Tangent and normal vectors can help us plot make interesting parametric functions.
A sine curve on a circle
Suppose you wish to make 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!
Plot the curve “wrapped” around a circle
of radius .
To do this, start by setting This will produce our circle of radius .
Now we need to add our sine curve. To do this, we compute the unit tangent vector
and the unit normal vector:
We’ve plotted our circle of radius with some unit tangent and unit normal vectors
for your viewing pleasure: 
To add the sine curve, quite litterally, add it in: We can confirm our construction
with SAGE:
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.
You have been given a curve in space, say We want to make a tube around it of
radius .
Here is the idea, we want to take our curve and attach two orthogonal
vectors to every point 
We’ll start by computing the unit tangentunit normalunit binormal
vectors. Write with me From these vectors we can obtain the unit tangentunit
normalunit binormal
vectors by differentiating: Finally we can obtain the unit tangentunit normalunit binormal
vectors via the cross product: Looking again at
we see the unit normal vectors in red, and the unit binormal
vectors in blue. So we now can plot our tube where runs from to :
We can check our work with SAGE.