Curvature - Three Dimensional Space

In this part we want to briefly discuss the curvature of a smooth curve (remind that for a smooth curve we require ^→r′ (t) is continuous and ^→r′ (t) ≠ 0 ).  The curvature measures how fast a curve is changing direction at a specific point.

There are various formulas for finding out the curvature for a curve. The formal definition of curvature is,

k = |d T^→/ ds|

in which  T^→ stands for the unit tangent and s is the arc length. Remind that we saw in a preceding section how to reparameterize a curve to acquire it into terms of the arc length.

Generally the formal definition of the curvature is not simple to use so there are two alternate formulas that we can utilize. Here they are.

Κ = ||^→T' (t)|| /|| ^→r' (t)||

Κ = || ^→r' (t) * ^→r''(t)|| /|| ^→r' (t)||³

These may not be particularly simple to deal with either, although at least we don't need to reparameterize the unit tangent.