** Knot insertion** is one of the several fundamental geometric
algorithms that permit to modify the defining parameters of a surface

Due to the ** fundamental identity**
(

For a B-spline or a NURBS surface, one can insert a new knot in the *u*-
or the *v*-direction. If the new knot is inserted in the
*u*-direction (*resp.*, *v*-direction), the above knot
insertion process will be applied to all sets of control points that
define isoparametric curves in the same direction. Let us use the surface in
the default scene two as an example. Since we need to see the impact on
control points, we choose to display knot curves without the surface.
The surface in default scene two is a NURBS surface with 4 rows and 5
columns and the degrees in the *u*- and *v*- directions are
2 and 3, respectively.

To insert a new knot 0.25 in the *u*-direction, we need to move the
*uv*-indicator so that its *u* value is 0.25. Since the degree
in the *u*-direction is 2, three consecutive control points involve
in the knot insertion process.

Now, select **Advanced Features**,
followed by **Knot Insertion**, followed by
**on U**. Then, the *u* value of the
current position of the *uv*-indicator is inserted as a new knot into
the knot vector in the *u*-direction. The right figure below shows
this new knot. After corner cutting, on each row an existing control point
(those marked with white rectangles above) is removed and replaced with
two new control points, which are marked in yellow rectangles below.
Also note that a new knot curve corresponding to the new knot
*u* = 0.25 is shown.

Let us insert *v* = 0.8 into the knot vector in the *v*-direction.
Move the *uv*-indicator until its *v* value is equal to 0.80.
Then, select **Advanced Features**,
followed by **Knot Insertion**, followed by
**on V**. Since the surface is of degree
3 in the *v*-direction, four control points in each "column"
involve in the knot insertion process, and two of them will be cut and
replaced with three new control points. The control points that will be
cut are shown in white circles in the above figure. After the new knot is
inserted, we have the following (left) figure in which the new control points
are shown in yellow circles. Note also that the new knot *v* = 0.8 is
shown in the ** Tracing Window**, and the new knot curve is shown
in red.

As mentioned earlier, knot insertion does not change the shape of the give surface (this can be seen in the above three figures); but, the number of control points is increased.

If the new knot value is equal to an existing one, the new (and of course the existing) one becomes a multiple knot. Otherwise, the new knot is a simple knot. Note that increasing the multiplicity of a knot decreases its continuity on the surface.