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?
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$\begingroup$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.
$\endgroup$ 6 $\begingroup$There are two ways looking at it depending on context.
First, by Euler's Law the normal curvature can be considered in a variable plane with respect a direction so that there is no torsion. There are two directions in which it is possible.There is a major or principal normal curvature $k_1$ and a minor normal curvature $ k_2$ perpendicular to it. Each is placed /represented on either side of Mohr's circle. In bending their product is constant. The sense is same for synclastic surfaces and opposite for anticlastic surfaces. So in the former you are referring to unit principal $k_1$ vector.
Second, the order of vectors in applying right hand rule to take vector cross product has to be considered.Third vector is cross of first two, has two senses as vectors.
$$ TNB\quad NBT \quad BTN $$
$$ B\, X \,T = N $$
So by convention the correct principal direction of a convex surface normal is inward.
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