The Museum of HP Calculators
Planimeter Applet Guide
Description of the planimeter mechanically and mathematically, and the
use of the applet.
Structure of the Instrument
Tracer Arm Movement
Tracer Arm Rotation
Doubling Back Movement
Position of the Pivot
Path of the Pivot
Measuring Twice or in the Opposite Direction
The Zero Circle
Corrections for Rotation of the Tracer Arm
Mathematical Expressions of Areas
Formula for Area
Formula for the Zero Circle
Proof of the Zero Circle
Formula for the Wheel Reading
Description of the Wheel Reading
Significance of the Wheel Reading
Geometric Interpretation of the Wheel Reading
Purpose of the Java Applet
Organization of the Applet Window
Colors in the Drawing Area
Some Areas Not Represented
Table of Area Colors
Planimeter Consists of Narrow Lines
Numeric Values of Areas
Selection of the Sample
Selection of the Drawing Mode
Tracing in the Drawing Area
Move and Resize Modes
Resizing Effects in the Drawing Area and Textboxes
Negative Numbers in Textboxes Implement Special Features
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 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
10. Mathematical Definitions
To prove these ascertains mathematically, consider the following definitions:
a - the area swept by the pole and tracer arms
s - the distance rolled by the wheel
w - the distance of the wheel from the pivot, WF, in the direction of
t - the distance of the tracer from the pivot, TF; length of the tracer
t - the angle of the tracer arm (in radians, relative to the angle
at the start of the tracing)
p - the distance of the pivot from the pole, FP
p - the angle of the pole arm (in radians, relative to the angle
at the start of the tracing)
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
13. Mathematical Expressions of Areas
These are the areas under consideration:
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.
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
½(t – w)²t = The area swept (but not measured by #2
above) between tracer and wheel due to rotation of the tracer arm.
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
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
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
18. Description of the Wheel Reading
From this and the previous description, a wheel reading is:
The area swept by both arms,
minus the area swept by the pole arm,
minus the area between tracer and wheel due to rotation of the tracer arm,
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
||With Planimeter Lines
||sample + swept + rotated
||swept + rotated
||sample + rotated
||sample + swept
||sample + swept
||sample + rotated
||swept + rotated
||sample + swept + rotated
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
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
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.
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
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
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, Author
Larry's Planimeter Platter
back to the planimeter applet
back to the main exhibit hall