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VF.3 Arc Length and Curvature

The length of a space curve is defined as follows. Suppose that the curve has the vector equation , or, the parametric equations , , where ', and are continuous. If the curve is traversed exactly once as increases from to , then it can be shown that its length is:

or

Notice that both of the arc length formulas can be put into the more compact form of:

Also notice that the equation for arc length in 3-D is very similar to the equation for arc length in 2-D.

Here is a picture to help you visualize the concept of arc length.

Notice that the following equations are equal:

 

and

If we were to use the formula for arc length to determine the length of these curves they would be the same, because they represent the same curve. These two equations are called parameterizations of the curve r. Now suppose that r is any function represented by , and at least one of the components is defined as one to one. Now use that equation to define an arc length function s as:

This, s(t) is the length of the part of the equation between r(a) and r(t). If we differentiate both sides of the equation above using Part 1 of the Fundamental Theorem of Calculus, we obtain:

It is often useful to parametrize a curve with respect to arc length because arc length arises naturally from the shape of the curve and does not depend on a particular coordinate system. If a curve r(t) is already given in terms of a parameter t and s (t) is the arc length function given by the equation (a), then we may be able to solve for t as a function of s: t = t(s). Then the curve can be reparametrized in terms of s by substituting for t: r = r(t(s)). Thus, if s = 3 for instance, r(t(3)) is the position vector of the point 3 units of length along the curve from the starting point.

Curvature

The unit tangent vector, if you recall, gives the direction of the space curve at a given point and is given by the following formula:

In the following figure you can see that sometimes the direction of the unit tangent vector changes very quickly and sometimes it hardly changes.

The curvature of the space curve at a given point is a measure of how quickly the curve changes direction at that point. More specifically, we define it to be the magnitude of the rate of change of the unit tangent vector with respect to arc length.

The curvature of a curve is:

Using the chain rule we come to a simpler version of the equation of a curvature:

Although the formula above can be used to compute all cases of curvature, the following formula is often more convenient to apply:

The curvature of the curve given by the vector function r is

Theorem

For a special case of a plane curve with equation , we can use the following equation to find the curvature

The Normal and Binormal Vectors

At a given point on a smooth space curve r(t), there are many vectors that are orthogonal to the unit tangent vector T(t). We single out one by observing that, since for all t, we have T (t)T' (t) = 0. Therefore T'(t) is orthogonal toT (t). Notices that T'(t) is itself not a unit vector. But if r' is also smooth, we can define the principal unit normal vector N(t) (or the unit vector) as

Also, the binormal vector is defined as the cross product between the unit tangent vector and the normal vector. It is perpendicular to both T and N and is also a unit vector.

 

The tangent, normal, and bi-normal vectors are useful for making some special planes. One of these planes is called the normal plane, and this plane is formed by the normal and binormal vectors. This plane contains all lines that are orthogonal to the tangent vector.

Another plane describe by these lines is called the osculating plane. This plane is determined by the tangent and normal vectors. The word "osculating" is Latin and has a meaning of kiss. This plane uses that name because it just "kisses" the line from which it is formed.

The circle that lies is the osculating plane, has the same tangent as the line it is being formed with, and has a radius , is called the osculating circle (or the circle of curvature). This circle has the same tangent vector, normal vector, and curvature as the line at the point it is being formed from.

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