Planimeter Applet Guide

Description of the planimeter mechanically and mathematically, and the use of the applet.
  1. Structure of the Instrument
  2. The Wheel
  3. Tracer Arm Movement
  4. Tracer Arm Rotation
  5. Doubling Back Movement
  6. Position of the Pivot
  7. Path of the Pivot
  8. Measuring Twice or in the Opposite Direction
  9. The Zero Circle
  10. Mathematical Definitions
  11. Areas Measured
  12. Corrections for Rotation of the Tracer Arm
  13. Mathematical Expressions of Areas
  14. Formula for Area
  15. Formula for the Zero Circle
  16. Proof of the Zero Circle
  17. Formula for the Wheel Reading
  18. Description of the Wheel Reading
  19. Significance of the Wheel Reading
  20. Geometric Interpretation of the Wheel Reading
  21. Purpose of the Java Applet
  22. Organization of the Applet Window
  23. Colors in the Drawing Area
  24. Some Areas Not Represented
  25. Table of Area Colors
  26. Planimeter Consists of Narrow Lines
  27. Numeric Values of Areas
  28. Selection of the Sample
  29. Selection of the Drawing Mode
  30. Tracing in the Drawing Area
  31. Tracing Modes
  32. Clearing
  33. Move and Resize Modes
  34. Resize Modes
  35. Resizing Effects in the Drawing Area and Textboxes
  36. Negative Numbers in Textboxes Implement Special Features
  37. Scaling Modes
  38. Ruler Mode

1.  Structure of the Instrument

A polar planimeter is an instrument that measures area.  The user traces the outline (perimeter) of an area on paper and the instrument reports the measurement by mechanical or digital means.  The mechanism of a polar planimeter is two arms with four significant points among them.  Point P is the pole, a fixed point.  Point T is the tracer, which is moved by the user around the perimeter of the area.  Point F is the pivot point, at the joint between the two arms.  The arm joining points P and F is the pole arm.  The arm joining points F and T is the tracer arm.  The arms are of fixed length during a measurement.  (Many models allow the tracer arm length to be adjusted between measurements; a few allow the pole arm to be adjusted.)  The position of point F, therefore, is a consequence of the positions of P and T and the lengths FP and TF.  The other significant point is W, the position of the wheel.   The wheel is attached to the tracer arm, so W is in a fixed location relative to T and F.  The wheel must roll perpendicular to TF; that is, the wheel's axis must be parallel to TF.  Consequently, the component of the position of W perpendicular to TF is irrelevant, so it is usually simpler to say W is on the line TF.

2.  The Wheel

The wheel measures and records area in the same way a wheel can measure distance: by counting the number of turns (and fractions of a turn) of the wheel.  The wheel of a planimeter physically measures distance, but the measurement reported is scaled (multiplied) by the length of the tracer arm.  The wheel is designed to grip the paper and turn on its axis when moved perpendicular to the tracer arm, and to skid over the paper without turning when moved parallel to the tracer arm.

3.  Tracer Arm Movement

When the tracer arm is moved perpendicular to its own length, it is obvious that the wheel correctly measures the rectangular area swept by the arm, although the measurement reported might be positive or negative, depending on the direction the arm is moved.  When the arm is moved in any straight line, it is also clear that the wheel correctly measures the parallelogram swept by the arm , because the wheel rolls only for the altitude of the parallelogram.  And these ideas can be extended to any translation of the tracer arm, if there is no doubling back over the area.  Rotation of the tracer arm and doubling back are more complicated.

4.  Tracer Arm Rotation

In simple cases, the tracer arm has no net rotation.  The user traces the outline of an area, beginning and ending at the same point, so the tracer arm returns to its original angle.  Any measurement taken due to rotation is undone by the time the tracing is finished because when the wheel rolls backwards, the count of turns is diminished.  Therefore any erroneous measurement that is taken is automatically cancelled when the measurement is repeated in the opposite direction.

5.  Doubling Back Movement

In doubling back over an area swept by the tracer arm, the measurement of that area is cancelled, but this can leave some area swept with the arm moving in one direction only and some other area swept with the arm moving in the opposite direction only.  The pole arm conveniently accounts for this, ensuring that the pivot point F travels a single path (along a circle) so the net movement of the pivot is only the difference between its beginning and ending positions.  And in simple cases (when the pivot returns to its starting position), the pivot has no net movement.  As with the tracer arm, the pole arm has no net rotation, due to the way two circles intersect.

6.  Position of the Pivot

If the user returns the tracer to its starting point, and the pole has not moved, then there are only 2 possible positions for F, corresponding to the tracer arm bent counterclockwise or clockwise relative to the pole arm.  Therefore, the arms must not be allowed to change from one direction to the other during a measurement, which could only occur during a measurement by allowing the arms to become parallel and therefore at their limit.  Changing the direction the arms are bent between measurements is the method used with the compensating planimeter.

7.  Path of the Pivot

In simple cases, when the user traces the outline of a sample area, part of the time the tracer arm is moving in one direction, sweeping the sample area and some other area, bounded in part by the arc swept by the pivot.  And part of the time the tracer arm is moving in the opposite direction, sweeping only the other area.  This cancels the measurement of the unwanted area, leaving only the sample area swept one time in one direction.  Although the user is tracing only the outline, the user is also dragging along the rest of the tracer arm which is inherently sweeping area.  And the far end of the tracer arm is tied down to the pole arm, so ensuring that the pivot finds its way back to its starting position when the tracer returns to its own starting position.

8.  Measuring Twice or in the Opposite Direction

Nothing about the planimeter ensures that the user returns to the starting point, or prevents an area from being measured twice or in the opposite direction.  A user is expected to mark or remember the starting point.  Several separate areas can be totaled, or one area traced several times can give more accuracy (averaging several measurements).  Measuring in the opposite direction can be exploited to conveniently measure an area with a hole inside.  A perimeter can be traced in two directions, and these will give oppositely signed readings on any planimeter.

9.  The Zero Circle

The only practical case of the tracer and pole arms not returning to their original angles is when they make one full revolution.  In this case, the user has returned the tracer to its starting point, but the planimeter's pole is inside the area.  The planimeter does not read correctly, but the error is a function of the dimensions of the planimeter, not the area traced.  This error is known as the zero circle because if the tracer arm were adjusted relative to the pole arm so that the wheel always pointed directly toward the pole, then any movement of the pole arm would produce no measurement because the wheel would skid only.  Yet the tracer point could trace a complete circle, the zero circle, around the pole.

10.  Mathematical Definitions

To prove these assertions mathematically, consider the following definitions:

11.  Areas Measured

Due to a movement of the tracer, the movement can be divided into infinitesimally small parts, and the area swept by both arms can be resolved into two components and summed: the area swept due to movement of the pole arm only (keeping the tracer arm parallel to its initial angle) and the area swept due to movement of the tracer arm only (without moving the pole arm).  The area swept due to the pole arm only can be further resolved into two parts and summed: the area swept by the pole arm (due to its rotation) and the area swept by the tracer arm (moving parallel to itself).

12.  Corrections for Rotation of the Tracer Arm

The area swept due to rotation of the tracer arm includes a measurement of area by the wheel which is the same as if the arm moved parallel to itself, except that the arm is rotated on the pivot point.  To correct for this without disturbing the wheel measurement, the arm can be rotated about the wheel, introducing two more components: the area that was swept but not measured between wheel and tracer and the area that was measured but not swept between wheel and pivot, both due to rotation of the tracer arm.

13.  Mathematical Expressions of Areas

These are the areas under consideration:
  1. ½p²p = The area swept by the pole arm due to its rotation.  Resolved into infinitesimally small movements, each is a triangle.  So the area is ½ base × altitude.  The base is p.  The altitude is the distance along the circumference of the circle (but essentially vertical because of the small size) and is p·p.
  2. ts = The area swept by the tracer arm moving parallel to itself.  When the tracer arm moves this way, the wheel correctly measures distance perpendicular to the tracer arm.  And this is the area reported by the wheel.
  3. ½(t – w)²t = The area swept (but not measured by #2 above) between tracer and wheel due to rotation of the tracer arm.
  4. ½w²t = The area not swept (but measured by #2 above) between wheel and pivot due to rotation of the tracer arm.

14.  Formula for Area

So the area swept by both arms, a, is the sum of these except #4 is negative:
a = ½p²p + ts + ½(t – w)²t – ½w²t

15.  Formula for the Zero Circle

This formula reduces to a = ts when p and t are zero.  That is, the area is the wheel reading when the tracing of a perimeter is completed and the user returns the tracer to the starting point.  And in so doing, the unwanted area swept in both directions is cancelled.  But if the planimeter is inside the perimeter, p and t are each 2·pi and (p² + (t – w)² – w²)·pi or (p² + t² – 2tw)·pi must be added to the wheel reading (ts) in order to get the correct area (a).  These are formulas for the zero circle.  A perimeter can be traced in two directions, giving oppositely signed readings, and the user must add or subtract the area of the zero circle accordingly.

16.  Proof of the Zero Circle

The zero circle is provably also the circle formed by rotating the pole arm with the tracer arm adjusted relative to the pole arm so that the wheel produces no measurement.  In this configuration, the wheel skids and does not turn and because it is mounted perpendicular to the tracer arm, there is are right angles PWT (pole, wheel, tracer) and PWF (pole, wheel, pivot).  (Though the wheel is not necessarily along the line TF, the point W is assumed to be since the position of the wheel in the direction it rolls is of no interest.)   By the Pythagorean Theorem, right angle PWT gives PT² = PW² + WT² and right angle PWF gives PF² = PW² + WF².  The radius of this zero circle is PT, so the area is pi·PT², or by substitution pi·(PF² – WF² + WT²).  PF is p; WF is w; and WT is t – w.  So the area of the zero circle is the same as given in the preceding paragraph.

17.  Formula for the Wheel Reading

While the formula for area is the one ultimately needed, it is poorly organized.  The formula has three other area terms, and a wheel reading mixed in.  Also, the formula has been applied magically, when the tracing is complete.  Less mysterious would be a trivial rearrangement that leaves the wheel reading as output variable:
ts = a – ½p²p – ½(t – w)²t + ½w²t

18.  Description of the Wheel Reading

From this and the previous description, a wheel reading is:
  1. The area swept by both arms,
  2. minus the area swept by the pole arm,
  3. minus the area between tracer and wheel due to rotation of the tracer arm,
  4. plus the area between wheel and pivot due to rotation of the tracer arm.

19.  Significance of the Wheel Reading

On the face of it, there is nothing new here.  But the point of view has changed and there is no reliance on the arms returning to a starting point (though the user is welcome to do so).  The areas are described in geometric terms only.  The wheel reading is the numeric consequence.

20.  Geometric Interpretation of the Wheel Reading

The four areas are two pairs.  The area swept by both arms minus the area swept by the pole arm leaves the area swept by the tracer arm.  The difference of the two areas between tracer and wheel and between wheel and pivot, due to rotation of the tracer arm, are collectively the area that would be covered between tracer and pivot if the tracer arm were detached and rotated about the wheel instead of the pivot.  If the wheel is between pivot and tracer, then these are two distinct sectors of a circle, but of opposite sign.  But if the wheel is not between, the one area cancels out part of the other leaving only one sector (bounded by two radii).  So the four areas can often be regarded as only two: the area swept by the tracer arm, and the area that would be covered between tracer and pivot if the tracer arm were rotated about the wheel instead of the pivot.

21.  Purpose of the Java Applet

The Java applet planimtr demonstrates the planimeter, these geometric areas, and the wheel reading as the user traces a number of samples, and allows the planimeter's dimensions to be altered and measurements scaled.

22.  Organization of the Applet Window

The applet window consists of a drawing area on the left and a number of controls on the right.  Of the controls, the top set report the numeric values of the various areas and the wheel reading; a middle control selects the sample to be traced, and the bottom set allow the user to select the “drawing tool” (the interpretation of mouse input) and/or enter dimensions from the keyboard.

23.  Colors in the Drawing Area

The drawing area uses a system of primary colors, combined and inverted, to show a planimeter (or at least a schematic of a planimeter) and the significant geometric areas, which can and do overlap.  Ordinarily, the background is white and the sample is magenta.  When the user traces with the planimeter (by dragging with the mouse), other colors appear.  The area swept by the tracer arm is cyan over the background, and blue over the sample.  In addition, the area that would be covered between tracer and pivot if the tracer arm were rotated about the wheel is yellow over the background and red over the sample.  And where both kinds of areas coincide is green over the background and black over the sample.

24.  Some Areas Not Represented

Because of the way the colors combine, there is no representation of an area's direction or sign (positive vs. negative).  Also, an area that appears twice in the same place (either by sweeping the area twice or by rotating the planimeter into a second circle) has no color.  The second tracing cancels the first.

25.  Table of Area Colors

Areas represented by Color
Color   Without Lines With Planimeter Lines
white   (none) sample + swept + rotated
magenta   sample swept + rotated
cyan   swept sample + rotated
yellow   rotated sample + swept
blue   sample + swept rotated
red   sample + rotated swept
green   swept + rotated sample
black   sample + swept + rotated (none)

26.  Planimeter Consists of Narrow Lines

The planimeter itself is black over the background, and in general inverts the color of the areas it overlays.  The planimeter consists of narrow lines and circles, so it is distinguished by shape rather than color.  Lines form the pole and tracer arm.  Circles surround the pole, pivot, tracer and wheel points.  The circle around the tracer point is further distinguished with two crossing lines that form convenient crosshairs.  The wheel circle is crossed with a single line perpendicular to the tracer arm (and parallel to the direction a wheel would roll).  It is not permitted (nor practical) for any of these points to coincide except wheel with pivot or wheel with tracer.  In either of these cases, the planimeter has only three distinct points.  Wheel with pivot is the same as wheel alone.  Wheel with tracer is a circle with a single crossing line parallel to the tracer arm.

27.  Numeric Values of Areas

On the top right of the applet window are signed numeric values of areas, beginning with the four areas of the current tracing, in order as given in the formulas above: both arms, pole arm, tracer-wheel and pivot-wheel.  The sum of the four areas is the wheel reading and is next from the top.  Next is the zero circle (which is a function of planimeter dimensions only and does not change due to tracing), then two more numeric values: the wheel reading plus the zero circle, and the wheel reading minus the zero circle.  Ordinarily, the planimeter is outside the sample, so the wheel reading should be the area of the sample, but if the planimeter is inside then the zero circle must be added or subtracted from the wheel reading.  Which of these three values (that include the wheel reading) is appropriate is for the user to determine.  The applet supplies samples and maintains the simulation of the planimeter and the areas, but the user must interpret the results.  The unit of measure of all areas is square distance.  The distance unit of measure is a function of the scale factor on the bottom right of the applet window.  The sign of the area values is also a function of controls on the bottom right of the applet window and how a perimeter is traced in the drawing area.

28.  Selection of the Sample

The middle control on the right side of the applet window is a drop down box which selects the sample shown in the drawing area.  Most of these are contrived, and the alleged numeric value of the area is also provided.  The samples are provided as graphics images.  The applet makes no attempt to interpret their meaning or how well the user traces them.  The contrived samples are in magenta on a white background.  There are also a small number of samples in full color, which mix in a complicated way with the graphics of the simulated planimeter and areas.  In general, the planimeter lines invert all colors; the area swept by the tracer arm inverts red; and the area covered if the tracer arm were rotated about the wheel instead of the pivot inverts blue.  These external samples are defined by parameters to the applet.  The name of the parameters is param<num> where <num> begins with zero and increases.  The value of the parameter is the URL of a graphics file of appropriate format.

29.  Selection of the Drawing Mode

On the bottom right of the applet window are controls that allow the user to select the “drawing tool” (the interpretation of mouse input) and/or enter dimensions from the keyboard.  This consists primarily of a set of mutually exclusive checkboxes to select among the major mode of the applet  (similar to the hieroglyphics often seen on toolbars).  Selecting a checkbox controls the meaning of mouse input.  It also unlocks an adjacent textbox for keyboard entry.  This locking feature is confusing because the textboxes are never hidden (nor do they change color) and so it is unclear when a box is unlocked if the user is accustomed to stronger feedback.  It is also very common to select a checkbox for one mode, enter text from the keyboard, forget to select a checkbox for another mode and then expect to use the mouse with the other mode.  Despite these difficulties, the applet always accepts mouse and keyboard input for the current mode, and a mode change can only be done by selecting a checkbox.

30.  Tracing in the Drawing Area

The first three checkboxes mean that dragging the mouse over the drawing area causes the tracer point to move with the mouse pointer, simulating the planimeter, and graphically and numerically displaying the areas.  There are several reasons why this might not work.  The tracer point can not reach the mouse pointer if it is too far from the pole point.  The tracer point can not reach further than the sum of the lengths of the tracer arm (tracer to pivot) and pole arm.  Even to do this, the arms would have to be parallel, a condition in which a planimeter measures poorly, so the reach is limited to 15° from parallel.  Similarly, the tracer point can not reach the mouse pointer if it is too close to the pole point.  The tracer point can not reach inside the difference in the lengths of the two arms, and to do this the arms would be parallel, so are limited to 15° from this.  Another reason the tracer point might not meet the mouse pointer is slow processing speed.

31.  Tracing Modes

The first three checkboxes differ from one another only in the meaning of pressing down the mouse button.  The first checkbox, “clear and trace,” clears the areas, graphically and numerically (except the area of the zero circle), whenever the (left) mouse button is pushed down.  This is the normal mode of the applet.  The second checkbox, “continue trace,” does no clearing when the mouse button is pushed.  The applet continues as if the mouse were dragged from wherever was to wherever the user pushes the mouse button.  This mode is especially useful if you accidentally release the mouse button during “clear and trace” (because you don't want your previous tracing cleared).  Select “continue trace” before trying to continue tracing.  The third checkbox, and the last of this group, is “discontinuous trace” which is the same as “clear and trace” except the wheel reading is not cleared.  This mode allows several discontinuous areas to be traced and summed in the wheel reading.  Still, if the planimeter is moved (by moving the pole) between discontinuous tracings, the wheel reading is cleared.

32.  Clearing

Selecting any checkbox besides the first three immediately clears the areas, graphically and numerically (except the area of the zero circle).

33.  Move and Resize Modes

The next four checkboxes move and resize the planimeter itself.  The first of these, “move,” moves the planimeter's pole to wherever the user pushes the mouse button.  Other than the obvious use, this mode can fetch the planimeter if has been lost or scrolled out of sight in the drawing area.

34.  Resize Modes

The next three checkboxes, “move pivot,” “move tracer,” and “move wheel,” allow the user to change the  dimensions of the planimeter itself, either using the mouse or by entering a numeric value with the keyboard.  “Move pivot” resizes the pole arm (between the pole point P and pivot point F, length p) and leaves only the pole unmoved, as the tracer and wheel go along for the ride.  “Move tracer” resizes the tracer arm (between F and tracer point T, length t), moving only the tracer point.  “Move wheel” resizes the segment FW along the tracer arm (between F and the wheel W, length w), moving only the wheel.

35.  Resizing Effects in the Drawing Area and Textboxes

None of these changes the angles of the arms (especially to provide a means of getting a negative length).  Whenever the user presses the mouse button, the position of the mouse pointer in the direction of the arm (pole arm for “move pivot” and tracer arm for the others) sets the position of the point.  No point moves perpendicular to the arm.  All of these include a textbox in which a numeric value may be entered directly.  (First select the checkbox, then select the textbox and then key in the value.)  The textbox (and graphical planimeter) is automatically updated when the mouse is used.  If the keyboard is used to change a textbox, the graphical planimeter is updated when the Enter key is pressed or the mode is changed by select a checkbox.

36.  Negative Numbers in Textboxes Implement Special Features

The unit of measure in these three textboxes is pixels.  It is not permitted (nor practical) for the tracer (t) or pivot (p) value to be less than 10 (in absolute value).  Negative numbers in these three boxes implement three special features.  If the tracer (t) and wheel (w) values are of opposite sign, the wheel is on an extension of the tracer arm behind the pivot (the pivot is between wheel and tracer).  This is the usual way planimeters are constructed; nevertheless, it is generally understood that the length of the tracer arm remains t, the distance between F and T.  If the pivot (p) and tracer (t) values are of opposite sign, then the tracer arm (F toward T) is bent clockwise relative to the pole arm (P toward F).  If they are of the same sign, then the tracer arm is bent counterclockwise relative to the pole arm.  If the tracer (t) value and the scale factor (described below) are of opposite sign, then the wheel reading increases when a perimeter is traced clockwise.  If they are of the same sign, the reading decreases when tracing clockwise.

37.  Scaling Modes

On the bottom right of the applet window, the last three checkboxes, “ruler,” “scale factor,” and “scale name,” set the unit of measure for the numeric values of area on the top right of the applet window.  Most importantly, the “scale factor” gives the value to scale the area values.  The scale factor is a linear measure, while the area values are two dimensional.  A scale factor of one is a measure of pixels in the drawing area.  The last checkbox, “scale name,” is provided so this can be recorded in the adjacent textbox, which can contain anything, unlike all the other textboxes which contain numbers.    And unlike all the other checkboxes, “scale factor” and “scale name” select modes that ignore mouse input in the drawing area.

38.  Ruler Mode

The uppermost of the last three checkboxes, “ruler,” allows the user to trace a scale given graphically in a map or drawing.  When this mode is selected, a line can be traced in the drawing area, by pushing down the mouse button at one end, dragging the mouse, and releasing the button at the other end of the line.  Internally, this line represents some standard number of pixels.  Then from the keyboard, a number is entered from the keyboard into the textbox adjacent to “ruler.”  The number is the length of the line in the unit of measure of interest.  This gives a scale factor, which is automatically updated in the “scale factor” textbox, whenever the line is traced, the Enter key is pressed, or a checkbox is selected.  If the user subsequently enters a “scale factor” directly, that is the scale factor.  The traced ruler line is displayed in the same color as the planimeter itself.  To erase the line, press and release the button without moving the mouse.  This gives a zero length line, from which no scale factor can be determined, regardless of what is entered in the ruler textbox.

Larry Leinweber, Proprietor

The Planimeter Applet

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