## The Mathematics of Surveying: Part II. The PlanimeterThe second of two columns on the mathematics of surveying (the first is here). ...Bill Casselman John Eggers
A planimeter is a table-top instrument for measuring areas, usually the areas of irregular regions on a map or photograph. They were once common, but have now largely been replaced by digital tools. The following picture gives some idea of the setup. The
The next figure gives you a better view of the mechanism.
We call the
Here is the diagram of the original planimeter, from the article by Jakob Amsler that introduced it:
How can such a simple thing measure areas?## Geometry of the planimeterThe clockwise movement of the planimeter is the direction opposite to what mathematicians have decided should be positive rotation. Rather than violate this convention, we are going to work from now on with a mathematician's planimeter, in which you move counter-clockwise. Like other mathematicians' fantasies, there are none on the planet! There are some restrictions on how to place the planimeter with respect to the curve you want to trace. The carriage can be slid along the tracer arm, but in all cases the length of the pole arm. This means that the tracer can never get within a distance r of the pole. On the other hand, when it is fully extended, the tracer can never reach beyond r - l. So the curve to be traced must lie within the annulus between two circles, one with radius r + l, the other r - l.r + l
In fact, it should become clear in a moment that the arms should never be completely extended, so the curve to be traced must lie completely inside the annulus. Furthermore, normally the pole is placed on the outside of the curve.
For a given point in the annulus there are exactly two possible configurations of the planimeter that place the tracer on that point. Choosing one point or the other means choosing a sign for a square root. We call this choosing an The next thing to do is to understand a few things about the motion of the measuring wheel. As the following picture shows, if the wheel travels in a straight line a distance where θ = C/R is its radius.R
So it is really true that the rotation of the wheel and the distance traveled by the tracer arm have some relationship to each other. But this relationship is a bit subtle. If the arm just moves straight ahead a distance , but if it shifts parallel to itself the area swept out will be lC. In the first case, a point on the circumference of the wheel will move distance 0. In the second case, the wheel will not move at all. And if the arm translates obliquely, the wheel will rotate a distance equal to the altitude of the parallelogram covered by the arm. In all cases where the arm translates parallel to itself, the area swept out by the portion of the tracer arm between the pivot and the tracer will be equal to C, where lC is the distance measured by the rotation of the wheel. CThis is the basic fact relating the arm's motion to the motion of the measuring wheel.
The way to phrase this precisely is that no matter how the the arm moves, the distance measured by the wheel is the path integral
where Γ is the path traveled by the point of the arm where the measuring wheel is attached, is the unit vector pointing in the direction of travel (so that, for example, if the arm is moving parallel to itself the dot product of t and n is 0).t## Guldin's TheoremNext we are going to try to make the behaviour of the planimeter intuitively clear, but first we are going to look at a special kind of planimeter, and in this case prove a more general result. Suppose we take a single freely moving arm of length right in its center.R
Then we move the arm around on the plane. If the measuring wheel rotates through a total angle of .C = θR
Area here is interpreted with a sign. If the arm just rotates around its center, one half of the arm goes forward and the other backward, and the two cancel. We have already seen that Guldin's assertion is valid in the case the arm just translates. Of course this doesn't always happen - the arm may rotate as it moves as well as translate. But we can see what is happening by chopping up the swept area as follows:
Because the measuring wheel is in the center of the arm, as the arm rotates, it squishes the little blocks on one side as it expands them on the other. These effects cancel each other out exactly.
A completely rigorous proof can be given by using the formula for change of variables in a double integral and the expression for wheel travel as a path integral. Now suppose the wheel is placed somewhere else. Say its position is is the center of the arm, and c a vector along the arm. The length of v will remain fixed, say at ρ. The path that the wheel follows is v. The path integral is nowc(t) + v(t)
The first integral is the distance the wheel would travel if it were at the center of the arm. In the second, the vector , and n has constant length. The vector v(t) moves around on a circle of radius equal to ρ. Therefore the dot product of v(t) and n is just the signed length of v'(t), and the second integral is equal to ρ times the total rotation of the arm. Hence:v'(t)
You can see immediately one simple case of this by rotating the arm around its center. Combining this with Guldin's Theorem, we see that in all cases:
C = C_{0} + ρ θArea swept out = l C _{0} = l C - l ρ θ ## The full resultGuldin's Formula gives a signed area - if you sweep backwards over an area, the wheel goes backwards and you cancel area you have already covered. If we apply this to the case where the free arm comes back exactly to where it started, we see that
In the case of the polar planimeter, the bottom of the arm is restricted to an arc of the circle of radius is the area traced out by the tracer. Furthermore, normally the pole lies outside the region to be measured, and in this case the total amount of rotation of the arm has to be l C. So in this case we have0
Area of the region traced = lCHere is the figure included by Jakob Amsler, the inventor, in his original paper on the instrument he invented:
It seems pretty clear from this that Amsler derived his construction through some form of Guldin's theorem. ## Planimeters and Green's TheoremAs we have already mentioned, having chosen the orientation of the planimeter, the planimeter configuration is a continuous function of the tracer position. Say we choose the positive orientation. Then we can attach to each point of the annulus a unit vector
How the measuring wheel is going to respond to motion along the curve in the next instant depends on the angle between this vector and the unit tangent vector of the curve. At
But since every point of the annulus corresponds to a unique positive configuration of the planimeter, we can assign a vector
More precisely, Green's Theorem tell us that
where
This doesn't seem to get us very far. What ought to happen is that the curl is a constant and calculate its curl, but that isn't very enlightening. We can take advantage of another fact, however. The vector field has circular symmetry, which means that it is determined by what it is on one radius. The cosine law gives us a simple formula for the circumferential component.n
It follows from simple geometry in this figure that the circumferential component of
f(ρ) = cos(γ) = (ρ^{ 2} + l^{ 2} - r^{ 2})/(2 ρ l)The real point of Green's Theorem is that in order to verify that the integrand is
Then
l (ρ+dρ)( f(ρ+dρ)- f(ρ) ) dθ or (θ/2) ( (ρ + dρ)^{ 2} - ρ^{ 2})which is the area of the region Ω. ## Other kinds of planimetersGuldin's Theorem implies that the motion of a measuring wheel will tell you the area traced out by the tracer whenever the arm with a measuring wheel on it traces out a curve but has one end restricted to a one-dimensional curve. This happens, for example, with the
## To find out more- still makes and sells planimeters.
**Differential and Integral Calculus**Volume II, R. Courant, Blackie & Son, 1936. The section on Guldin's Formula (pp. 294-298) offers an explanation of how the planimeter works.- Amsler's original article, Vierteljahresschrift der Naturforschenden Gesellschaft in Zuerich, 1856. This is missing the diagrams, but they are here:
Our thanks to Donna Sammis of the Stony Brook University Library for locating the article and to her husband Robert for supplying photographs of the figures. The company that Amsler founded produced instruments well into the 20th century. This photograph shows the logo, on a pantograph version of the planimeter: - "About Planimeters," EngineerSupply. The company has also posted a video on YouTube.
Bill Casselman John Eggers Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society's MathSciNet can be used to get additional bibliographic information and reviews of some these materials. Some of the items above can be accessed via the ACM Portal , which also provides bibliographic services. |
Welcome to the These web essays are designed for those who have already discovered the joys of mathematics as well as for those who may be uncomfortable with mathematics. Search Feature Column Feature Column at a glance |