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15.4 Computation of Curvature

Computation of curvature and the various directions of interest with respect to the curve is quite straightforward, given a parametric representation of the curve, so that we can instruct a spreadsheet to compute everything here along our curve with a handful of instructions.

The vector d T ^ d s , by the chain rule is d T ^ d t d t d s and d t d s is the reciprocal of d s d t which is | v ⟶ | and also can be written as | d r ⟶ d t | .

Since T ^ is the unit vector in the direction of d r ⟶ d t , it is d r ⟶ / d t | d r ⟶ / d t | .

The derivative of this latter expression with respect to t is, by the quotient rule

d T ^ d t = d ( d r ⟶ / d t | d r ⟶ / d t | ) / d t = d ( d r ⟶ / d t | v ⟶ | ) / d t = d 2 r ⟶ d t 2 | v ⟶ | − v ⟶ d | v ⟶ | d t | v ⟶ | 2

We can identify d 2 r ⟶ d t 2 with the acceleration of the motion which we denote by a ⟶ ( t ) .

We need here differentiate | v ⟶ | with respect to t , which, given that | v ⟶ | is ( v ⟶ · v ⟶ ) 1 / 2 obeys d | v ⟶ | d t = a ⟶ · v ⟶ | v ⟶ | .

Putting all this together we find

d T ^ d s = 1 | v ⟶ | d T ^ d t = a ⟶ | v ⟶ | − v ⟶ ( a ⟶ · v ⟶ ) | v ⟶ | | v ⟶ | 3 = a ⟶ ( v ⟶ · v ⟶ ) − v ⟶ ( a ⟶ · v ⟶ ) ( v ⟶ · v ⟶ ) 2

This result looks somewhat messy but it actually not so bad. Recall that a ⟶ ( v ⟶ · v ⟶ ) − v ⟶ ( a ⟶ · v ⟶ ) ( v ⟶ · v ⟶ ) is the projection of a ⟶ normal to v ⟶ . Therefore we have here that d T ^ d s is the projection of a ⟶ normal to v ⟶ divided by the square of the magnitude of v ⟶ .

Thus the curvature κ , which is the magnitude of this vector, is the component of a ⟶ normal to v ⟶ divided by the square of the magnitude of v ⟶ .

κ = | a ⟶ ( v ⟶ · v ⟶ ) − v ⟶ ( a ⟶ · v ⟶ ) v ⟶ · v ⟶ | v ⟶ · v ⟶

Consider the example of the helix: x = cos ⁡ t , y = sin ⁡ t , z = t .

We can compute: v ⟶ = d r ⟶ d t = ( − sin ⁡ t , cos ⁡ t , 1 ) and a ⟶ = d v ⟶ d t = ( − cos ⁡ t , − sin ⁡ t , 0 ) .

Here a ⟶ and v ⟶ are perpendicular, so that we get κ = | a ⟶ | v ⟶ · v ⟶ = 1 2 for all values of t .

The center of curvature is at the reflection of the point on the curve at which we compute it, through the z axis, that is, at the point with coordinates ( − cos ⁡ t , − sin ⁡ t , t ) , a distance 2 (or 1 κ ) from ( cos ⁡ t , sin ⁡ t , t ) in the direction of the projection of a ⟶ normal to v ⟶ .

Exercises:

15.1 Show that the curvature of a circle is 1 r . (This proves that the radius of curvature is 1 κ .)

15.2 Find the curvature κ , position r , and center of curvature at t = j 100 for j = 0 to 700 for the following curve (using a spreadsheet)

x = cos ⁡ t , y = cos ⁡ 2 t , z = sin ⁡ 3 t .

Chart it as best you can.

15.3 Set this curve up on the applet. Where is the curvature greatest.

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