# Planimeters

## Planimeters

A planimeter is an analogue device for extracting numerical data from a drawing or a map. The simplest device is one for measuring the length of a route on a map. It consists of a small wheel connected to a counter, and is simply rolled along the route, keeping it aligned so that the plane of the wheel points along the direction of motion (i.e. there is no sideways, skidding, motion). These are very cheap to make, and have been given away by garages, for example, as a "free gift" when you buy a road atlas. The device is most useful if the size of the wheel, and any gearing, is chosen so that the reading is directly in miles or kilometres.

Another quantity that is useful to know is the area of a region on the map. Running a simple measuring wheel around the boundary does not give the area, as you can see by looking at the two shapes in Fig. 1.

Fig. 1. Illustrating that area is not the same as perimeter

These both have an area of 12 squares (count them), but their perimeters are 14 units (for the one on the left) and 16 units (for the one on the right). So something a bit more complicated than a measuring wheel is required.

Fig. 2. shows a rod AB with a wheel mounted at its centre. The contact surface of the wheel is made from polished metal. This gives sufficient friction to roll along the surface when the motion is at right angles to the rod, but allows the wheel to slide, without turning, when the motion is parallel with the length of the rod. If the rod moves between two positions AB and CD as shown, then the area ACDB "swept out" by the rod is measured by the rotation of the wheel.

Fig. 2. A rod with a wheel at its centre moves from AB to CD. Note that this is a plane diagram, looking down at the sheet of paper (it is not a three-dimensional view).

To see why this is so, imagine that the rod first moves from AB to EG. It is clear that the area of the rectangle AEGB is equal to the length of the rod multiplied by the distance moved by the wheel, which is its circumference multiplied by the number of rotations (including fractions of a rotation). If the rod then moves from EG to FH, the wheel doesn’t rotate any further, as the motion is just one of sliding, and the area AFHB is equal to AEGB (this follows because the triangles EFA and GHB have equal areas). If the rod now swings around from FH to its final position CD, then again the wheel doesn’t rotate, and the area is unchanged (since the extra area added by the movement of H to D is subtracted by the movement of F to C).

Now consider the motion shown in Fig. 3. One end of the rod, A, is made to move along a straight line, while the other end, B, traces out a closed curve.

Fig. 3. The end A of the rod AB moves along a straight line while the other end B traces out a closed curve

During a complete circuit of the closed curve, the roughly triangular area to the left of the curve is swept out twice, once upwards and once downwards, and these two areas cancel out. The interior of the curve, however, is only swept out once, in an upward direction, so that the reading recorded on the wheel is a measure of the area inside the curve. The case described here, where the end A is constrained to move along a straight line, is known as a linear planimeter. This restriction is not necessary, however, as all that is required is that the end A should trace out a curve with zero area. A common alternative choice is for the end A to move to-and-fro along an arc of a large circle, giving the design known as a polar planimeter. For the case of the linear planimeter, an alternative demonstration of its principles is given later, in which it will be seen that the measuring wheel can be fixed anywhere without changing the result, provided that its axle is parallel with AB.

As well as being fitted with a revolution counter, the wheel carries a graduated cylinder, allowing 1/100ths of a revolution to be recorded, and a vernier scale can improve the resolution to 1/1000th of a revolution. In practice, the accuracy will be limited by the operator’s skill in tracing around the curve.

## The Exhibit

The planimeter displayed here has an uncertain history. It was used in the Physics Department (Swansea University), possibly for measuring the area damaged by sparking on electrical contacts, from photographic enlargements, or it could have been used for more general purposes by Dr M R Hopkins, who was interested in metrology and had worked in that field at the National Physical Laboratory and at the British Iron and Steel Research Association (BISRA) which was located at Sketty Hall. Before that, the planimeter was on the inventory of the Royal Aircraft Establishment at Aberporth on Cardigan Bay, but we have no record of the use that they made of it. It is a more elaborate instrument, known as a quadriplanimeter. This is rather difficult to describe in non-mathematical terms. Briefly, the area A inside a curve, as shown in Fig.4, can be considered to be composed from a large number of very small areas, represented by dA (where "d" stands for "differential" – i.e. a very small piece, or "element" of the area). The total area is the sum of all the small pieces, leading to the formula

A = ò dA,

where ò is the integral sign introduced by Newton, and is a stylised letter S, standing for "sum". On the quadriplanimeter, the second dial records the first moment of area, defined as

M1 = ò y dA,

where y is the distance from the element of area dA to a reference line.

Fig. 4. The area A is shown divided up into small elements dA, each being a distance y from the reference line of the quadriplanimeter.

The quantity M1 can be used to calculate the volume of a figure of revolution, as shown in Fig. 5; the volume swept out when a plane curve is rotated around an axis in its plane (which must coincide with the reference line of the planimeter) is simply 2p×M1. The ratio of the two quantities, M1/A, gives the position of the centre of area from the reference line. We can only speculate on how the instrument was used at Aberporth, because much of the work done there was classified. The centre of area might have been used to calculate the centre of buoyancy of a boat’s hull, and the volume of a figure of revolution could have been involved in the design of rockets, shells, fuel tanks, etc.

Fig. 5. Showing how a solid of revolution is generated by rotating a closed curve around the reference line.

Because of the presence of the reference line, the quadriplanimeter has to be based on the linear, rather than the polar, design. The required modification of the basic instrument is shown in Fig. 6. The main rod is of length a, the end A (see also Fig. 3) moves along the reference line, while the end B traces around the curve to be measured. An additional rod is placed with one of its ends, C, also moving along the reference line at a fixed distance b from the point A. This second rod is also of length b and its other end, D, has a sliding connection on the first rod. Simple geometry shows that the angle between CD and the reference line is twice the angle between AB and the reference line. These angles are shown as q and 2q in Fig. 6.

Fig. 6. A second rod CD enables the instrument to measure the first moment of area.

The second rod CD carries a measuring wheel, similar to that on the first rod, but with its axis aligned at right angles to the rod (in other words, with the plane of the wheel parallel with the rod). Suppose the point B moves around a small rectangle with sides dx and dy as shown. Consider first what happens to the basic planimeter wheel mounted on the long rod. The contributions from the vertical sides of the small rectangle will cancel out (since the layout of the rods will be identical at both sides of the rectangle) so the net result arises from the two horizontal sides. As the end B moves to the right along the bottom side of the rectangle, the motion is partly rotation and partly sliding. The amount of rotation is the component of dx resolved along the direction of the wheel, which is -dx sinq, and we can choose the convention that this is given a negative sign because the direction of rotation corresponds to a downwards movement in the diagram. When the end B moves back along the top of the rectangle, the angle q will be increased to (q + dq), and the contribution to the reading on the wheel will be +dx sin(q + dq). The resultant movement is therefore

dx[sin(q + dq) - sinq] = dx[sinq cos(dq) + cosq sin(dq) - sinq].

Since dq is assumed to be very small, we can approximate cos(dq) = 1 and sin(dq) = dq, giving the resultant movement as

cosq dx dq = dx (dy/a) = (1/a)dA,

since dy = a dq cosq and dxdy = dA, the element of area. Since an arbitrary region with area A can be divided into a large number of elementary rectangles, the result generalises to any region defined by a closed curve. This is an alternative proof of the result obtained earlier, which showed that the first dial records the area, and that the area is a times the distance rolled by the wheel.

Now consider the second wheel, which is mounted on the rod CD, with its plane of rotation aligned along CD. As before, the contributions to the rotation of this wheel arising from the vertical sides of the rectangle cancel out, leaving just the horizontal sides. Along the bottom side, the distance moved, resolved along the direction of the wheel, is dx cos(2q), while along the top side, the distance moved is -dx cos 2(q + dq), so that the resultant movement is

dx[cos(2q) - cos(2q + 2dq)]

= dx[cos(2q) - cos(2q)cos(2dq) + sin(2q)sin(2dq)]

= 2 sin(2q) dx dq, using the same approximations as before

= 4 sinq cosq dx dq

= (4/a2y dA,

where y = a sinq is the distance of the element dA above the reference line. The wheel on the rod CD therefore measures the first moment of area, and its value is a2/4 times the distance rolled by the wheel. In the planimeter in our collection, the calibration constants are 0.1 for the area and 0.4 for the first moment.

The question naturally arises: Why is it called the first moment? That is because there is a whole series of moments. The second moment, defined as

M2 = ò y2 dA,

is of interest to physicists and engineers. It gives the moment of inertia of a rotating body (which is needed in order to calculate its angular momentum and kinetic energy) and it gives the stiffness of a structural beam of arbitrary cross-section. If you are aware of a planimeter that can measure the second moment (and what it is called) then please let us know! For notational consistency, we can also represent the area A as M0, but there is not much point in replacing the simple description "area" by the rather more lengthy "zeroth moment of area".

It might be thought that a process that relies on the combined rotation and sliding of a wheel cannot be very reproducible, so the planimeter was tested on an arbitrary closed curve. A simple choice is a semicircle of radius r = 10 cm, with its diameter along the reference line of the instrument. We carefully trace around the curve, recording the reading on the dials at the start and finish of each circuit, and subtracting them. As a check on accuracy, we do this a few times, preferably starting each circuit at a different point. The readings are given in the table, for three anticlockwise circuits.

 Dial 1 Dial 2 Start Finish Diff.(a) Start Finish Diff.(b) A(c) M1(d) 3020 1443 1577 2354 4022 1668 157.7 667.2 1457 9890(e) 1567 5754 7414 1660 156.7 664 3023 1455 1568 3277 4950 1673 156.8 669.2
Notes:
1. This dial counts downwards for an anticlockwise circuit
2. This dial counts upwards for an anticlockwise circuit above the reference line
3. The calibration factor for dial 1 is 0.1
4. The calibration factor for dial 2 is 0.4
5. This measurement involved the dial passing through zero = 10,000

The simple planimeter (dial 1) gives the area, which should be ½pr2 = 50p = 157.1 cm2. The actual results, from Table 1, are: 157.7, 156.7 and 156.8, giving an average of 157.2 cm2, which is within 0.1% of the correct result.

On dividing the first-moment reading by the area, we get the distance of the centre of area of the semicircle from the reference line (this is why a semicircle was chosen – there would not be much achievement in finding the centre of a circle!) For each of the three sets of readings, we get 4.2308, 4.2374 and 4.2679 cm. The actual distance, from calculation, should be 40/(3p) = 4.244 cm, so we have obtained values within 0.2 – 0.5% of the true result. The value of M1 itself should be 2/3 r3 = 666.7, and the average of the three measurements is 666.8, which is only 0.015% too large. The conclusion is that planimeters can be very accurate.

## Uses

Planimeters have been used in cartography, ship design (as suggested above), local government, surveying, and sometimes just as a mathematical tool to calculate the area inside some curve. It is still possible to purchase a recent model, and the main difference is that the circular and cylindrical scales have been replaced by digital displays. Even that degree of modernisation is superseded by using a digital form of a map, input from a scanner or already stored on a computer, clicking on a curve of interest, and getting the area printed automatically – plus the first moment, second moment, … as required.

C.J.Evans, Dec.2008