Bezier Curve Properties

Several important properties of Bezier curves are summarized here. Many of these can be understood by examining the Bernstein blending functions used to define Bezier curves.

1. The k-th derivative at the start (end) of a Bezier curve depends only on the first (last) (k+1) control points. Two obvious special cases:
   • k=0: The Bezier curve starts at the first control point and stops at the last control point. (In general, it will not pass through any other control point.)
   • k=1: The vector tangent to the Bezier curve at the start (stop) is parallel to the line connecting the first two (last two) control points.
2. A Bezier curve will always be completely contained inside of the Convex Hull of the control points. For planar curves, imagine that each control point is a nail pounded into a board. The shape a rubber band would take on when snapped around the control points is the convex hull. For Bezier curves whose control points do not all lie in a common plane, imagine the control points are tiny balls in space, and image the shape a balloon will take on if it collapses over the balls. This shape is the convex hull in that case. In any event, a Bezier curve will always lie entirely inside its planar or volumetric convex hull.
3. Closely related to the previous is the fact that adjusting the position of a control point changes the shape of the curve in a "predictable manner". Intuitively, the curve "follows" the control point. In the image below, see how a curve defined in terms of four control points (the magenta curve) changes when one of its control points is moved to the right, yielding the modified (cyan) curve.
4. There is no local control of this shape modification. Every point on the curve (with the exception of the first and last) move whenever any interior control point is moved. This property can also be observed in the image shown in the previous item.
5. Also related to property #2 is the fact that Bezier curves exhibit a variation diminishing property. Informally this means that the Bezier curve will not "wiggle" any more than the control polygon does. In other words, the curve will not wiggle unless the designer specifically introduces wiggling in the control polygon. More formally, the variation diminishing property can be stated as follows: any straight line will intersect legs of the control polygon at least as many times as it crosses the Bezier curve itself. See the example below which illustrates the property with a degree 12 Bezier curve:
6. The effect of control point P_i on the curve is at its maximum at parameter value t = i/n. Among other things, this somewhat ameleriorates problems related to the fact that there is no local control (property #4).
7. Bezier curves exhibit a symmetry property: The same Bezier curve shape is obtained if the control points are specified in the opposite order. The only difference will be the parametric direction of the curve. The direction of increasing parameter reverses when the control points are specified in the reverse order.
8. Bezier curves are invariant under affine transformations, but they are not invariant under projective transformations.
9. Bezier curves are also invariant under affine parameter transformations. That is, while the curve is usually defined on the parametric interval [0,1], an affine transformation mapping [0,1] to the interval [a,b], a≠b, yields the same curve.