The planimeter #

Suppose that you are nineteenth-century land surveyor and are tasked with finding the area of a plot of land, say something like New Zealand:

How would you proceed? Until things like maps and blueprints became digitized sometime late in the twentieth century, determining the area of an irregularly-shaped region drawn on a piece of paper was a common and an important task in a variety of settings, including land surveying, architecture, and engineering. This lab will introduce you to a decidedly nineteenth-century solution to this problem that uses the then-newly-developed technology of vector fields and path integrals.

1. Background #

Computing area with Green’s Theorem #

Our solution will come from a surprising application of Green’s Theorem and a nineteenth-century mechanical device. But first let us set the stage with some mathematics. Consider the vector field \(\vec{F}(x,y) = x \vec{j}\). According to Green’s Theorem, for any simple closed curve \(C\) that is oriented counter-clockwise and encloses a region \(R\), we have \[\int_\mathcal{C} \vec{F} \cdot d\vec{r} = \int \hspace{-.1in} \int_R curl \; \vec{F} \; dx \; dy.\]

For our particular choice of vector field, we have \(curl \; \vec{F} = 1\), which gives Green’s Theorem a rather unique interpretation: the right-hand-side of the equation above represents the area of the region \(R\) and leads us to the following:

An upshot of this observation is the following. Suppose that we are able to construct a mechanical device that computes the path integral of a vector field with \(curl\) equal to one. Then our measurement problem will be solved! In fact, one such device was proposed in 1854 by the mathematician Jakob Amsler-Laffon. We will study how well it works in this lab.

The polar planimeter #

A polar planimeter, as shown below, is a simple mechanical device consisting of two arms, a graduated wheel, a counter and vernier scales, a tracing point, and an anchor point.

Detailed use instructions appear in this video, but here is a quick guide on how to use it:

1. Begin by placing the planimeter on the same piece of paper as the region whose area is to be measured. Make sure the anchor point is outside the region and set the counter and the vernier scales to zero.

2. Place the tracing tip at a point on the region’s circumference and begin tracing the curve with the tracing point of the planimeter. As you do this, the graduated wheel will turn and the counter and vernier scales will keep track of the number of the rotations made the planimeter’s graduated wheel. The graduated wheel will sometimes turn forwards, sometimes backwards, and sometimes it will stay still. The scales will keep track of the signed number of rotations.

3. Continue and trace one complete revolution around the region and note the final reading from the counter and vernier scales. Each individual device comes with a scaling factor. Multiply your reading by this scaling factor. The result purports to be the area of the region.

We will dissect what is going on, and whether the polar planimeter actually does what it claims to do, in the sections below.

2. A mathematical model #

First, let’s create a mathematical model of the planimeter and the problem it is trying to solve. Let \(\mathcal{C}\) be a simple closed curve in the plane that is oriented counter-clockwise and write \(\mathcal{R}\) for the region it encloses. Our task is to compute the area of \(\mathcal{R}\).

Tracing arm and pivot point #

Let us set the anchor point at the origin, and introduce the following conventions:

• let \((x,y)\) be the coordinates of the tracing point which will circumnavigate the curve \(\mathcal{C}\),
• and let \((a,b)\) be the coordinates of the pivot point where the arms of the planimeter join.

Note that both \(a\) and \(b\) will change as \(x\) and \(y\) change, so we can think of both as functions of \(x\) and \(y\). Finally, let us assume that both arms of the planimeter have unit length.

The animation above shows the relative positions of the two arms as the planimeter traces out the circumference of the region and how \((a,b)\) changes as we change \((x,y)\).

	Solve[{a+b==1, a-b==17},{a,b}]

Graduated wheel #

The planimeter’s graduated wheel is mounted perpendicular to its tracing arm; the exact location depends on the device manufacturer. Since the planimeter ultimately counts how many times this wheel rotates, it seems appropriate for us to figure out an answer to the following:

To begin, we consider two special cases. In the first, suppose that the motion of the tracing point is exactly in the direction of the tracing arm, as in the animation below:

It should be clear that in this case the wheel will not turn at all. On the other hand, if the motion of the tracing tip is perpendicular to the tracing arm, the wheel will turn by an amount proportional to the distance traveled by the tracing point; the proportion will depend on exactly where the wheel is mounted along the tracing arm:

If the wheel is halfway between the pivot and the tracing point, the constant of proportionality will be one-half. If it is a tenth of the way, the constant will be one tenth.

What if the tracing point is being moved some other direction than the two above? How much will the graduated wheel turn? Read on.

Planimeter vector field #

Things will be a little easier if we introduce a new vector. Let \(\vec{F}\) be a unit vector perpendicular to the tracing arm and recall that:

• if the tracing point moves perpendicular to \(\vec{F}\) then the graduated wheel does not spin, and
• if the tracing point moves exactly in the direction of \(\vec{F}\), then the graduated wheel spins by an amount proportional to the speed of the tracing point.

In fact, as the coordinates \((x,y)\) of the tracing point change, so does the perpendicular vector \(\vec{F}\), so in fact what we have is not a single vector, but a vector field!

Definition:

To define a vector at a point in the plane, first arrange the planimeter so that its tracing point is at \((x,y)\). Then draw a unit vector starting at the tracing point perpendicular to the tracing arm. Since there are two possibilities, pick one, just make sure to do so in a consistent manner. This process defines a vector field \(\vec{F}(x,y)\) whose graph appears below:

We are now ready to return to the question of what happens to the graduated wheel if the tracing point moves in a general direction, say \(\vec{v}\). How quickly will the graduated wheel spin?

The final exercise justifies our introduction of the vector field \(\vec{F}(x,y)\).

3. A formula for \(\vec{F}(x,y)\) #

We are ready to address the main goal of this lab: does the planimeter actually work? In the previous section, we found that the planimeter does in fact compute a path integral. It is a path integral of a very particular vector field \(\vec{F}(x,y)\). Follow the outline of these problems to complete our work.

Finally, to make the planimeter usable, we need to know the constant of proportionality: what number should be multiply our answer by to obtain the area of the region. This turns out to be quite simple. Each device would come with a constant that was calibrated at the factory. The calibration sheet for my planimeter is a small piece of paper with different values for different settings for the device: