When working with space curves, why is it called the "Principal Unit Normal Vector"? I know there are two. So how do we know which one is the principal one? Also, is there a principal unit binormal vector and principal unit tangent vector? Why or why not? What makes the naming scheme need the word principal for some and not others? And what does "principal" mean?
[Math] What does “Principal” mean in “Principal Unit Normal Vector”
differential-geometrymultivariable-calculusterminology
Best Answer
Roughly, the principal unit normal vector is the one pointing in the direction that the curve is turning. It's the one obtained by a particular formula - the formula you've presumably been taught.
There's no principal unit tangent or binormal. The tangent doesn't have a "principal" because while there are indeed two options, one is forward and one is backward according to the parameterization. We never care about the backward one, so the "unit tangent vector" is always the one pointing forward along the curve, by convention. We could take the other convention, but that would be silly.
For the binormal, there's only one option once you've decided which tangent vector and which normal vector you're using; so again, when we say "binormal" we mean the one generated by the standard formula, using the unit tangent and the principal unit normal.
Neither approach works for the unit normal. There are two options for the unit normal, even once you've picked your unit tangent; and both options are perfectly reasonable, we just have to pick one. So we picked one, in the most straighforward way available, and called it the "principal" unit normal.