Planimeter App Download
Divina Hujer
The original polar planimeter was invented in 1854 by JacobAmsler, a Swiss mathematician and inventor of many measuringinstruments. It was so much simpler, easier to use, and more accurate than previouslyinvented planimeters that the latter quickly became obsolete. Further modifications of his basic design were made only to improve itsaccuracy. The linear planimeter works on the same basic principleas the polar planimeter, and is simply a variation that allows theareasof long, skinny regions to be measured.
This planimeter is on permanent loan from a member our biologydepartment. If you are looking for a planimeter, good places to check are physics,chemistry, and engineering departments. They are almost alwaysavailableat reasonable prices on eBay.
The following picture gives some idea of the setup. The pole arm rotates freely around the pole, which is fixed on the table. The tracer arm rotates around the pivot, which is where it joins the polar arm. You trace a curve in the clockwise direction with the tracer, and as you do so the measuring wheel rolls along, and the total distance it rolls is accumulated on the dial. The support wheel keeps the thing from flopping over. At the end, you read off a number from the dial, and after multiplication by a factor depending only on the particular configuration of the planimeter, you get the area inside the curve.
The 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 l of the tracer arm is less than that the length r of the pole arm. This means that the tracer can never get within a distance r - l 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.
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 orientation for the planimeter. It is positive if the square root is positive. Once an orientation has been chosen, it will remain the same unless the arm is fully extended. This must never happen. As long as your curve is entirely within the annulus, the configurations of the planimeter will vary smoothly and uniquely with the path of the tracer.
Next 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 l and attach to it a measuring wheel of radius R right in its center.
In the case of the polar planimeter, the bottom of the arm is restricted to an arc of the circle of radius r with center at the pole, hence l C 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 0. So in this case we have
As 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 n, the one pointing counter-clockwise and perpendicular to the tracer arm at the tracer.
But since every point of the annulus corresponds to a unique positive configuration of the planimeter, we can assign a vector n to every interior point of the annulus, and it therefore defines a vector field. The curve Γ is the boundary of its interior Ω, and by one of our assumptions this is contained entirely in the region where n is defined. Green's Theorem tells us that the path integral around the boundary of this region is also equal to a certain integral over Ω: Therefore
Guldin'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 rolling planimeter, in which the pivot is restricted to a straight line by being on a rolling cylinder.
The planimeter is a drafting instrument used to measure the area ofa graphically represented planar region. The region being measured mayhave any irregular shape, making this instrument remarkably versatile.In this age of CAD and digital images, I suspect that the planimeter isheading toward obsolescence, but not just yet. They are still being manufactured.
When I first used a planimeter, I was somewhat troubled by the factthat I did not understand how it worked. Oh sure, I don't understand howmy car works either, but the planimeter essentially has only three movingparts. That makes its mechanism considerably less complex than a typicaldoorknob. After eighteen years of sleepless nights, I decided to buy aplanimeter and figure it out.
This is my planimeter. I got it at a Snohomish antique mall.It was made in Germany and distributed by Keuffel and Esser. The date on the bottom of the box says1917. It is in perfect working condition, and it is nearly identical tothe instruments I was using in the early 80s. The brass cylinder is anchoredto the table with a point, like a compass point. It pivots, but does notslide. The elbow joint bents and slides freely. The pointer on the otherend is used to trace the perimeter of the region. Near the elbow is a wheel,which simply rolls and slides along the tabletop. The scale is on the wheelitself, so it tells how far the wheel has turned. Sure enough, that numberis proportional to the area of the region. The conversion factor dependson the scale of the drawing or photograph.
With this modification, we havedesigned a different type of planimeter. This is not the same as the polar planimeter,but it should work. The blue arm has been eliminated from the mechanism.Point B slides along a straight groove. As before, point Ctraces the region in a clockwise direction. Let's see what happens when we trace a very simple region, rectangle PQRS, two sides of whichare parallel to the path of point B. Start at point P andmove in a clockwise direction. Again, if you have Java, click on the sketchto see a simulation.
Shortly after this web page was created, I found a planimeter that operates just this way. It is called an Amsler integrator. It is very large, and it was used primarily for measuring cross-section areas on boat plans.
I know, I know. It looks like a lot of hand waving so far. The planimeterin this model does not even have the same construction as the one in thepicture. I said that I would get back to that, and I will, but not rightnow. Also, I said that this instrument would measure the area of irregularlyshaped regions, but I only proved that my modified instrument (which is called a linear planimeter) would workfor a rectangle that is oriented square with the runner. What about thatirregular region?
Now, using the linear planimeter, trace one rectangle. Lift the planimeterso that the wheel does not move. Place it down on a second rectangle, andtrace it. Continue doing this until all of the rectangles have been traced.The scale will show an accurate area for the region.
What we need to be tracing is the perimeter courses, highlighted inred here. Of course, it would be more accurate if we fill in those gapswith smaller rectangles. As we fill more and more of the gaps, this redboundary takes the shape of the original boundary. In other words,we could save a lot of work by forgetting all of the rectangles and simplytracing the boundary. That is exactly how a planimeter is used.
This part is being saved for last because it requires calculus. If you are willing to accept my word that the length of the blue anchor arm does not matter, then the explanation above may give you a conceptual understanding of the planimeter.
In this sketch we have a planimeter tracing a boundary. Notice that the path of point B is restricted to a circle. For now, forget about the area of the region. Concentrate on the area that is covered by the green arm. Take the arbitrarily small subregion covered as the planimeter goes from C to C'. Call its area dw.
By integrating as the planimeter traces the entire region, we can get a formula for W, the area of the region crossed by the green arm when the planimeter makes one complete circuit around the region.
The first term in this formula integrates the angle change as the pointer traces the entire region. Since the planimeter begins and ends in the same position, the net change in the angle is zero. Therefore, the changes in the scale reading caused by rotation have no influence on the final reading.
Of course, what we computed was the area crossed by the green arm, not the area of the region that was traced. Think again. When the arm is sliding backwards, the scale is moving backwards. In that case, the area it is covering is subtracted from the total. When the planimeter traces a region, the arm makes a broad sweep, then backs over the part that is outside of the region.
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