Instance of Bezier curves:
Some instance of Bezier curves.
Bezier Case 1
For n = 1 . B0, 1 (u) = 1 - u, and B1, 1 (u) = u.
C (u) = (1 - u) P0 + u P1, which is the equation of a straight line among P0 and P1.
Bezier Case 2
For n = 2. Now we obtain C (u) = (1 - u)2 P0 + 2u (1 - u) P1 + u2 P2, which is a parabolic arc from P0 to P2.
Whereas the control points may be in 3-D space, the curve lies entirely in the plane described by the three control points. Also, the curve is approximated fairly nice by the polygon composed by the control points (the control polygon).
Bezier Case 3
For n = 3. This is a very commonly utilized form. It might represent a fairly complex set of shapes of curves, as seen in the figures below. Notice down how shape of the control polygon approximates the shape of the curve. The convex hull property is true (as it is for all Bezier curves). A loop in the control polygon can or cannot lead to a loop in the Bezier curve - as seen in the instance below. By proper selection of control points, you might get a cusp (point where the derivative is not defined).
Instinctively, Bezier curves follow the shape of the control polygon. The reason is seen from an analysis of the basic functions. At (u = 0), the curve is influenced purely by the first control point, P0. At (u = 1), it is purely effected by Pn. In the middle, each control point, Pi, 'influences' or 'attracts' the curve most while the value of the parameter u is closest to i / n. To convince yourself regarding this, draw the functions (a) Bi,2 for i = 0, 1, 2 on a single graph; (b) Bi,3, for i = 0, 1, 2, 3 on a single graph.