Examples of Bezier curves:

Bezier Case 1

For n = 1 . B_{0, 1} (u) = 1 - u, and B_{1, 1} (u) = u.

C (u) = (1 - u) P₀ + u P₁, that 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, that is a parabolic arc from P₀ to P₂.

When the control points might be in 3-D space, the curve lies wholly in the plane specified by the three control points. The curve is also approximated quite fine by the polygon build by the control points (the control polygon) - as seen in the instance of the figure below.

For n = 3. It is a very commonly utilized form. This can represent a quite complex set of shapes of curves. Notice down how the shape of the control polygon approximates the shape of the curve. The convex hull property is accurate (as this is for all Bezier curves). A loop in the control polygon might or might not lead to a loop in the Bezier curve - as illustrated in the instance below. By appropriate selection of control points, you may obtain a cusp (point where the derivative is not defined).

Instinctively, Bezier curves follow the shape of the control polygon. The purpose is seen from an analysis of the fundamental functions. At (u = 0), the curve is affected purely by the prime control point, P₀. At (u = 1), this is purely effected by P_n. In the middle, each control point, P_i, 'attracts' or 'influences' the curve most while the value of the parameter u is closest to i / n. To convince yourself regarding this, draw the functions (a) B_i,2 for i = 0, 1, 2 on a single graph; (b) B_i,3, for i = 0, 1, 2, 3 on a single graph.