Tangent and normal vectors can help us make interesting parametric plots.

A sine curve on a circle

Suppose you wish to draw a sine curve on a circle like this:

How do you do this? Well, a general method for placing one curve along another is to use unit tangent and unit normal vectors!

Thickening a curve

Suppose you have a vector-valued function that defines a curve in space, and you want to build a parameterized surface that looks like a “thickened” version of the curve. In other words, we want to convert a curve like

into a thickened “tube” like

To plot a “tube” around a vector-valued function , we need three handy vectors:

  • The unit tangent vector:
  • The unit normal vector:
  • The unit binormal vector:

Let’s see these vectors in action with our next example.