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!

and the unit normal vector:

We’ve plotted our circle of radius with some unit tangent and unit normal vectors for your viewing pleasure:

We can confirm our construction by making a graph:

### 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

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.

Here the function runs along the center of the tube, and and create a moving axis, where a circle is drawn. Putting this all together we get a tube drawn around . We can check our work with the following interactive: