The hodograph of a curve is actually its first derivative. The hodograph
(*i.e.*, first derivative) of a Bézier (*resp.*, B-spline)
curve of degree *p* is a Bézier (*resp.*, B-spline) curve
of degree *p*-1 whose control points can be computed easily
from the given control points. As a result, computing the second derivative
of a Bézier or a B-spline curve is a simple task, because one can
construct the control points of the hodograph and then construct the
hodograph of the hodograph. In fact, it is not difficult to fine a formula
for higher order derivatives.

Selecting **Hodograph (First Derivative)**
and **Second Derivative** will open the
hodograph and second derivative windows, respectively.
Select **Dismiss** to close the window.
These windows have two more buttons.
**Magnify** and
**Shrink** can be used to zoom in
and zoom out with respect to the coordinate origin. If a curve has
many control points, it is possible that some control points in the
hodograph and second derivative windows are too populated to be
shown clearly. Thus, **Magnify**
and **Shrink** are useful in this
and other cases.

In the following, the left figure is a Bézier curve of degree 4 with
*u* = 0.42. The middle one is the hodograph window and
the right one is the second derivative window. In all windows,
the control points are clearly shown. As *u* moves on the
given curve **p**(*u*), the points on
**p**'(*u*) and **p**''(*u*) are shown
on-the-fly. The tangent (*resp.*, second derivative)
vector is the vector from the coordinate origin to
**p**'(*u*) (*resp.*, **p**''(*u*)).

However, rational Bézier and NURBS curves do not have such an elegant property. Their hodograph and second derivative curves are no more rational Bézier and NURBS curves and as a result no control points are shown in the hodograph and second derivative windows.

The above shows the hodograph and second derivative of a rational
Bézier curve. In this case,
**Shrink** is needed to bring
the hodograph and second derivative curves into the window area.

Further discussion can be found in
**The First Derivative (Hodograph) and Second
Derivative**.