A planimeter is an instrument for measuring area [the two-dimensional counterpart to the opisometer, a device for measuring length]. Typically, it is used on maps, or similar diagrams.

The planimeter is a simple-looking device. It is a 'V'-shaped device; two rigid arms meet at a pivot. The arms need not be the same length, and the pivot allows the angle between the arms to change. At the end of one arm is the 'anchor', at the end of the other arm is the 'tip' (if the arms are not equal length, the arm with the tip is always shorter than the arm with the anchor). The anchor is to be fixed to some point on the map outside the region to be measured so the tip can trace the perimiter of the region. The planimeter can rotate about the anchor, but the anchor must not move on the map when it is in use. Partway between the pivot and the tip is an opisometer-like device which is attached to a mini-odometer [the odometer measures in units more like centimetres, or in fact square centimeters, than miles; it is part of the planimeter so you don't have to roll the wheel backwards like on a conventional opisometer]. This opisometer is fixed at a certain point along the arm and it does not slide up or down, and the axle of the wheel is parallel with the arm. The pivot allows the tip-arm to rotate 0 to 180 degrees, so that either the tip-arm is always clockwise from the anchor-arm or the tip-arm is always anticlockwise from the anchor-arm. [Which way the angle goes depends on the design of the particular planimeter, the pivot is rather like an elbow in that it does not allow the tip-arm to 'bend backwards', and you can get a planimeter which works like your right arm or one which works like your left arm.]

As unintuitive as it sounds, the number of revolutions the wheel makes when the tip makes one circuit around a shape is directly proportional to the area of that shape. The wheel will roll smoothly when the arm is rotated about the pivot, keeping the anchor-arm steady, but when the anchor-arm is rotated the wheel will usually drag across the page. [If the angle between the arms is 90 degrees, the wheel drags without turning, if the arms are collinear the wheel will rotate without dragging - otherwise, it drags a bit and turns a bit at the same time.] This is by design, the opisometer is only supposed to measure the component of the tip's movement which is normal to (perpendicular to) the arm.

To the math: Let φ be the angle that the anchor-arm makes with North (or any fixed direction), positive angles going anticlockwise. Let θ be the angle between the arms, 0 (folded back on the anchor-arm) to π (pointing directly away from the anchor), positive angles again going anticlockwise. Imagine the opisometer is at the tip for now (that it isn't really is dealt with later). Let A be the distance from the anchor to the pivot (i.e. the length of the anchor-arm), and let B be the distance from the pivot to the tip (i.e. the length of the tip-arm).

If φ is kept constant, and θ is changed, the opisometer travels in an arc centred at the pivot. The arc is on a circle of radius B, the angle of the arc is Δθ, and the opisometer's wheel (perpendicular to the tip-arm) rolls along the arc without slipping. Thus, the opisometer will roll a distance of B * Δθ.

If θ is kept constant, and φ is changed, the opisometer again travels in an arc, this time centred at the anchor. The radius this time is not so easy to calculate; trust me, it's sqrt((A ^ 2) + (B ^ 2) - (2 * A * B * cos(θ))) [a quick check shows at θ = 0 the radius is (A - B), and at θ = π the radius is (A + B) - just as you'd expect]. Just call the radius R for now. The angle of the arc is Δφ, but the opisometer does not necessarily roll smoothly along the arc - in fact, the only conditions under which the wheel does roll smoothly is if θ is 0 or π. If θ is (π / 2), the wheel won't roll at all, it will just be dragged across. The angle the opisometer's wheel makes with the direction of motion is equal to the angle from the pivot to the anchor to the tip, call it α. By the law of cosines, (cos(α) / A) = (cos(θ) / R), so cos(α) = ((A * cos(θ)) / R).

Put this all together, and you get that the distance the opisometer travels is the radius of the arc times the angle of the arc times the cosine of the angle between the wheel and the direction of travel =
(R * Δφ * ((A * cos(θ)) / R)) =
(A * cos(θ) * Δφ)
Conveniently, the Rs cancel out.

Imagine a person traces the outline of a shape from time t1 to t2. Let φ(t) be the angle φ at time t, and let θ(t) be the angle θ at time t. With the above information, it can be seen that the planimeter is simply a mechanical integrator which integrates the function ((B * dθ(t) / dt) + (A * cos(θ(t)) * dφ(t) / dt)) with respect to t from t1 to t2. But if the tip makes one circuit of the perimeter of the area, then at t2 the tip is in the same place it was at time t1, so φ(t2) = φ(t1) and θ(t2) = θ(t1). Further, since the tip-arm cannot make a full revolution about the pivot, the (B * dθ(t) / dt) part of the integral is 0.

It's still very unclear how the integral of (A * cos(θ(t)) * dφ(t) / dt) is proportional to the area.

Imagine the planimeter starts at position (φ1, θ1). Rotate about the elbow, in, by Δθ [Δθ > 0] to θ2 [i.e. θ2 = (θ1 - Δθ)]. Rotate about the anchor by Δφ [Δφ > 0] to φ2 [i.e. φ2 = (φ1 + Δφ)]. Rotate about the elbow back to θ1, then rotate about the anchor back to φ1. The shape traced out is a curved quadrilateral (and the tip goes anticlockwise around the perimeter).

The first arc will clock up (B * -Δθ), the third arc will add (B * Δθ), so the two cancel each other out. This is an instance of the (B * dθ(t) / dt) part of the integral mentioned above totalling 0.

The second arc will clock up (A * cos(θ2) * Δφ). The fourth arc will add (A * cos(θ1) * -Δφ). This totals to (A * (cos(θ2) - cos(θ1)) * Δφ).

The radius of the outer arc of the quadrilateral (the arc furthest from the anchor) is R1 = sqrt((A ^ 2) + (B ^ 2) - (2 * A * B * cos(θ1)). The radius of the inner arc is R2 = sqrt((A ^ 2) + (B ^ 2) - (2 * A * B * cos(θ2)). The area between the two is (((R1 ^ 2) - (R2 ^ 2)) * Δφ) = (2 * A * B * (cos(θ2) - cos(θ1)) * Δφ).

So for this particular shape, the odometer shows (A * (cos(θ2) - cos(θ1)) * Δφ), and the area covered is (2 * A * B * (cos(θ2) - cos(θ1)) * Δφ), or (2 * B * the value on the odometer).

Now consider any shape. Find a large curved quadrilateral that fits in the shape; measure it with the planimeter and you have an estimate of the area. Find a second curved quadrilateral which shares an arc, or a part of an arc, with the first and which fits in the remainder of the shape. Run the planimeter over this, add the result to your first estimate and you get a better estimate. Now consider: the tip of the planimeter has travelled anticlockwise about each quadrilateral, which means it has travelled along the arc they share twice - once in each direction. Well, obviously that arc contributes nothing to the final answer, so there's no point in even moving the tip over that arc. The tip could be traced around the outline of the two quadrilaterals, anticlockwise, once, to give the same result. Ever more accurate estimates of the area of the shape can be made by adding more quadrilaterals sharing parts of arcs with previous quadrilaterals, and tracing the outline of the resulting shape. The more quadrilaterals are added, the closer the outline traced will match the border of the original shape - in the limit, tracing the tip precisely along the border of the shape has the effect of summing the area of all the infinitesimal quadrilaterals it would take to fill the shape exactly.

Finally, I shall address the issue that the opisometer is not at the tip of the planimeter's arm. The purpose of the opisometer is to measure the component of the tip's movement [due to rotation about the anchor] which is perpendicular to the tip-arm. As long as the axle of the opisometer's wheel is parallel to the arm, the wheel itself can be anywhere along the arm and it will measure exactly the same thing. [As for the other term being integrated; wherever the opisometer is on the arm, as long as it doesn't move along the arm the total distance travelled due to rotation about the pivot will be 0.]

Of course, on a planimeter, the odometer is calibrated so that it includes the factor of (2 * B); you can read the area directly off the odometer display without having to do any calculations.

Pla*nim"e*ter (?), n. [Plani- + -meter. Cf. Planometer.]

An instrument for measuring the area of any plane figure, however irregular, by passing a tracer around the bounding line; a platometer.


© Webster 1913.

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