Bézier, rational Bézier, B-spline and NURBS surfaces are
parametric surfaces. More precisely, for a point (*u*,*v*) in
the domain [0,1]×[0,1], it corresponds a unique point
**S**(*u*,*v*) on the surface. As (*u*,*v*) moves
in the domain, **S**(*u*,*v*) traces out the surface.
The seven items in the top of the ** Tracing Window** are
designed for this purpose.

Clicking on **Tracing Point** activates
the simplest tracing activity. You should see a red sphere, the
** tracing point** on the surface. As the

There are three ways of tracing the surface:

- Clicking on
**UV**allows to move the*uv*-indicator freely in the domain. - Clicking on
**U**restricts the movement in the*u*-direction. More precisely, the value of*v*is fixed to the current*v*value of the*uv*-indicator. As the*uv*-indicator moves, only the value of*u*changes. - Clicking on
**V**restricts the movement in the*v*-direction. More precisely, the value of*u*is fixed to the current*u*value of the*uv*-indicator. As the*uv*-indicator moves, only the value of*v*changes.

Click **here** for
de Casteljau's algorithm for curves, and
**here** for
de Casteljau's algorithm for surface. Also,
click **here** for de Boor's
algorithm for curves, and
**here** for
de Boor's algorithm for surfaces.

To activate de Casteljau's or de Boor's algorithm, click on
**Calculation**. By default,
the processing order is **V to U**;
but, clicking on **U to V** will change the
processing order. Note that buttons **U**,
**V** and
**UV** apply to both de Casteljau's and
de Boor's algorithms, since these three buttons provide three different ways
of tracing a surface.

If **V to U** is selected, this means the
de Casteljau's or de Boor's net for (*u*,*v*) is constructed by
applying de Casteljau's or de Boor's algorithm to each row, which defines
a curve in the *v*-direction, and computing a point corresponding to
the value of *v* in (*u*,*v*). Thus, for each row we have
an intermediate point. These intermediate points define a curve in the
*u*-direction. Applying de Casteljau's or de Boor's algorithm
again yields a point on the surface corresponding to (*u*,*v*).
In the following figure, which is created by clicking on
**V to U**, the red control nets are
the control nets in the *v*-direction, while the blue control net is
in the *u*-direction. Since this is a NURBS surface, de Boor's
algorithm is used and as a result, the number of control nets in the
*v*-direction (four, marked in white) is less than the number rows
(five).

If **U to V** is selected, the order of this
computation is reversed. The intermediate points are in the
*u*-direction while the final tracing point is computed in the
*v*-direction.

As the *uv*-indicator moves in the domain, the tracing point and the
computed control net moves as well.

Click **here** to download the surface
(in file **tracing.dat**) used
on this page for your practice.