# Clamped, Open and Closed Surfaces

By default, the surface shown on the drawing canvas is clamped. More precisely, the surface is tangent to the plane determined by control points 0,0, 0,1 and 1,0 at 0,0, the plane determined by 0,n-1, 0,n and 1,n at 0,n, the plane determined by m-1,0, m,0 and m,1 at m,0, and the plane determined by m,n-1, m,n and m-1,n at m,n, where m+1 and n+1 are the number of rows and columns, respectively. In the following, the left figure is the control net, and the right one shows the clamped surface. Click here to download a copy of file tube-1.dat for your practice.

In addition to clamped surface, this system supports open and closed surfaces. In the Tracing Window, there are six options to choose:

For each direction (e.g., the u- or v-direction), one can choose Clamped, Open or Closed. The option Open means that the surface is not tangent to the control points in that direction. For example, if we choose Open in the u-direction and Clamped in the v-direction, we shall have the left figure below. The surface in the open direction (i.e., in this case, the u-direction) becomes shorter, while the surface maintains its "full length" in the other direction. On the other hand, if we choose Open in the v-direction and Clamped in the u-direction, the surface in the v-direction becomes shorter.

A surface is closed in a direction if the surface warps into a tube in that direction. If we choose Closed in the u-direction and Clamped in the v-direction, we shall have the left figure below. The surface wraps into a tube in the u-direction and fully "extended" in the other. If we choose Closed in the v-direction and Clamped in the u-direction, the surface wraps around becoming a long tube.

A surface can have one direction closed and the other direction open. In this case, the closed direction wraps into a tube while the open direction becomes shorter comparing with the clamped case.

Of course, a surface can be closed in both directions and in this case the surface becomes donuts-like. The following left is the above surface with both directions closed. The right figure shows some of its isoparametric curves.

The following figures are generated from file tube-2.dat. Click here to download a copy. The figures, from left to right, show clamped in both direction, closed in the u-direction, closed in the v-direction, and closed in both directions.