# Ruled Surfaces

Given two curves *C*_{1}(*u*) and
*C*_{2}(*v*), the ruled surface is the surface
generated by connecting line segments between corresponding points, one on
each given curve.
More precisely, if *t* is a value in the domain [0,1] of both curves,
a segment between *C*_{1}(*t*) and
*C*_{2}(*t*) is constructed. This segment is
usually referred as a *ruling* at *t*. As *t* moves
from 0 to 1, the ruling at *t* sweeps out a surface and this is the
ruled surface defined by curves *C*_{1}(*u*) and
*C*_{2}(*v*).

To design a ruled surface, select
**Advanced Features** followed by
**Cross Sectional Design**. This will
bring up the curve system. In the curve system, just design two NURBS
curves that are not coplanar, and then select
**Techniques** followed by
**Generate Ruled Surface**. The surface
system will display a ruled surface defined by the two curve in the curve
system.

Note that these two curves do not have to be of the same degree with
the same number of control points and the same number of knots.
Note also that before sending the created surface to the surface system, the
curve system will modify the two curves. As a result, the number of control
points and the number of knots, and their positions shown in the surface
system, in general, will not be the same as those in your original design.

Many commonly seen and useful surfaces are ruled surfaces. Here are
a few examples. Since circles will be frequently used, in case you forgot
how to design a circle, click **here** to
download a file (**circle7.dat**) in which
a circle is defined as a NURBS curve of degree 2 using seven control points:

### Example 1: Hyperbolic Paraboloid

The well-known saddle surface (*i.e.*, hyperbolic paraboloid) is
generated by two segments as a ruled surface. The following is the two line
segments in the curve system, each of which is defined as a NURBS curve of
degree 3 with four control points. As long as the control points are
collinear, the curve is a line segment.
Click **here** to download a copy of this file
**hp.dat**.

After sending this design to the surface system, we shall see the following
hyperbolic paraboloid on the drawing canvas. The right figure shows clearly
a few rulings.

### Example 2: Cylinder

Cylinder is another well-known ruled surface. It is generated from two
circles. In the curve system, one can create a circle in the *xy*-plane
and save it. Then, import this circle back into the scene. This will
produce two circles with the same orientation. Translating one of them in
the *z*-direction would produce the desired circles.
Click **here** to download a
copy of this file **cylinder.dat**.

After sending this design to the surface system, we shall see the following
cylinder on the drawing canvas. The right figure shows clearly a few rulings.

### Example 3: Hyperboloid of One Sheet

A variation of the cylinder yields a hyperboloid of one sheet. What we
need to do is rotating one of the two circles about the *z*-axis some
degree. Click **here** to download a
copy of this file **hy.dat**.

After sending this design to the surface system, we shall see the following
hyperboloid of one sheet on the drawing canvas. The right figure shows
clearly a few rulings between the two given circles.

### Example 4: A Ruled Surface Generated from Two Curves

The following are two NURBS curves. The one in the foreground is of degree 2
defined by five control points, and the one in the background is of degree 2
defined by four control points. Click
**here** to download a copy of this file
**other.dat**.

After sending this design to the surface system, we shall see the following
ruled surface on the drawing canvas. The right figure shows clearly a few
rulings between these two curves.

### Example 5: An Interesting Surface

Let us design a helix-like surface. In the curve system, we first put
a line segment in the *z*-axis as a degree 3 NURBS curve defined by
four control points. Then, we put 21 control points in a spiral-like
curve surrounding the *z*-axis. This is a NURBS curve of degree 3.
These two curves are shown below. Click
**here** to download a copy of this file
**helix.dat**.

After sending this design to the surface system, we shall see the following
ruled surface on the drawing canvas. The right figure shows clearly a few
rulings between these two curves.