
                      +---------------------------------+
                      |                                 |
                      |   POLAR PROGRAM TUTORIAL FILE   |
                      |                                 |
                      +---------------------------------+


The purpose of this file is to provide a tutorial lesson on most of the basic
features of the program called POLAR.  While it is possible to learn how to
use the program by reading all the built-in help screens contained within
each menu, if you are a first-time user, we suggest you read this tutorial
while first running the program, and then later you can digest all the
information contained in the help functions within each menu in the program.
You can import this file, POLAR.TXT, into any word processor and then print
it so you can read the hard copy output while you run the program.

To begin the tutorial you must have access to the program file POLAR.EXE.  We
assume this file is in the current directory on the currently selected disk
drive.



PRELIMINARIES
=============


KEYBOARD CONVENTIONS
--------------------

As part of this tutorial we need to give directions on which keys to press on
your keyboard.  We will enclose in angle brackets single keystrokes that you
should type.  For example, if we ask you to type the first three letters
of the alphabet we will show <A> <B> <C>.  If we ask you to press the control
key, the alternate key, the backspace key, the space bar, or the enter (or
return) key, we will show <CTRL>, <ALT>, <BACKSPACE>, <SPACE> or <ENTER>.

Each enclosure in angle brackets should refer to exactly one keystroke or one
character.  On most keyboards, to type in a left or right parenthesis requires
using a <SHIFT> key in conjunction with another key.  We will NOT show the
shift key as a separate keystroke in this tutorial because we consider
entering a parenthesis as entering a single character.  Other characters like
<+> or <*> can be entered in two different ways on some keyboards, one way
uses the <SHIFT> key, the other way does not.  In general, we would only show
the single character in angle brackets and we leave it up to you to decide
whether or not the <SHIFT> is needed to enter the character.

These keyboard conventions should make clear exactly how many and which keys
you press.  If it is necessary to press two keys simultaneously we will show
a connecting plus sign between the keystrokes.  This is done primarily with
the Control key <CTRL>, and the Alternate key <ALT>.  For example, when we
show <CTRL>+<W> it means you should press both the Control key and key W at
the same time.  As another example, <ALT>+<X> is used in the Main Menu to
exit from the program.


DISPLAY STRINGS
---------------

We also need to indicate the contents of text strings you might see on the
display screen.  Such text parts will always be displayed in double quotes in
this tutorial file.  You will not see the double quotes on the screen, and the
screen may contain other text parts that we do not show in this file.  The
double quotes are simply a convenient way to indicate parts of what you may
see on the display.


ADVICE FOR NOVICES AND EXPERTS
------------------------------

This tutorial file assumes you have the mathematical background required to
understand the features that will be demonstrated.  You may find some sections
more applicable to novices than experts, or vice versa, depending on your
background and experience.  If you encounter an example that is beyond your
understanding, you can either skip that example, or you can press the keys and
view the results, even though you may not fully comprehend the output.  This
tutorial does not discuss techniques on how to best use or apply the available
features.  It only serves to demonstrate the basic features and capabilities
which you can learn to apply to solve problems that are of interest to you.


GETTING STARTED
===============

To begin running the POLAR program type the command:

                          <P> <O> <L> <A> <R> <ENTER>



A FIRST EXAMPLE
===============

You can't perform any really useful operations (except reading the Main Menu
help screens) without first keying in a function.  So perform the first menu
item by pressing
                                      <K>

to key in a new function.  An edit box will appear on the screen into which
you are to type your mathematical formula.  The first example function we will
enter is the cardioid  R = 3*(1+cos(@)).  In this tutorial file we will use
the letter @ to denote an angle.  This is the ASCII symbol that comes closest
to matching the Greek letter theta or phi.  On most keyboards this angle
letter is entered using <SHIFT>+<2>.

The asterisk character * is used to to indicate multiplication.  Note that
you do not type in the R = part of the formula since this program knows R is
a function of @.  Type

             <3> <*> <(> <1> <+> <C> <O> <S> <(> <@> <)> <)> <ENTER>

After pressing ENTER a brief message should appear indicating your function
formula has been accepted and the new formula should now appear at the top of
the Main Menu display screen.


                                "R = 3*(1+COS(@))"


We will now go the Graph Menu by pressing key

                                      <G>

You should see a new menu labeled as the "Graph Menu".  Press key

                                      <G>

a second time and you should see the graph of a cardioid that corresponds to
the above formula.

It is normal for the program to sound a short beep to indicate it is finished
making a graph.  After admiring the graph press either

                                <ESC> or <ENTER>

and you should return back to the Graph Menu.  Entering and graphing most
polar functions is about as simple as this first example.



CHANGING THE POLAR GRID COORDINATE BACKGROUND
==============================================

Some people prefer to see radial lines and circles as the background over
which their polar graphs are drawn.  If you press key P in the Graph Menu you
can change the type of polar grid over which your graph is drawn.  The default
grid shows the rectangular X-Y axes.  Each time you press key P the grid
background changes values between rectangular, polar, and both rectangular and
polar grids.  For now, press

                                      <P>

Having pressed key P once, the polar grid value should be set to "Polar".
Press

                                      <G>

to make the graph, and you should note the polar grid background over which
the graph is drawn.  After the graph is complete press

                               <ESC> or <ENTER>

to return to the Graph Menu, and once there, press

                                      <P>

again to change the polar grid value to "Both".  Then press

                                      <G>

to make a new graph and note the background for the graph.  This time you get
the polar grid together with the rectangular axes.

We prefer the regular rectangular coordinate axes for our background graphs
because the other grid backgrounds can tend to make some graphs look crowded.

Press
                               <ESC> or <ENTER>

to return to the Graph Menu and you can leave the polar grid value in the
state you prefer.  If you like the plain coordinate axes press

                                      <P>

one more time and the indicated value should show "Rect." for rectangular
axes only.

To return to the Main Menu from the Graph Menu press

                                <ESC> or <ENTER>
again.




SWITCHING BETWEEN R EQUALS and R^2 EQUALS
=========================================

You should now be back in the Main Menu and see the formula


                              "R = 3*(1+COS(@))"

at the top of the screen.

From the Main Menu press key
                                      <S>

to switch the graph formula to the form


                             "R^2 = 3*(1+COS(@))"


POLAR can graph either the radius, or the radius squared, as a function of
the angle @.  Some interesting polar graphs such as lemniscates use this
second form.

Press key
                                      <G>

twice to return to the Graph Menu and make a new graph based on the new form
of the equation.

As the new graph is made you may note it seems as if two points get plotted
at the same time.  The reason of course is that the radius can now take on
simultaneous positive and negative values.  This new graph makes two smaller
loops inside a larger one.

From the Main Menu you need only press key S to switch between these two
forms of polar equations.

To return to the Main Menu from the Graph Menu press

                                <ESC> or <ENTER>
twice.

Once back in the Main Menu press
                                      <S>

to switch the formula back to its default form in which the radius is not
squared.

                              "R = 3*(1+COS(@))"




PERFORMING INTEGRATION TO FIND AN AREA
======================================

Instead of going to the Graph Menu we are going to the Integration Menu.
Press key
                                      <I>

and you should see a new menu labeled as the "Integration Menu".

The default limits of integration let @ vary from 0 to 360 degrees which
corresponds to one complete revolution.

We are going to integrate the cardioid over the @-range from 0 to 360.

The number of subintervals should already have a default value of 30 which
is a little too small for our example.  So press

                                      <N>

to change the number of subintervals, and when prompted type

                              <1> <0> <0> <ENTER>

Note the new value of delta-@ = 3.6 which means we will generate a circular
sector every 3.6 degrees.  Then press key

                                      <A>

to calculate the area.  The next graph made should now show 100 circular
sectors which approximate the signed area inside the cardioid.


                           "Midpoint Riemann Sum"

                         "Area = 42.397552587884258"


Press
                               <ESC> or <ENTER>

once and you will see the above value re-displayed in the Integration Menu.

The reason for re-displaying the integral result is so you can see the
function formula, the limits of integration and all the integral parameters
in one screen.  Press

                               <ESC> or <ENTER>

one more time to return to the Integration Menu.




AN ARCLENGTH APPROXIMATION
==========================

The POLAR program can be used to calculate arc lengths.  Since we still
have the cardioid entered with one complete revolution as represented by the
limits of integration, lets try to calculate its arc length.

From the Integration Menu press
                                      <N>

to change the number of subintervals, and when prompted type

                                <1> <0> <ENTER>

Our first approximation will be with 10 arcs.

Press                                 <R>

to begin the calculation of the arclength around the cardioid.  The screen
should first show the graph and then it will draw the approximating arc
length elements.  As you can see from the graph, there is considerable
space between the straight line segments and the curved sections on the
cardioid.


                          "Arc Length Riemann Sum"

                        "Length = 23.1447079417700194"


Press                          <ESC> or <ENTER>

twice to return to the Integration Menu.

To make a better arc length approximation we will double the number of
subintervals from 10 to 20.

From the Integration Menu press
                                      <N>

to change the number of subintervals, and when prompted type

                                <2> <0> <ENTER>

Press                                 <R>

to calculate the arclength.  You should see a new graph with a better
approximation to the true arclength.


                          "Arc Length Riemann Sum"

                        "Length = 23.7803280069283225"


But we can still do better.  So we will change the number of subintervals to
100.

Press                          <ESC> or <ENTER>

twice to return to the Integration Menu.

From the Integration Menu press
                                      <N>

to change the number of subintervals, and when prompted type

                              <1> <0> <0> <ENTER>

Press                                 <R>

to calculate the arclength.  You should see a new graph with a better
approximation to the true arclength.


                          "Arc Length Riemann Sum"

                        "Length = 23.9911223721488482"


This time you probably can't tell any difference between the approximating
arcs and the graph of the cardioid.  The exact value of the true arc length
is 24.

Press

                               <ESC> or <ENTER>

three times to return to the Main Menu.




THE GRAPH/TANGENT/NORMAL TRACE MODES
====================================

When a graph is made, you can enter any one of three Trace Modes to move
along the graph one point at a time or to experience the variation in the
tangent or normal lines to the graph.

Press
                                      <G>

twice, the first time you should go to the Graph Menu and the second time
should cause the graph to be drawn.  While looking at the graph press key

                                      <T>

and you should see a tangent line drawn on the graph, starting at the @-value
which is the midpoint of the formula's current domain.

The coordinates of the current point of tangency are displayed at the bottom
of the screen, and the top of the screen shows the equation of the tangent
line in y=mx+b form.


                    "Y = -0.0314262660643684*X + 0.

                            @ = 180.    R=0.
                            X = 0.      Y=0."


Both the rectangular and polar coordinates of the point of tangency are given
at the bottom of the screen.

You may press either the left or right cursor arrow keys to cause the tangent
line to move across the graph.  As you move the point of tangency and the
tangent line you should note the coordinates of the point of tangency at the
bottom of the graphics screen get updated, as does the actual equation of the
tangent line that appears at the top of the screen.  The polar angle as well
as the rectangular coordinates are constantly updated.

Think of the cursor keys as increasing or decreasing @.  In fact, the up and
down arrow keys can be used as well as the left and right arrow keys.  When
the cursor keys on the numeric keypad are used then pressing a SHIFT key in
conjunction with any of the arrow keys causes a larger amount of movement,
for each keypress.  Try moving the tangent line along the curve by repeatedly
pressing the up cursor arrow key with the <SHIFT> key.  Then move in the
opposite direction by using the down arrow key.

After moving the tangent line around, press

                                      <N>

to switch to showing the normal lines to the graph.  Then continue to move
the current point around the graph using the cursor keys to view the tracking
of the normal lines to the graph.  You can press keys

                                 <T> or <N>

to flip the current line between a tangent and a normal.

Next, press                           <G>

to enter the Graph Trace Mode.  Again, press left or right or up or down
arrow keys to increase or decrease the @-parameter.  You should see a single
point trace out the polar curve one point at a time.

Finally, press

                               <ESC> or <ENTER>

to exit from the Trace Mode.


Then press
                               <ESC> or <ENTER>

two more times to quit viewing the graph and return to the Main Menu.




THE COORDINATE TRACE MODE
=========================

When you use POLAR to graph a function, the program generates the graph and
then waits for you to press a key while you are looking at the graph.  If the
key you press is either the  ESC  key or the  ENTER  key then you are simply
returned immediately to the Graph Menu.  If instead, you press key C, you will
enter what is called the Coordinate Trace Mode.  This is one of the most
useful modes of the entire POLAR program.

To demonstrate this mode, we assume you still have the function formula


                                "R = 3*(1+COS(@))"


displayed at the top of the Main Menu display screen.  Press key

                                      <G>

to enter the Graph Menu and press
                                      <G>

a second time to actually make the graph of the cardioid.  The program
should sound a short beep when the graph is finished.

Now press key
                                      <C>

and you should be in the Coordinate Trace Mode.  You will know when you are
in this mode because the bottom of the graphics screen should show the X and
Y coordinates of a point that is initially in the exact center of the display
screen.  The angle and radius of this point are also displayed.  Now, using
your numeric keypad, press key

                                      <3>

and you should see the point cursor move away from the origin down into the
fourth quadrant.  Continue pressing key

                                      <3>

until the trace point is approximately near the point X=1 and Y=-1.  You will
probably not be able to hit this exact point, due to the limited resolution
of your graphics display screen.

Pressing key 3 on the numeric keypad causes the trace point to move in a
southeast direction.  Pressing keys 8, 2, 6, and 4 would cause movement in the
north, south, east, and west directions, corresponding to the direction of the
arrows on those keys.  Pressing key 7 would cause a movement in the northwest
direction, and pressing key 9 would make the movement northeast.  Key 1 moves
southwest.  So the arrangement of the keys on your numeric keypad serves as a
convenient way to represent movements in directions corresponding to the
sides and corners of the rectangular arrangement of the keys.

Pressing key 5 in the center of the numeric keypad causes the cursor to be
centered on the graphics screen.  Sometimes the trace point is hard to see
because it overlaps some other graphic object that is already drawn on the
screen.

If you have a large 101-key keyboard with separate cursor arrow keys, you can
use the extra arrow keys to move the trace point in a direction corresponding
to the direction marked on each key.

Pressing a SHIFT key in conjunction with one of direction keys causes the
trace point to move 5 pixels at a time instead of one pixel at a time.  This
makes for a little faster movement for longer distances.  This SHIFT key
feature works only with any of the numeric keypad keys.

As you continue to press any cursor key, the coordinate trace mode point
cursor should continue to move in the direction corresponding to the key, and
more important, the coordinates at the bottom of the screen will continually
be updated to correspond to the position of the cursor point you see.

Now we are going to try to use the trace point to find one of the y-intercepts
of the graph of the function.  Use the cursor keys to move the trace point
over the point on the Y-axis that corresponds to @=90.

The true point of intersection should occur where X=0 and Y=3., but the
limited resolution of your display means you will probably not see these exact
values displayed for the point's coordinates.




ZOOMING IN ON A POINT
=====================

Press key
                                      <5>

on your numeric keypad to center the cursor on the display screen.  Then press
the key for the lower case character z

                                      <z>

The graphics screen window should change and a new graph will be made which
encompasses an area that is 1/4 of the previous graphics screen.  The new
center of this screen will be the old Coordinate Trace Mode point, and you
will remain in the Coordinate Trace Mode.  Press lower case character z a
second time

                                      <z>

and you will continue to zoom in on the origin.  Each time you zoom in, a
completely new graph is drawn.

Pressing lower case z makes the new graphics window smaller and corresponds to
zooming in on a smaller portion of the graph.  Pressing upper case Z performs
a zoom out operation that makes the size of the XY-plane window larger.  You
should remember the relation between the size of the letters z and Z to help
you decide how to zoom in or out.  In either case, the Coordinate Trace Mode
point remains as the point of central focus, and in fact this point will be
the new center of the new graphics screen after zooming either in or out.

Now press the Escape key

                                     <ESC>

twice.  The first time you exit the Coordinate Trace Mode, and the second time
you return to the Graph Menu.  Note the range of values for the current XY-
plane window X & Y minimums and maximums.  While looking at these values,
press
                                  <CTRL>+<W>

and you should see the window min/max values returned to their default values
where X ranges between -7 and +7 and Y ranges between -5 and +5.

Then press key
                                      <G>

to make a new graph and you should see the graph has returned back to its
original state before we began zooming in.




ZOOMING IN USING A BOXED WINDOW
===============================

Press key                             <C>

to re-enter the Coordinate Trace Mode.

Now use the cursor keys to move the Coordinate Trace Mode point to where
it is near the point X=-2 and Y=+2.  The placement does not have to be exact.
Then press key
                                      <X>

to mark the current point as an anchor point (either upper or lower case X
will do).  Whenever you are in the Coordinate Trace Mode and you press key X
you mark an anchor point.

Now move the cursor in a southeast direction until it is near the point X=+2
and Y=-2.  As the cursor moves you should see a rectangle drawn which has the
anchor point you marked as its upper left corner.  The rectangle which you are
defining can be used to zoom in that part of the graphics screen enclosed
inside the rectangle.  When the lower right corner is near X=+2 and Y=-2,
press key
                                      <Z>

(lower or upper case is not relevant with the boxed rectangle) and a new
graphics window should be drawn which shows the interior of the rectangle,
but this interior now fills the entire graphics screen.  The position of the
new Coordinate Trace Mode point is in the center of the new graph window, and
you remain in the Coordinate Trace Mode with the point's coordinates displayed
at the bottom of the screen.

After marking an anchor point, but while still defining the zoom-in rectangle,
you can press the Escape key to abort both the anchor point and the zoom-in
rectangle.  However, you remain in the Coordinate Trace Mode.

Now press
                               <ESC> or <ENTER>

two times to return to the Graph Menu.  Once in the Graph Menu press

                                  <CTRL>+<W>

to put the window back to its default position and size.




USING THE LINE DRAWING MODE
===========================

Now that you know about the Coordinate Trace Mode and the Tangent Line Mode
we are going to try the Line Drawing Mode to manually approximate the
tangent line to a graph, at a particular point on the graph.  Press key

                                      <G>

to make the graph of the polar curve  R = 3*(1+COS(@))  and then press key

                                      <C>

to go into the Coordinate Trace Mode.  Use the cursor keys to move the trace
point out in the first quadrant until it is near where X=3.62 and Y=3.62.
At this point @=45.  Again, the placement does not have to be exact.

To start the Line Drawing Mode at the current cursor point, press key

                                      <L>

and you should see a line drawn on the screen through the trace point.  The
equation of the line, in y=mx+b form, should be at the top of the screen.
When you first enter this mode, the line should have an approximate slope of
m=1.0.

The Coordinate Trace Mode point serves as an anchor point, through which the
line is drawn.  Now you can use the up or right cursor keys to increase the
slope or use the down or left arrow keys to decrease the slope of the line.

Try decreasing the slope of the line by pressing the down arrow key.  Each
keypress causes a change in the angle by about 1 degree, or 5 degrees if you
simultaneously press a SHIFT key.  This SHIFT key action only works with the
arrow keys on the numeric keypad.

Keep decreasing the slope until the line nearly matches what should be a
tangent line to the graph.  The true tangent line has the equation that is
approximately  y = -0.42X + 5.15.

Once you are satisfied with your tangent line approximation, press key

                                      <P>

to cause a radical change in the line.  Pressing key P makes a perpendicular
line through the anchor point, i.e., the new line has a perpendicular slope
compared to the line slope at the time you press key P.  In this case the new
line should approximate a normal line to the curve.

Now press key
                                      <P>

a second time and the previous tangent line should be restored.

As you increase or decrease the slope of the line, the line may become nearly
vertical, and the slope may abruptly change between a large positive number
and a large negative number.  In fact, the slope is usually calculated to be
in the range between -200 and +200, both of which represent nearly vertical
lines.

That is about all there is to the Line Drawing Mode.  If you press either

                               <ESC> or <ENTER>

you will exit the Line Drawing Mode and you can confirm this has happened
because the equation of the line at the top of the graphics screen will
disappear.  However, you will remain in the Coordinate Trace Mode and you
should still see the coordinates of the trace point at the bottom of the
screen.  Press

                               <ESC> or <ENTER>

again, and you will exit the Coordinate Trace Mode and just see the graph.

So press
                               <ESC> or <ENTER>

one more time to return to the Graph Menu.




MULTIPLE GRAPHS AND FINDING POINTS OF INTERSECTION
==================================================

Press key
                                      <G>

to make a new graph, and after you hear the beep when the graph is complete,
press key
                                      <S>

and the picture will be saved in memory.  After seeing a brief message that
the graph has been saved in memory, the graph will re-appear.


Press                          <ESC> or <ENTER>

once to return to the Graph Menu.

The graph of the cardioid has been saved and we are now going to change the
current function formula to the circle: R = 2.5

Press
                                      <K>

to key in a new function formula.  The bottom portion of the screen will clear.
A hidden feature of the POLAR program is the ability to change the function
formula at any time from within any menu by pressing key K.  Once you get past
the Main Menu you don't have to go all the way back to the Main Menu just to
key in a new function formula.


When prompted press

                              <2> <.> <5> <ENTER>


You should see a new formula at the top of the Graph Menu screen.


                                  "R = 2.5"


Press key
                                      <G>

to graph the circle.  You should see a circle centered at the origin with
radius = 2.5.

Then press
                               <ESC> or <ENTER>

to return to the Graph Menu.  The graph we just made was formed on a blank
background because the Background Graph should be turned off.  Press key

                                      <B>

to change the Background Graph value to  "On".


Then press key
                                      <G>

to make a new graph and as the new graph is made, you should note that the
initial graph background is the old cardioid that was saved in memory, and
the new circle is then graphed on top of the old background graph that was
saved.

We wish to find the point of intersection of the circle and the cardioid that
lies in the 2nd quadrant, near the point X=-0.41 and Y=2.46.  Press key

                                      <C>

to enter the Coordinate Trace Mode.  Move the cursor to the point of
intersection of the two graphs and read the coordinates at the bottom of the
screen.  To two decimal places the coordinates should approximate X=-0.41 and
Y=2.46, but the limited resolution of your display screen may prevent you from
seeing these exact values.

Now press
                               <ESC> or <ENTER>

two times to return to the Graph Menu, and once there, press

                                      <B>

to turn the Background Graph  "Off", and press

                                      <G>

to make a new graph which now only shows the circle.  The cardioid remains
saved in the background, until you quit the program, or until you save a
new graph which overwrites the existing background graph.  Any number of
polar graphs may be overlayed on top of each other and saved in the
background.

Now press
                               <ESC> or <ENTER>

twice to return to the Main Menu.




FINDING THE MAX/MIN EXTREMA
===========================

Another feature of this program is the ability to automatically find the
maxima and minima of any polar curve.  We assume the current function is
the same as used in the last example above.  The top of the Main Menu screen
should show


                                   "R = 2.5"


Press                                 <E>

to get to the menu labeled the Extrema (Max/Min) Menu.


The default search interval for @ is [0,360] and the default number of
samples is 360.  These default values are adequate for the extrema example
we wish to explore.

The visual search attribute should be turned on by default.  Press

                                      <S>

to begin the search.


The program should first graph the function and then it will show a trace
point move across the graph, with @ restricted to the search interval.  The
interval will be subdivided into 360 equally spaced points and the program
will determine the absolute extrema of these sampled points.  The sampling
includes both the X-coordinates and the Y-coordinates.  The program beeps
when the search ends, and will show two Y-coordinates at the bottom of the
graphics screen.


                             "Y = 2.5       Y = -2.5"


The value on the left is the maximum Y and the value on the right is the
minimum Y.  If you look closely at the graph you should also be able to see
the two points on the graph that are marked as the extreme points.  Note
that these are absolute extrema for the entire polar curve which is a circle.
Also note the markings of the minimum and maximum X-values on the X-axis.

Press                          <ESC> or <ENTER>

once and you will return to the Extrema Menu where you should see the complete
coordinates of the four points that have been found.


        "Minimum X = -2.5              Maximum X = 2.5
                 @ = 180.                      @ = 0.
                 Y = 5.3021595361E-11          Y = 0.

         Minimum Y = -2.5              Maximum Y = 2.5
                 @ = 270.                      @ = 90.
                 X = -7.9532393854E-11         X = 2.651079768E-11"


These values are subject to interpretation under the presence of normal
round-off error.  The three values represented in scientific notation are
really 0.

Press                          <ESC> or <ENTER>

again to completely return to the Extrema Menu.


For the second extrema example we will turn off the visual display attribute.
Press
                                      <V>

to turn off the visual display option.  The program will perform the search
without going into its graphics mode.  Press

                                      <S>

to begin the search in text mode.  You should note a small symbol at the
bottom of the screen that looks like it is spinning while the search
continues.  When the search ends the program will sound a short beep and you
should see the same four points returned as shown above.  Press

                               <ESC> or <ENTER>

two times to return to the Main Menu.



THE DISCRETE VERSUS CONNECTED GRAPH TYPES
=========================================

When graphing any polar curve you can have the program make the plot of the
graph in one of two ways.  The default graph type is called discrete and
this means the individual points on the graph are plotted one by one with
no particular connection between one point and the next.  To see this
attribute in a slightly emphasized form enter the function R=5*SIN(3*@).

Press                                 <K>

to key in a new function then type

                  <5> <*> <S> <I> <N> <(> <3> <*> <@> <)> <ENTER>

The status line at the top of the screen should show

                             "R = 5*SIN(3*@)"

Press                                 <G>

to go to the Graph Menu.

Press                                 <S>

to change the starting and ending @-values and when prompted type

                          <ENTER> <1> <8> <0> <ENTER>

Then press                            <G>

to make the graph.

If you look carefully at the graph you will see a noticeable amount of space
between one point and the next.  The graph is made up of a series of discrete
points.

We will now change the graph type to the connected type.  Press

                                     <ESC>

to return to the Graph Menu and once there press

                                      <T>

to change the graph type.  You should see the graph type appear as

                                  "Connected"

The next graph will be made by connecting each individual dot to the next
dot with a solid line.  Usually the connecting line segments are so short
they are not noticeable and the polar graph appears as a smooth curve.
Press
                                      <G>
to see the new graph.

You should notice how this graph looks more solid than the discrete graph.

Sometimes it is more desirable to view a continuous graph in its connected
form.  But there are other times when a graph is more properly made by using
the discrete type.

To see the discrete type one more time press

                                   <ESC> <T>

to change back to the discrete type and press

                                      <G>

to make a new graph.  Notice the denseness of the points on this graph.
Press
                                     <ESC>
once to return to the Graph Menu.



CONTROLLING THE SPEED AND DENSITY OF THE GRAPH
==============================================

An alternative to using the discrete graph type is to make the graph samples
more dense by specifying a smaller delta-@ value.  The current delta-@ value
should be set for every 1 degree.  We will now change this to plot a point
every 1/3 of a degree.

Press                                 <N>

to change the number of @ samples and when prompted type

                          <ENTER> <1> </> <3> <ENTER>

You should see delta-@ = 0.3333333333333 and the number of @ samples is 540.

Press                                 <G>

to make a new graph.

This time the graph will take longer to plot because many more points are
being plotted.  The graph will also look more dense because the points are
spaced closer together because this graph is made using a smaller delta-@
value.

Next we will make an even more dense graph.

Press                              <ESC> <N>

to return to the Graph Menu and change the delta-@ value.  When prompted
type
                       <ENTER> <1> </> <1> <0> <ENTER>

The delta-@ value should now be 0.1 which means a point will be plotted
every 1/10 of a degree.  The number of @ samples is now 1800.  Next press

                                      <G>

to make a new graph and as the graph is made notice the speed at which the
points get plotted.  The new graph should be very dense.

To vary the speed and density of the plot even more, press

                   <ESC> <N> <ENTER> <1> </> <3> <0> <ENTER>

which now makes the delta-@ value = 1/30 of a degree.  Press

                                      <G>
to make the graph.

This time you will notice it takes much longer to complete the graph, and the
points that get plotted are even more dense than before.

Thus you can simultaneously vary the speed and density by making a larger or
a smaller delta-@ value.  The larger the delta-@ value the faster the graph
is made but the less dense the points on the graph.  The smaller the delta-@
value the slower the graph is made but the more dense the points on the graph.
You can also directly specify the number of @ samples and ignore the delta-@
values.  In each of the above examples we let the program automatically
compute the number of @ samples each time we pressed ENTER when prompted for
the number of @ samples.

Press                                <ESC>

twice to return to the Main Menu.




GRAPHING WITH A PARAMETER
=========================

The next feature of the POLAR program we will demonstrate involves the use
of what is called a graph parameter.  For this program the graph parameter
is an auxiliary variable that is denoted by the letter P.  By controlling the
use of P you can make a series of related graphs.  We will first demonstrate
the use of P by keying in the function Y = P + 2*COS(@).



Press                                 <K>

to key in a new function and when prompted type

               <P> <+> <2> <*> <C> <O> <S> <(> <@> <)> <ENTER>


The status line at the top of the screen should show

                                "R = P+2*COS(@)"


Now press                             <G>

to get to the Graph Menu and once there press

                       <S> <ENTER> <3> <6> <0> <ENTER>

which sets the @-domain interval from 0 degrees to 360 degrees.

Press                                 <N>

to change the number of @ samples and when prompted type

                         <3> <6> <0> <ENTER> <ENTER>

which makes one sample point every one degree.


Next press                            <Q>

to get to the Parameter Menu.

The Parameter Menu shows the starting and ending values for P.  The delta-P
value gives the spacing between P-values over this interval and the number of
P samples tells how many times the parameter P will be sampled over the
given interval.  The number of P-values and the delta-P value go together.
Usually you can ignore the delta-P value and just enter the number of samples
that you want.  As each of these two quantities are edited the other quantity
is automatically changed.  The number of samples will usually be between 5
and 20.

Press                                 <S>

to change the starting and ending values of P and when prompted type

                            <2> <ENTER> <4> <ENTER>

The new P domain interval is defined starting at 2 and ending at 4.

Next press                            <N>

and when prompted type
                              <8> <ENTER> <ENTER>

You should now see the delta-P value is 0.25 and the number of P samples is
8.

The use of the parameter should already be turned on so press

                                      <G>

to make a graph using the P parameter.

You should see a series of 9 graphs, all of which are called limacons.

Each limacon is an odd-ball oval shape with a dimple on the left side.  As
the parameter increases the dimples become less pronounced.  In fact, the
very first graph is a cardioid (P=2).  The larger the value of the parameter
P the less the pronouncement of the dimple.  For this example the larger
values of P correspond to making larger radii.

By varying the range of the parameter P and the number of parameter samples
you can show any series of related graphs.

The variable P can appear in more than one place in your function formula,
but there is only one parameter.  This parameter is always denoted by P.
Except for showing a series of related graphs, P is not intended for any
other use within this version of POLAR.  Unless P appears at least once in
your function formula, P will be ignored.

Press                           <ESC> <Q> <P>

to return to the Parameter Menu and turn the use of the parameter off.  Now
whenever the radius function gets evaluated P will act as a constant whose
value is the starting P-value.

Then press
                                     <ESC>

two more times to return to the Main Menu.



MAKING A TABLE OF FUNCTION VALUES
=================================

The most mundane feature of POLAR is the ability to make a table of function
values.  This is not a significant feature, but since it is a Main Menu item
we will give one example of its use.

Actually, there are times when it is desirable to see a table of function
values.  But perhaps most users will only use this feature as a check on
whether they have correctly entered a long complicated formula.

For our example we will use the simple formula R = @.


From the Main Menu press

                                      <K>

and when prompted, type in the formula as

                                  <@> <ENTER>

The top line in the Main Menu screen should show:


                                    "R = @"


Now press
                                      <T>

and you should see the "Table of Values Menu".  To make a table with the
default values, press
                                      <S>

and when prompted, enter the starting value as @=0.  Type

                                  <0> <ENTER>

and the program should make a table of values.

Earlier we mentioned that although the human interface in POLAR only uses
angles considered to be in degrees, internally, POLAR computes with angles in
radians.  Thus the table you are looking at should show 1 degree increments,
but the values are in radians.

The table is made by starting with the @-value you give, and @ is incremented
line by line, using the value of delta-@.  The delta-@ value can be a negative
value if you want to make a table with values of @ that decrease.  To evaluate
a function at only one exact point make delta-@ equal to 0.

You make a new table every time you change the starting value.  Now press

                                <ESC> or <ENTER>

and you should return to the Main Menu.




MORE POLAR CURVE GRAPHING EXAMPLES
==================================

The following examples are just a few of the many interesting graphs that you
can make using the POLAR program.  By now you should be able to enter the
function formulas and set the associated graphing parameters on your own.

For all of these examples you should set the XY-plane window to its standard
size with X in the range between -7 and +7 and with Y in the range between
-5 and +5.


                                         Starting      Ending     Delta-@
         Polar Function Formula              @            @        value
      ------------------------------     ---------     ------     -------

 1.   R   = 3/COS(@)                         0          180         1/2

 2.   R   = 15*SIN(@)*SQR(COS(@))            0          180         1/2

 3.   R   = 2*COS(@)*(8*SQR(SIN(@))-3)       0          180         1/2

 4.   R   = 0.2*@                            0          1440         2.

 5.   R   = 5*SIN(@)/@                     -360         360         17/10

 6.   R^2 = 16*COS(@/360)*COS(@/4)          -90         360          1.

 7.   R^2 = 5*COS(@/360)-1/COS(@)            1.         360          1.

 8.   R   = 3*SIN(30*@/17+PI/30)+1           1          6120         1.





CONCLUSION
==========

This concludes the POLAR program tutorial.  If you haven't done so already,
you can now read the help information contained within each main program menu
item.  Most of the basic features have been covered here, but you will gain
more insight by reading all the help information available to you.  If after
all this you still have questions, you can contact the author at the address
given below.


From the Main Menu press
                                   <ALT>+<X>

to quit and exit from the entire POLAR program.


The POLAR program is periodically updated to make improvements, add new
features, (and sometimes to correct bugs!).  You may also wish to contact the
author to check if you have the latest version of the program.  The author
also invites your comments about how you liked the program and will consider
any suggestions you may wish to offer for making the program even more useful.




OTHER PROGRAMS
==============

If you enjoy using the POLAR program you may be interested to know there is
a whole suite of mathematical programs made by the author of POLAR.  These
programs are intended to help motivate an interest in mathematics and computer
science.  Some of the titles of these programs and a brief description of each
is given below.


 1. MATRIX - a program that teaches row operations with matrices.  Features
    include fraction mode, decimal mode, solves linear systems, inverses,
    determinants, sets of basis vectors, eigenvectors and eigenvalues,
    Gram-Schmidt orthogonalization, and the simplex algorithm.


 2. YFUNX - a program for graphing and analyzing functions in rectangular
    form, Y=F(X).  Includes coordinate trace and tangent/normal line modes,
    zooming in and out, scalable axes, graph parameter variable. Numerical
    integration features standard algorithms plus Gaussian Quadrature and the
    Romberg algorithm.  Animation features include plane areas, plane arc
    length, 3D volumes (disks & cylindrical shells) and 3D surface areas.
    Newton's method and the method of successive bisections are for solving
    F(X)=0.  Automatically finds maximum/minimum extrema.  All algorithms
    may be demonstrated in either graphics or text modes.


 3. POLAR - a program for graphing and analyzing functions in polar form,
    R=F(@) or R^2=F(@).  Similar to YFUNX, includes coordinate trace and
    tangent/normal line modes, zooming in and out, scalable axes, and a graph
    parameter variable.  Numerical integration for polar areas and arc length.
    Automatically finds maximum/minimum extrema over any section of a curve.


 4. PARAM - a program for graphing and analyzing functions in parametric form,
    X=F(T) and Y=G(T).  Similar to YFUNX, includes coordinate trace and
    tangent/normal line modes, zooming in and out, scalable axes, and a graph
    parameter variable.  Numerical integration calculates plane areas and arc
    length.  Automatically finds maximum/minimum extrema over any section of
    a curve.


 5. POLPM - a program for graphing and analyzing functions in polar
    coordinates, but that have been parametrized, say R=F(T) and @=G(T).
    Similar to the POLAR and PARAM programs, this program includes coordinate
    trace and tangent/normal line modes, zooming in and out, scalable axes,
    and a graph parameter variable.  Numerical integration for plane areas
    and arc length.  Automatically finds maximum/minimum extrema over any
    section of a curve.


 6. DIFEQ - a program related to 1st order differential equations.  Includes
    graphing the direction field and solves initial value problems using
    Euler methods and a 4th order Runge-Kutta method.  Includes coordinate
    trace mode, zooming in and out, and scalable axes.  Algorithms may be
    demonstrated in either graphics or text modes.


 7. CURVE3D - a program for making 3D graphs of curves given in the parametric
    form X=f(t), Y=g(t), and Z=h(t).  The resulting curve may be viewed from
    any position, and the drawing is a true-perspective 3D picture.


 8. SURF3D - a program to graph 3-dimensional surfaces of the form Z=F(X,Y).
    The resulting surface may be viewed from any position, and the drawing is
    a true-perspective 3D picture.  The surface may be displayed using lines
    of constant x, or constant y, or a fishnet.  Included is a hidden line
    algorithm for more realistic pictures.


 9. CFIT - a program which performs curve fits to data.  Includes linear
    regression for linear, exponential, logarithmic, and power functions.
    Graphs scatter diagrams and the fitted function curves and performs
    a statistical analysis, including an automatic best fit selection.  Data
    may be saved to or read from disk files.


10. GALTON - simulates coin tossing experiments related to probabilities and
    demonstrates graphically how the binomial distribution is related to the
    standard normal Gaussian bell-shaped curve.  Also compares stack counts
    with the numbers generated in Pascal's Triangle.  Either coins or
    ping-pong balls may be used in simulated experiments.  Variable number of
    rows of pegs, variable number of objects, variable left-right probability
    for generating skewed distributions.  Includes a single-step mode under
    full user control.


11. BUFFON - simulates needle dropping experiments related to probabilities
    used to approximate the number Pi.  Needles are randomly dropped on a
    grid of equally spaced parallel lines.  The length of each needle is 1/2
    that of the distance between the lines.  After dropping a large number
    of needles a count is made of the needles which cross a line.  Most
    needles do not touch or cross any line, but the ratio of the total
    number of needles dropped divided by the number of needles which cross
    a line approximates Pi.


12. PROPC - a symbolic logic program that calculates truth tables, analyzes
    tautologies, parses infix formulas and displays their Polish notation
    form, and generates Karnaugh maps from either tables or formulas.


13. RPNDEMO - a program which simulates how a calculator with RPN logic works.
    This program includes its own language and is similar in power to the
    HP-41 calculator.  Programs may be animated to show the internal workings
    of the machine.  Can also be used to teach assembly language concepts.


14. CALC - a reverse Polish logic calculator that operates on 5 data types.
    Included are real and complex numbers, fractions, binary integers and
    polynomials.  Special features include factoring integers and
    polynomials, analyzing repeating decimals and working with continued
    fractions.


15. LOAN - a finance program that handles the two standard cases of compound
    interest.  Uses the 5 standard financial variables n i PV PMT FV found
    on most financial calculators.  Can determine payment schedules for
    loans and annuities and can print amortization schedules for loans and
    interest earning schedules for lump sums and periodic payments.


16. FCARD - simple flash card type of program that can be used to memorize
    any simple series of facts, with one item per line of text.  Items can
    be presented in a random order with timing if desired.


17. THANOI - a game known as the Towers of Hanoi.  The game solution uses
    a recursive algorithm and the purpose of the program is to demonstrate
    the validity and simplicity of a recursive solution to a complex problem
    that would otherwise overwhelm a normal human being.


18. TRIANGLE - a simple program which solves triangle problems in which one
    is given 3 facts about a triangle and must solve for all the remaining
    parts.  Handles all 19 cases of triangle inputs and includes the Law of
    Cosines and the ambiguous case of the Law of Sines.  Can automatically
    determine when two valid triangle solutions exist.  Draws all triangle
    solutions to scale on a graphics screen and computes the perimeter and
    and the area in addition to finding and labeling all the sides and angles.


19. EXPMCON - a utility type of program that works with the above MATRIX
    program and the commercial scientific word processor called EXP.  This
    program converts MATRIX files from an ASCII format to the EXP format.


20. BMPLOT - a utility program that makes high resolution monochrome bitmap
    function plots, identical to the kinds of graphs made by the programs
    YFUNX, POLAR, PARAM, and POLPM.  The bitmaps may be read into other
    programs such as paint or drawing or desktop publishing programs which
    can be used to add labels and titles.  The monochrome bitmaps may be of
    any size or resolution so the output is compatible with virtually every
    printer and/or graphics environment.  The file formats supported include
    PCX, TIFF, and BMP.  The HP-GL/2 plotter language is an optional output
    to either a file or any HP-compatible plotter or PCL 5 LaserJet compatible
    printer.


21. XPRES - a program which computes integers with up to 20,000 digits per
    integer.  This RPN calculator is useful for computing exact values of
    factorials, permutations, combinations, and powers of integers.  For
    example, you can compute the exact value of numbers like 1000! or the
    exact value of 2 raised to the 5,000th power.  Integers may be saved to
    or read from ASCII text disk files.


22. TURING - a program which simulates the operation of a Turing Machine
    which is an abstract model of a primitive digital computer.  In fact,
    the model is fundamental to all digital (logical) computations.  Such a
    machine was conceived by the British mathematician Alan Turing in 1935,
    long before digital computers became established.  Turing also worked on
    machines to break codes used by the German Enigma spy machine in World
    War II.  Three sample demonstration programs are included.


23. PTRIPLE - a program which generates and tests Pythagorean Triples.
    Three numbers, say a, b, c are a Pythagorean Triple if a^2 + b^2 = c^2.
    If the GCF among a, b,and c is 1 the triple is called primitive.  Every
    non-primitive triple is a multiple of a primtive triple.  This program
    works with both general and primitive triples and can make ranges of
    tables of triples in ASCII text files.


For more information about any of these programs you may contact the author.

   John Kennedy                 Voice Phone/Messages any time of day or
   Mathematics Department       night: (310) 450-5150  Extension 9721.
   Santa Monica College
   1900 Pico Blvd.              Internet E-Mail: jkennedy@netcon.smc.edu
   Santa Monica, CA  90405
   U.S.A.
