Relation of Roll of Wheel to Area Traced

CHAPTER III.

Theory of the Polar Planimeter.

Relation of Roll of Wheel to Area Traced.

In Fig. 3 of Plate XII, let ABC be any given figure of whose area pq is an element. Let P, W and T be the Pole, Wheel and Tracer respectively of the Planimeter, and let the Tracer move across the width of the element of p, the width of the element being very small.

Let OO be an arc of the Zero Circle, the upper width of the element being coincident with it at the point q. Let U be the small angle at the Pole subtended by the width of the Element pq, while the parts of the Planimeter are designated by the letter assigned them.

Then

Area pq = ½ U (p² + t² + 2 pt Cos a) – ½ U (p² + t² + 2 ft)
= Ut (p Cos a – f) ... (1) But the area pq is the area included between the portion of the periphery of the figure traced, the arc of the Zero Circle, and the radii drawn from the Pole to the beginning and end of the line traced by T. We have already found that the distance which the Wheel would roll and record after tracing the width of the element at b is: Distance rolled = U (p Cos a – f) ... (2)

Comparing Eqs. 1 and 2, it is seen that the area of the element is equal to the distance rolled by the wheel for the given tracing multiplied by the length of the tracer arm t.

As this is true for any element of area of the figure traced, it must be true of every element, and hence the total area of the figure outside the arc of the zero circle.

It will require but little consideration to show that the instrument deals in like manner with that portion of the traced figure included within the Zero Circle, and that the rolling of the Wheel resulting from the tracing of the periphery of any given closed figure when multiplied by the length of the Tracer Arm gives at once the area of that figure.

If c is the circumference of the Wheel and r the number of revolutions made by the Wheel during any given tracing, it is evident that the distance rolled by the Wheel, which we have also referred to as the “roll” of the wheel, is c × r. Hence, denoting the area of any figure by A, and the length of Tracer Arm by t, we have

A = c × r × t which is what we have already termed the “General Equation of the Polar Planimeter,” and from which is at once obtained the “General Principle” of the instrument.

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