The ** Tracing Window** appears when you start the surface system.
Or, it can be activated by selecting

The ** Tracing Window** has three parts. The top part contains
buttons for generating and tracing the current surface, the middle part
is the domain of the surface (

The knot vector in the *u*-direction is shown horizontally, while the
knot vector in the *v*-direction is shown vertically. The knots
of the current surface are shown in white color. For the knots in the
*u*-direction, their values are displayed vertically in two rows.
The first row gives the multiplicity of knots and the second row displays
the knot values. The knots in the *v*-direction are displayed in the
form of m (d.ddd), where
m is the knot's multiplicity and
d.ddd is the knot's value.

Surface IDs are also shown in the domain in black. If the current
surface is subdivided, each subdivision's domain has the same color used
in the subpatch. Suppose a surface with ID **k** is subdivided, yielding
two subpatches. The IDs of these two subpatches are **k0** and
**k1**. For example, if a subpatch **0100** is subdivided, the
two subpatches will have IDs **01000** and **01001**. Subpatch
**k0** inherits the color of the original patch. But, the original
surface ID will not be used. In this way, one can easily determine the
parent-child relationship among subpatches. The above figures show the
matching of colors in the domain and colors used for the subpatches.

To move into a subpatch, click anywhere in its domain. Or, you can also
drag the *uv*-indicator into its domain.

The last two rows of buttons allows you to pick a type of knot sequence
for each direction. A surface can be clamped, open, or closed in one or
both directions. Since a parametric surface is defined on two variables,
*u* and *v*, the *u*-direction
(*resp.*, *v*-direction) is the direction where *u*
(*resp.*, *v*) can change while *v* (*resp.*, *v*)
is fixed. You can click on **Clamped**,
**Closed**, or
**Open** to make the surface clamped, closed
or open in a direction. Please refer to
**Clamped, Open and Closed Surfaces** for the
meaning of a clamped, closed and open surface.

Selecting **Calculation** activates the
visualization of de Casteljau's or de Boor's algorithms. All intermediate
computation steps (*i.e.*, all intermediate control nets) of
de Casteljau's or de Boor's algorithm will be displayed. See
**Tracing the Surface - de Casteljau's and de Boor's
Algorithms** for the details. Normally,
**Tracing Point** should be selected
before selecting **Calculation**.
This will allow both the control nets and tracing point be shown at the same
time. However, this is not necessary. If only
**Calculation** is selected, you still can
see all control nets except the tracing point.

Selecting **V to U** instructs the system
to do the following:

- For each row of control points, which define a curve, use de
Casteljau's or de Boor's algorithm to compute a point on this
curve at the value of
*u*of (*u*,*v*). - The points computed in the previous step defines another curve.
Then, compute a point on this curve corresponding to the value of
*v*of (*u*,*v*). This is the corresponding point of (*u*,*v*) on the surface.

You can request only to trace an isoparametric curve in the *u*- or
the *v*- direction. Or, you can request to trace in arbitrary
direction without any restriction. Click
**U** (*resp.*,
**V**) allows to trace the surface
in the *u*-direction (*resp.*, the *v*-direction).
More precisely, clicking on **U**
will cause the current value of *v* fixed and tracing activity
only changes the value of *u*. Clicking on
**UV** allows you to trace the
surface in an arbitrary direction (*i.e.*, no restriction
in any direction).