtain from control point coordinates as
- đ
(0) = âđđ^{â˛}_{0 }+ đđ_{1} - đ
(1) = âđđ^{â˛}_{đâ1 }+ đđ - Also, Parametric second order derivatives of endpoints also obtained by control point coordinates as:
(0) = đ(đ â 1)[(đ^{â˛â˛}_{2 }â đ_{1}) â (đ_{1 }â đ_{0})](1) = đ(đ â 1)[(đ^{â˛â˛}_{đâ2 }â đ_{đâ1}) â (đ_{đâ1 }â đ_{đ})]- Bezier curve always lies within the convex hull of the control points.
- Moreover, Bezier blending function is always positive.
- Sum of all Bezier blending function is always 1.
- So any curve position simply the weighted sum of the control point positions.
- Bezier curve smoothly follows the control points without erratic oscillations.

**Related Terms**

Computer Graphics, Cardinal Splines, Cubic Spline Interpolation Methods, Hermit Interpolation

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