Bezier Curves

So far we have discussed how to draw a curve through a given set of points. These methods in many cases give excellent results and are particularly useful in describing the forms the foundation of which was obtained by experiments or calculations. However, the above methods, in particular, cubic splines, are inconvenient for interactive work. The direction and magnitude of tangential do not provide the necessary intuitive understanding of the curve because the relationship between a set of numbers and form of the corresponding curve is not obvious.

The Bezier curve is given by a polygon, as shown in Fig. 3.5.

Fig.3.5. Bezier Curve

Since the basis is the Bezier Bernstein, the following properties of Bezier curves are known:

· Basis functions are real.

· The degree of the polynomial that defines the portion of the curve is less by 1 than the number of points corresponding to the polygon.

· The basis of the curve shape contours the polygon.

· The first and last points of the curve coincide with the corresponding points defining the polygon.

· Tangent vectors at the ends of the curve in the direction coincide with the first and last sides of the polygon.

· Curve lies inside the convex hull of the polygon, i.e. within the large polygon built from the given points.

· The curve has the property of reducing variation. This means that the curve intersects any straight line not more than polygon.

· Curve is invariant under affine transformations.

Considering the above properties, we can easily predict the curve shape by the polygon form.

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