Vectors question: Is r'(t) technically a tangent (non-unit) vector to r(t)
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Vectors question: Is r'(t) technically a tangent (non-unit) vector to r(t)

[From: ] [author: ] [Date: 11-09-23] [Hit: ]
a helix) has constant torsion.Given two single-valued functions κ(s) and τ(s) given for s > 0, there exists exactly one space curve, determined except for orientation and absolute position, for which s is the arc length, κ is the curvature and τ is the torsion.......

τ = -N . B'

is called the "torsion", and is equal to the rate of change of the osculating plane. You can think of it as the amount by which the curve is twisting. A curve which twists the same amount (e.g. a helix) has constant torsion.

Given two single-valued functions κ(s) and τ(s) given for s > 0, there exists exactly one space curve, determined except for orientation and absolute position, for which s is the arc length, κ is the curvature and τ is the torsion. This is known as the fundamental theorem of space curves.

And if you thought that was remarkable... the whole thing generalises to higher dimensions and non-Euclidean spaces!

You've just opened a door to the wonderful world of differential geometry. You also may just have discovered your graduate school major. Or possibly not.

Answer to the additional details question:

I mean it can't be the zero vector. If |r'(t)| = 0, then r'(t)/|r'(t)| is (obviously) undefined.

-
it is true that dr / dt is a tangent vector as well as C dr / dt for any nonzero C

and v =dr / dt = | dr / dt ] T { definition of T }---->

d²r / dt² = a(t) = dv / dt T + v dT/dt = [ a_T] T + µ v N = [ a_T ] T + [a_N] N--->

. N = [ a - (a_T) T ] / | a - (a_T) T |
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