Tubes


Tube around a helix.
Click for VRML

Introduction:

One of the easiest ways to generate a surface from a space curve is to put a tube around the curve. This can be done relatively simply by introducing a new parameter theta, which can then be used to define a circle of radius r in the Frenet frame of the curve. In particular, we want the curve to be perpendicular to these circles at all points, so we can use the normal and binormal unit vectors as a basis for our circles. At each point along the curve, we can then simply draw this new circle and connect the points to form a tube. A logical extension of the tube, called a sea shell, allows the radius to change with the value of t.

Definition:

,

where x(t) is a curve, N and B are the normal and binormal vectors to that curve, and r is the radius of the tube.

Properties:

Tubes have the interesting property that their volume does not depend on what curve they surround. That is, the volume enclosed by a tube depends solely on the length of the curve and the radius of the tube.