CURSUS DIFFERENTIAALMEETKUNDE

3.* CURVATURE*

*Definition 21:* Given a curve with arc length as parameter: * x*(s). Let

We define the curvature vector to be the derivative of the tangent vector, that is

Now we choose for every s a vector * n*(s) (called principal normal vector) with length 1 which has the same direction or the opposite direction as

Note that this can be done in two ways. For planar curves we see:

The curvature vector becomes

The principal normal vector keeps its length 1 and stays at one side of the curve.

So for each s there is a number κ(s) so that __x__** ^{..}**(s) = κ(s)

We call κ(s) the curvature function of the given curve. The curvature changes its sign in an inflection point, and its absolute value is the length of the curvature vector.

We define the curvature radius R(s) to be 1/κ(s) (∞ in an inflection point), and the curvature center to be

*Explanation 22:* The curvature is a measure for the change of the direction of the tangent vector:

Let Δφ be the angle between * t*(s) and

||

So |κ(s)| = ||

*Explanation 23*: About the curvature circle.

The circle r(cos(t),sin(t)) becomes with arc length as parameter: (cos(s/r),sin(s/r)). Hence it follows that the curvature satisfies κ(s) = ||__x__** ^{..}**(s)|| = 1/r.

Now let

The curvature circle has parametrisation

*Problem 24*: Prove that s is also arc length of the circle * c*(s), and that

Here are two theorems about the curvature function that we don't prove. We can prove the second one by generalizing example 27 which comes immediately after it.

*Proposition 25*: κ(s) is invariant under motion (that is to say: invariant when the curve is stiffly moved through space).

*Proposition 26*: With any given curvature function there is a planar curve having this curvature. This curve is unique up to rotations and translations.

*Example 27*: (as an illustration of proposition 26).

Let κ(s)=s, and * x*(s) a planar curve with this curvature. (We are going to determine this curve.)

Because ||

Hence α(s)=(1/2)s

When we choose an other c this means we rotate the curve, when we choose an other d and d' we translate the curve.

This curve is called clotoid (spiral of Cormi). Since

For planar curves in the x,y-plane we always have: κ(s) is the derivative with respect to arc length s of the angle between the tangent vector of length 1 and the x-axis.

*Problem 28*: Determine the curvature of the cycloid (see problems 18 and 20) as a function of s.

*Problem 29*: Prove that the straight lines are the curves with curvature function κ(s)=0.

Thereafter, prove that, whenever the tangents of a curve go through a fixed point, or are all
parallel, this curve must be a straight line.