Moving Control Points

Changing the position of a control point will change the shape of the defined Bézier curve. Our question is:

Suppose a control point P_k is moved to a new position P_k + v, where vector v gives both the direction and length of this move. This is shown as follows:

Let the original Bézier curve be as follows:

Since the new Bézier curve is defined by P₀, P₁. ..., P_k+v, ..., P_n, its equation D(u) is

In the above, since only the k-th term uses a different control point P_k + v, after regrouping we know that the new curve is the sum of the original curve and an extra term B_n,k(u)v. This means:

More precisely, given a u, we have point C(u) on the original curve and D(u) on the new curve and D(u) = C(u) + B_n,k(u)v. Since v gives the direction of movement, D(u) is the result of moving C(u) in the same direction. The length of this translation is, of course, the length of vector B_n,k(u)v. Therefore, when B_n,k(u) reaches its maximum, the change from C(u) to D(u) is the largest.

The following figure illustrates this effect. Both the black and red curves are Bézier curves of degree 8 defined by 9 control points. The black one is the original curve. If its control point 3 is moved to a new position as indicated by the blue vector, the black curve changes to the red one. On each of these two curves there is the point corresponding to u=0.5. It is clear that C(0.5) moves in the same direction to D(0.5). The distance between C(0.5) and D(0.5) is the length of vector B_8,3(0.5)v = 8!/(3!(8-3)!)×0.5³(1-0.5)^8-3v = 0.22v. Hence, the distance is about 22% of the distance between the original control point 3 and the new control point 3 as shown in the figure below.

We can obtain one more important conclusion from the above discussion. Since B_n,k(u) is non-zero in the open interval (0,1), B_n,k(u)v is not a zero vector in (0,1). This means that except for the two endpoints C(0) and C(1) all points on the original curve are moved to new locations. Therefore, we have