Question

what is principal normal vector ? how its define ?

Answer 1

i think the principal normal vector to a curved surface; such as a sphere
is the normal that points outwards, rather than inwards

the unprincipled normal points inwards

Answer 2

For a vector function \(\mathbf r(t)\), I believe principle normal vector \(\mathbf N(t) \) would be: \[
\mathbf N(t) = \frac{\mathbf T'(t)}{\|\mathbf T'(t)\|},\quad \mathbf T(t)=\frac{\mathbf r'(t)}{\|\mathbf r'(t)\|},
\]

Answer 3

@wio what is the meaning of T'(t) how is it graphically this point confusing me

Answer 4

The \(\mathbf T(t)\) is the tangent vector.
It is like the velocity of a vector, if \(t\) is the time. It tells you the direction and magnitude that the particle is traveling in at that moment.

So \(\mathbf r(t)\) tells the position while \(\mathbf T(t) = \mathbf r'(t) \) tells the direction of motion.

Answer 5

Should have put \(\mathbf T(t) =\frac{\mathbf r'(t)}{\| \mathbf r'(t)\|} \). The magnitude of \(\mathbf T(t)\) is always 1 as a unit vector.

Answer 6

So in short \(\mathbf r'(t)\) is velocity, and \(\mathbf T(t)\) is just a unit vector tangent to the particle's path.

Answer 7

On the other hand \(\mathbf N(t)\) tells you the direction normal (perpendicular) to the particle's path. It ultimately tells you the direction that the particle is turning toward. So it will tell you if it is turning left or right. If the particle were not turning at all, then \(\mathbf T(t)\) would be constant with respect to \(t\) and \(\mathbf N(t)\) would be zero.