A hunk of helix:

A few utilities...

These two are clear:

Next, I scratch out a little function that accepts a vector-valued function and a parameter, and returns the tangent vector, normal vector, etc., all at once. Most of what you can see in the function is probably obvious, as it is straight from the definitions. What might not be familiar is what I called centerOfCurvature, but this is simply the center of the osculating circle. Calling it for the moment, it should be clear that

,

where are the function, its radius of curvature, and its normal vector, resp.

Below I calculate the different points around the circle itself. Thanks to the convenient properties of the unit tangent and unit normal vectors, a point on the circle is given by

Note that the minus sign is due to the fact that we need a vector pointing  from the circle's center to the point on the curve; the normal points opposite that.

Grab a bunch of points on the circle and connect them using the Line function.

This is just to add color and such.

Helix with osculating circle

Another curve

Just some dopey curve.