The line perpendicular to the tangent, and lying in the osculating plane (the plane through "three consecutive points" on the curve), is called the Principal Normal. For example, this curve lies completely in the plane (it's 2D), so the principal normal will always lie flat on the worksheet. The Unit Normal [math] {\small {\bf n}} [/math] may be chosen to face either direction along this line. Here are two ways we might choose [math] {\small {\bf n}} [/math]:

I'll use the first way; that is, I'll choose [math] {\small {\bf t},{\bf n},{\bf b}} [/math] consistently as a right-handed coordinate system. Note that, by continuous manipulation of this curve in 3 dimensions, I can still make [math] {\small {\bf n}} [/math] appear -- from this point of view-- on the other side of [math] {\small {\bf t}} [/math] (rotated by -90°). How? And what physical meaning is implied?* _____________ *For example, try assigning the curve a left/right, and a top/bottom.

The osculating circle is the circle which passes through three consecutive points on the curve.

As with all differential definitions, this is a statement about limits. If we take three points on the curve and bring them closer to one another in such a way that the distance between them can be made less than any given (finite, nonzero) quantity... is there still a unique circle defined by these three points? What properties does it share in common with the curve? It appears the answer is yes. And if we lift the curve off the page so it takes up space? How might such a relationship be defined, and checked?