Differential Geometry/Normal Line and Principal Unit Normal Vector

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Consider a curve C of class of at least 2 with the arc length parametrization f(s). The unit tangent vector is f'(s). Since f'(s)·f'(s)=1, we can differentiate this to obtain f'(s)·f(s)=0.

Therefore, if f(s) is not the zero vector, then it is a vector that is orthogonal to the unit tangent vector. The line passing through this vector and f(s) is the 'principal normal line of this curve at the point f(s). It is clear that this normal line is within the osculating plane because it is spanned by f(s). The unit vector within this line,

p(s)=\frac{f''(s)}{|f''(s)|},

is the unit principal normal vector of the curve C at the point f(s).

It is important to note that when you change the direction of the parametrization of the curve, the unit tangent vector also changes directions, but the principal normal unit vector does not.