Planimeter Applet Guide
Description of the planimeter mechanically and mathematically, and the
use of the applet.

Structure of the Instrument

The Wheel

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

Mathematical Definitions

Areas Measured

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

Tracing Modes

Clearing

Move and Resize Modes

Resize Modes

Resizing Effects in the Drawing Area and Textboxes

Negative Numbers in Textboxes Implement Special
Features

Scaling Modes

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:

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
the tracer

t  the distance of the tracer from the pivot, TF; length of the tracer
arm

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
arm.
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
the wheel.

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
– 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:

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
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, tracerwheel and
pivotwheel.
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
Return to Larry's Planimeter Platter