Engineering Math

9.4.1 Arc Length And Orientations

The application of the integral considered here is the concept of the length of a curve.

1. x_i is continuous on
   [a,b]
   .
2. x_i^′ exists and is continuous and bounded on
   [a,b]
   , with x_i^′
   (a)
   defined as the derivative from the right,

   xi(a-+h-)−-xi(a) hl→im0+ h ,

   and x_i^′

   (b)
   defined similarly as the derivative from the left.

3. For p
   (t)
   ≡
   (x1(t),⋅⋅⋅,xn(t))
   , t → p
   (t)
   is one to one on
   (a,b)
   .
4. |p′(t)|
   ≡
   ( ∑ ) ni=1 |x′i(t)|2
   ^1∕2≠0 for all t ∈
   [a,b]
   .
5. C = ∪
   {(x1 (t),⋅⋅⋅,xn (t)) : t ∈ [a,b]}
   .

The functions x_i

, defined above are giving the coordinates of a point in ℝⁿ and the list of these functions is called a parametrization for the smooth curve. Note the natural direction of the interval also gives a direction for moving along the curve. Such a direction is called an orientation. The integral is used to define what is meant by the length of such a smooth curve. Consider such a smooth curve having parametrization . Forming a partition of , a = t₀ < < t_n = b and letting p_i = ( x₁, , x_n ), you could consider the polygon formed by lines from p₀ to p₁ and from p₁ to p₂ and from p₃ to p₄ etc. to be an approximation to the curve C. The following picture illustrates what is meant by this.

Now consider what happens when the partition is refined by including more points. You can see from the following picture that the polygonal approximation would appear to be even better and that as more points are added in the partition, the sum of the lengths of the line segments seems to get close to something which deserves to be defined as the length of the curve C.

Thus the length of the curve is approximated by

∑n |p (tk)− p (tk−1)|. k=1

Since the functions in the parametrization are differentiable, it is reasonable to expect this to be close to

∑n |p′(tk−1)|(tk − tk− 1) k=1

which is seen to be a Riemannn sum for the integral ∫ _a^b

dt and it is this integral which is defined as the length of the curve.

Would the same length be obtained if another parametrization were used? This is a very important question because the length of the curve should depend only on the curve itself and not on the method used to trace out the curve. The answer to this question is that the length of the curve does not depend on parametrization. The proof is somewhat technical so is given in the last section of this chapter.

Does the definition of length given above correspond to the usual definition of length in the case when the curve is a line segment? It is easy to see that it does so by considering two points in ℝⁿ p and q. A parametrization for the line segment joining these two points is

fi(t) ≡ tpi + (1− t)qi,t ∈ [0,1].

Using the definition of length of a smooth curve just given, the length according to this definition is

∫ 1( n )1 ∕2 ∑ (p − q )2 dt = |p − q|. 0 i=1 i i

Thus this new definition which is valid for smooth curves which may not be straight line segments gives the usual length for straight line segments.

The proof that curve length is well defined for a smooth curve contains a result which deserves to be stated as a corollary. It is proved in Lemma 9.6.6 on Page 523 but the proof is mathematically fairly advanced so it is presented later.

When the parameterizations are equivalent, they preserve the direction of motion along the curve, and this also shows there are exactly two orientations of the curve since either g⁻¹ ∘ f is increasing or it is decreasing. This is not hard to believe. In simple language, the message is that there are exactly two directions of motion along a curve. The difficulty is in proving this is actually the case.

Proof: Formula (9.7) is obvious because f⁻¹ ∘ f

= t so it is clearly an increasing function. If f ∼ g then f⁻¹ ∘ g is increasing. Now g⁻¹ ∘ f must also be increasing because it is the inverse of f⁻¹ ∘ g. This verifies (9.8). To see (9.9), f⁻¹ ∘ h =∘ and so since both of these functions are increasing, it follows f⁻¹ ∘ h is also increasing. ■

The symbol ∼ is called an equivalence relation. If C is such a smooth curve just described, and if f :

→ C is a parametrization of C, consider g ≡ f, also a parametrization of C. Now by Corollary 9.4.3, if h is a parametrization, then if f⁻¹ ∘h is not increasing, it must be the case that g⁻¹ ∘h is increasing. Consequently, either h ∼ g or h ∼ f. These parameterizations, h, which satisfy h ∼ f are called the equivalence class determined by f and those h ∼ g are called the equivalence class determined by g. These two classes are called orientations of C. They give the direction of motion on C. You see that going from f to g corresponds to tracing out the curve in the opposite direction.

Sometimes people wonder why it is required, in the definition of a smooth curve that p^′

≠0. Imagine t is time and p gives the location of a point in space. If p^′ is allowed to equal zero, the point can stop and change directions abruptly, producing a pointy place in C. Here is an example.

In this case, t = x^1∕3 and so y = x^2∕3. Thus the graph of this curve looks like the picture below. Note the pointy place. Such a curve should not be considered smooth.

So what is the thing to remember from all this? First, there are certain conditions which must be satisfied for a curve to be smooth. These are listed above. Next, if you have any curve, there are two directions you can move over this curve, each called an orientation. This is illustrated in the following picture.

Either you move from p to q or you move from q to p.

Note that Example 9.4.6 is an example of a piecewise smooth curve although it is not smooth.