## 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 1/100^{th}s of a revolution to be recorded, and a vernier scale can improve the resolution to 1/1000^{th} of a revolution. In practice, the accuracy will be limited by the operator’s skill in tracing around the curve.