Bezier Curves and Surfaces
- 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.
- 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 p0 to pn where pk = (xk, yk, zk).
- 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. 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) = p0
- Moreover, It always passes through last control point i.e. p(1) = pn
- Parametric first order derivatives of a Bezier curve at the endpoints can obtain from control point coordinates as
- 𝑝′(0) = −𝑛𝑝0 + 𝑛𝑝1
- 𝑝′(1) = −𝑛𝑝𝑛−1 + 𝑛𝑝
- Also, Parametric second order derivatives of endpoints also obtained by control point coordinates as:
- ′′(0) = 𝑛(𝑛 − 1)[(𝑝2 − 𝑝1) − (𝑝1 − 𝑝0)]
- ′′(1) = 𝑛(𝑛 − 1)[(𝑝𝑛−2 − 𝑝𝑛−1) − (𝑝𝑛−1 − 𝑝𝑛)]
- 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.