Bezier Curves:-

Bezier curve is an another approach for the construction of the Curve.A Bezier curve is determined by a defining polygon. Bezier curves have a number of properties that make them highly useful and convenient for curve and surface design. They are also easy to implement. Therefore Bezier curves are widely available in various CAD systems and in general graphic packages. In this section we will discuss the cubic Bezier curve. The reason for choosing cubic Bezier curve is that they provide reasonable design flexibility and also avoid the large number of calculations.

In general, a Bezier curve section can be fitted to any number of control points. However, as number of control points increases, the degree of the Bezier polynomial also increases. Because in a Bezier curve a degree of a polynomial is one less than the number of control points used. For example. three control points generate a parabola four points generate cubic curve and so on. This is illustrated in Fig. (19)

The Bezier curves can be specified with boundary conditions, with a characterizing matrix or with blending functions. Out of these, blending function specification is the most convenient way for general Bezier

consider that the curve has n +1 control points Pk (xk, yk, zk) ... where k varies from 0 to n. The co-ordinates of these control points can be blended to produce position vector p(u), which gives the path of an approximating Bezier polynomial function between p0 and pn .The position vector can be given by.

Bernstein polynomials:-

The Bezier blending functions BEZk,n (u) are the Bernstein polynomials. They are specified as,

Equivalently, we can define blending functions with the recursive calculation as,

Properties of Bezier curve:-

1. The basis functions are real.
2. Bezier curve always passes through the first and last control points i.e. curve has same end points as the guiding Polygon.
3. The degree of the polynomial defining the curve segment is one less than the number of defining polygon point. Therefore, for 4 control points, the degree of the polynomial is three, i.e. cubic polynomial.
4. The curve generally follows the shape of the defining polygon.
5. The direction of the tangent vector at the end points is the same as that of the vector determined by first and last segments.
6. The curve lies entirely within the convex hull formed by four control points.
7. The convex hull property for a Bezier curve ensures that the polynomial smoothly follows the control points.
8. The curve exhibits the variation diminishing property. This means that the curve does not oscillate about any straight line more often than the defining polygon.
9. The curve is invariant under an affine transformation.

In cubic Bezier curve four control Points are used to specify complete curve. Unlike the B-spline curve, we do not add intermediate points and smoothly extend Bezier curve, but we pick four more points and construct a second curve which can be attached to the first. The second curve can be attached to the first curve smoothly by selecting appropriate control points.

Fig. (20) shows the Bezier curve and its four control points. As shown in the Fig. (20), Bezier curve begins at the first control point and ends at the fourth control point. This means that if we want to connect two Bezier curves, we have to make the first control point of the second Bezier curve match the last control point of the first curve. we can also observe that at the start of the curve, the curve is tangent to the line connecting first and second control points. Similarly at the end of curve, the curve is tangent to the line connecting the third and fourth control point. This means that, to join two Bezier curves smoothly we have to place the third and the fourth control Points of the first curve on the same line specified by the first and the second control points of the second curve.