Length Of A Curve

We begin with the simplest version of arc length, and one that requires no calculus. The transcendental number n is defined as the ratio of the circumference of any circle to its diameter, n = C/D = C/2r, where r is the

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* r

Figure 5.2 In this figure, a one-radian angle is the value of 0 when s = r = 1.

radius. Therefore, C = 2nr. This suggests a particularly pleasant unit of angle measure (much nicer than the degree, which is completely unnatural). One radian of angle is defined to be the unique angle such that a wedge of that angle subtends one unit of length on a circle of radius 1 (Figure 5.2).

Now, still on the unit circle, suppose that a ray makes an angle of 0 radians with the positive x-axis. Define the coordinates of the intersection of this ray with the unit circle to be (cos 0, sin 0). This defines two functions from the interval [0, 2n] into the interval [-1, 1]. The unit circle is then, by definition, parameterized by x (t)=cos(t), y(t)=sin(t), t e[0,2n]. (5.12)

Usually, arc length for a curve in the plane is computed by (1) choosing a sequence of points lying along the curve, (2) drawing the straight line between each two successive points, (3) computing the length of each segment using the Pythagorean theorem, (4) adding up the lengths of all the segments, and, finally, (5) examining the limit as the length of each segment goes to zero simultaneously while the number of segments approaches infinity. Schematically, the division of a curve into segments is shown in Figure 5.3.

Each of the segments in the approximation may be viewed as the hypotenuse of a triangle. One might denote the horizontal and vertical sides of each triangle by dx and dy respectively, and the hypotenuse by ds. The Pythagorean theorem states:

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a =

xa b =

xn

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Figure 5.3 The division of a curve into segments in order to calculate arc length.

Adding up the contributions from all the segments and taking the limit as the number of segments goes to infinity is called "integration," and will not be treated in full detail here (see Spivak, 1980). To see that our procedure really works, however, let's use it to calculate the length of the unit circle, which we already know to be C = 2nr.

Approximate the circle by N equally spaced line segments. The jth line segment has its initial point at and its final point at 2n j/N radians. Therefore, the Nth segment has its endpoint at 2n radians, corresponding to a full revolution around the circle. For N = 4, the segments form a square inscribed within the circle.

The first segment goes from the point (1,0) to the point (cos(2n / N), sin(2n / N)), so the length of this (and hence every) segment is since there are N segments. For small x, however, 1 - cos x is approximately equal to x2/2. Therefore,

The approximation that we used, 1 - cos x ~ x2/2, gets better and better as x gets smaller. Specifically, we used it for x = 2n / N, which gets smaller as N (the number of segments) goes to infinity! So our procedure gives 2n as the limiting value for the length of the unit circle, just as we suspect.

Length(segment) = ^(l - cos(27t / N)f + sin(27T / N)2 = ^2-2cos(2jr/N).

Therefore, the total length of all the segments is

If this seems mysterious, plug Equation 5.14 into a calculator with N = 500, and compare it to the numerical value for 2n, which is about 6.28.

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