The Learning Curve/The Learning Curve (Weird Science, 1996).iso/mathematics/plotters/complexplot/english/complexplotter.doc

Text File | 1993-04-11 | 25.0 KB | 722 lines

    Documentation of ComplexPlotter

    This is a translation of the documentation of the german version
    of ComplexPlotter. The original text contains a lot of mathematical
    expressions, and I often could only guess how they are translated.
    So please excuse the mistakes.


    1. Introduction

    This program is a "complex function plotter", with which sets of
    complex numbers can be pictured by a function.

    Because you now probably think "oh no, that´s too complex for me",
    I want to describe this: Complex Numbers can be compared with
    two-dimensional points (x- and y-coordinate are called real and
    imaginary part), and so sets of complex numbers can be interpreted
    as drawings or pictures. On complex numbers you can use functions
    in the same way as on the commonly used real (or rational) numbers.

    Most of the functions which are used for real numbers are defined
    for complex numbers, too (for example the sinus- or the exponential
    function). As the real functions distribute all numbers (of the
    definition set) to other numbers, the complex functions put
    two-dimensional points to other points. From the original
    picture is created a new picture. And this is the core of the
    matter: this function plotter is a flexible tool for picture
    metamorphosis.

    For example you can distort pictures, bend them, "mirror the
    inner parts out", enlarge them at different points etc. The
    results are often interesting, or even amusing.

    To input the original drawing (also called definition area) the
    program has an editor, in which you for example can paint
    freehandedly with your mouse. Then there are two figure generators
    (cartesic and polar nets), which are useful for more systematic
    studies about what a specific function does. Drawings can be saved
    to disk. Also you can copy the drawing which was already changed
    by a complex function back to the editor to change it again (if
    you repeat this there can be some kind of film effect, and you
    can for example watch the convergency of the function).

    So this program is suitable for experiments and play-around, but
    for distinct mathematic tasks as well (e.g. study of convergency,
    fix points, singularities, periodicy etc).

    Just to make it even more useful it is possible to emulate other
    types of function plotters, for example the "normal" plotter, which
    produces the well-known pictures of parabolas, sinus-waves etc.
    More in chapter "Emulation of other plotter-types" (4.2).

    The used mathematical background of course is no general knowledge.
    Who is interested in knowing more can try to read books of higher
    mathematics or function theory. But as well you can let that be and
    just experiment with this program (for comparison: the pictures
    of fractal geometry are widely known and liked, but how many
    people really understand, how they are created).



    2. How the program is used

    The program can be started from CLI with

        Plotter/English/ComplexPlotter

    or from the workbench (the program was written under version
    1.3 but should run under 2.0 or higher as well). It needs the
    "MathTrans.library" (it is to be found in the directory libs of
    your system disk). Then there should be at least 1MB RAM (I
    think it could run with 512KB if there is nothing else in memory,
    but I don´t guarantee).

    If the program doesn´t start either there isn´t the MathTrans.
    library or there isn´t enough RAM left.

    The program has three screen-sized windows, between which can
    be freely switched. In the first window "Main parameters" you
    can type the function to plot and the borders of the two other
    windows, where the original drawing is defined ("definition
    area"/"definition window") and the picture, which was changed
    by the function, is shown ("result area"/"result window").

    Every window has its own menu bar, which are activated by the
    richt mouse button, as usual. Some features can be accessed from
    all three windows, while most can be used from only one window.

    Most features can also be accessed by key-commands which are
    shown at the appropriate menu items (right amiga-key plus
    letter key).

    A typical function plot goes this way: first you type in the
    wanted function term and define the borders of the definition
    area. Then you switch to that window and choose a drawing
    (either by using the mouse, using the generators or loading from
    disk). Then you order the Amiga to "plot" the drawing and can then
    look at the result. You can choose to let determine the borders
    of the result window every time a plot is done by switching to
    "Automatic" in the main parameters window.

    After that you can zoom different parts of the result picture
    to watch details. You can copy the result picture back into
    the editor of the definition area in order to use that picture
    for a next plot. If you use the right function it is possible
    to create a film-like picture series.

    In the next sections of this documentation follows a complete
    list of the program´s features.



    2.1 Main parameters window

    Here you can define the complex function with which shall
    be worked. To do that you click on the marked input field
    and type in the function. Used as the variable is "z". More
    about the term syntax below.

    Secondly you can set the borders of the two other windows.
    Be sure that the right border has a greater value than the
    left one, and the top border has a greater value than the
    bottom border, otherwise you get an error message when
    switching to that window.

    For the result window you can choose if the borders shall be
    determined by the computer or the given values shall be used.
    Normally you´ll select "Automatic". This means, that every
    time the result window is selected its borders are calculated
    automatically. The numbers which are displayed in the main
    parameters window are changed accordingly. If you don´t like
    that, choose "Manual".

    The menu bar offers the following:


     1) Commands:

     a) Plot (Amiga-P)
        This starts the calculation of the result picture out of
        the definition drawing. After this the result window is
        activated to show the result.

     b) Quit program
        With this command you quit the program (after a safety
        check).


     2) Windows:

     a) Definition area (Amiga-D)
        To switch to the definition window where you define the
        original drawing.

     b) Result area (Amiga-R)
        To switch to the result window where you can watch the
        calculated picture (when already plot).

     c) Automatic (Amiga-A)
        Same as the button "Automatic" (see above).

     d) Manual (Amiga-M)
        Same as the button "Manual" (see above).


    2.2 Syntax of the floating point numbers and the functions

    Numbers can be typed in normal and exponential writing. They
    can range between +/- 1e-19 and +/- 9e18.

    Imaginary parts are marked by an appended "i". Or you can put
    "*i" behind or "i*" before the number, but with the first
    method the numbers are read directly as imaginary parts, whereas
    in the second and third method the program reads the number as
    a real part and multiplies it with i, which of course takes
    much more time.

    Here some examples:

        0.52578        equals .52578 or +.52578
        -598.1245
        1.42e6        equals 1.42*10^6
        7.34e-8        equals 7.34*10^-8
        6.345i        equals 6.345*i or i*6.345
        12+8i
        3-.5i

    There is a wide range of mathematical functions:

        ...+...         Addition
        ...-...         Subtraction
        ...*...         Multiplication
        .../...         Division
        ...^...         Potency
        re(...)         Real part of a complex number
        im(...)         Imaginary part of a complex number
        abs(...)        Absolute value ("length") of a complex number
        arg(...)        Argument ("angle") of a complex number
        con(...)        Conjugated complex number
        sqrt(...)       Square root
        exp(...)        Exponential function
        log(...)        Logarithm
        sin(...)        Sinus function
        cos(...)        Cosinus function
        tan(...)        Tangens function
        sinh(...)       Sinus Hyperbolicus
        cosh(...)       Cosinus Hyperbolicus
        tanh(...)       Tangens Hyperbolicus
        arcsin(...)     Arcus Sinus (inversion of sinus)
        arccos(...)     Arcus Cosinus (inversion of cosinus)
        arctan(...)     Arcus Tangens (inversion of tangens)
        arsinh(...)     Area Sinus (inversion of sinus hyp.)
        arcosh(...)     Area Cosinus (inversion of cosinus Hyp.)
        artanh(...)     Area Tangens (inversion of tanges Hyp.)
        int(...)        Rounded values of real and imaginary part
        sgn(...)        Signums of real and imaginary part
        swap(...)       Swaps real and imaginary part

    More about these functions in the third chapter.

    You can use the following symbols:

        z        The variable. It stands for the points of the
                definition area.
        x        Real part of the variable (equals re(z)).
        y        Imaginary part of the variable (equals im(z)).
        pi        The famous constant 3.1415927...

    Besides that you can use up to ten bracket levels. Spaces are
    skipped, and you can use capital letters as you like. Finally
    some examples (they demonstrate the syntax, but are not really
    for trying out):

        2*sin(z/(cos(z)+exp(z)))+6i
        (i*z+exp(z))*(z^2)
        (-z)^(x+y)
        abs(z)*exp(i*(arg(z)+pi/4))


    2.3 Definition window

    In this window the points, which are to be pictured by the
    complex function, are defined. For that an editor is implemented,
    and some basic figures can be used. All features of the editor
    are activated by the menu bar (or the equaling key-commands).

    On the top window bar the position of the mouse pointer is always
    shown in floating point format. If you, for example, move the
    mouse pointer to the right bottom edge, the right and bottom
    border values, which you have set in the main parameters window,
    are shown.

    Also the active drawing mode is shown, that´s normally "Freehand".
    In this mode you can draw freely within the window area. Other
    modes are "Lines", "Circles", "Fill" and "Zoom". In line mode
    you can draw lines: start and end point will be put into the
    definition area memory. In circle mode 36 points of the circle
    edge, and in fill mode of course the fill point will be memoried.

    When you have drawn some points, you can use the menu item "Zoom
    complete" to adjust the window borders, so that the drawing
    completely fits into the window (as is done in "automatic" mode).
    With "Zoom out" the window borders are pushed outward, so that
    the window area becomes larger, and the drawing looks smaller.
    Often it is good to watch the drawing from some distance to see
    it better.

    Last not least you can use "Zoom a part" to enlarge a part of
    the drawing by marking this part with the mouse.

    If you´ve done something wrong you can take back the last made
    part of the drawing by using "Undo". This is especially helpfull
    when using the fill mode, because a wrong set fill point can easily
    fill the whole window.

    With "Load" and "Save" you can load a drawing from disk or save
    the just created drawing. For that purpose a filerequester appears
    which should be easy enough to use.

    "Fishing net" and "Spiders web" are two figure generators which
    are especially useful when systematically studying the effects
    of a complex function. The fishing net (cartesic net) is an
    array of horizontal and vertical lines, of which the cross points
    are put into memory. In an extra parameter window you have to
    determine the position and size of the net. You also have to
    declare the number of horizontal and vertical sections (0-999).
    With a number of sections of 1 a bar, and with a number of sections
    of 0 a simple line appears. It´s easy to learn, just try around
    a bit.

    The spiders web looks as it is named. You define the center point
    of the net, the radius, the number of rays (3 to 99) and the number
    of circles (1 to 99). Here, too, is the best way of understanding
    the use of the parameters experimenting.

    Here the commands of the menu bar:

     1) Commands:

     a) Plot (Amiga-P)
        With this menu item the computation of the result picture
        is started. After that the result window is activated.

     b) Copy Res->Def (Amiga-K)
        With this the content of the result area is copied into
        the definition area. The old definition drawing will be
        overwritten.

     c) Clear ALL (Amiga-L)
        With this item the complete contents of the definition area
        will be erased (irrevocably) !

     d) Quit program
        To exit ComplexPlotter.


     2) Windows:

     a) Parameters (Amiga-E)
        This is to switch back to the main parameters window,
        e.g. to change the function.

     b) Result area  (Amiga-R)
        To switch to the result window.

     c) Zoom complete (Amiga-A)
        With this command the borders of the window will be put
        tightly around the drawing, so that it fits completely into
        the window. For example that´s useful to take back zooms
        done by the next to menu commands.

     d) Zoom a part (Amiga-M)
        If you want to work on a specific part of the drawing, you
        can choose this menu command and then mark this area. The
        window borders will then be set around the marked area.

     e) Zoom out (Amiga-Z)
        To get a better view of the whole drawing you can enlarge
        the definition area by moving the borders some way outward.
        A zoom complete with following zoom out often produces
        the best view of the complete drawing.


     3) Drawmode:

     a) Freehand (Amiga-1)
        This is the mode with which you can draw freely with the
        mouse. With the left mouse button the points or lines
        are set.

     b) Lines (Amiga-2)
        With this mode you can connect two points with a straight
        line. For that purpose you move the mouse pointer to the
        first point, press the left button, and move to the second
        point while still pressing the button. Then you release the
        button.

     c) Circles (Amiga-3)
        If you want to draw circles, you can use this drawmode. You
        choose the center point and then move to a point on the circle
        line while pressing the left mouse button. You should keep
        in mind that these are circles in the definition area, and
        can look like ellipses on the screen, just as the borders of
        the window are defined.

     d) Fill (Amiga-4)
        To fill bordered parts of the drawing you can use this mode.
        You should be carefull that the desired parts are really
        completely bordered, otherwise the whole window is filled.

     e) Undo (Amiga-U)
        With this menu item the lastly drawn figure is undone
        (freehand line, line, circle, fill, fishing net or spiders
        web).


     4. Figures:

     a) Fishing net (Amiga-5)
        With this item you activate the figure generator for cartesic
        nets. An extra parameter window appears, where you can define
        your desired net. The net will be generated if you click on
        the "Start"-button or finish the last parameter with the
        enter key. With the "Abort"-button the generation is canceled.

     b) Spiders web (Amiga-6)
        The generation of a spiders web (polar net) works in the same
        way as the generation of a fishing net.

     c) Load (Amiga-7)
        Of course you can save completed drawings. With this command
        you can load drawings from disk. An existing drawing will
        be overwritten. The filerequester has features to walk
        through directories, selection of the drive, scroll buttons,
        direct input of path name and file name, and supports
        double clicking of the file name.

     d) Save (Amiga-8)
        This item is for saving the drawings of the definition area.
        It is used as the load item.

    You should know that the drawing memory is limited to 3000 points,
    but that should be sufficient for most purposes.



    2.4 Result window

    In this window the plotting result is being displayed. When
    the automatic mode is selected the borders of this window will
    be adjusted to the size of the picture. The border values are
    displayed on the top window bar.

    There exist the same window commands as in the definition window,
    with the exception that the "Zoom a part"-feature is active all
    the time. This means that you can zoom to selected parts of the
    picture without calling a menu command, just by marking a rectangle
    with your mouse.

    With the Copy&Plot-feature you can copy the picture back into
    the definition area and compute a new plot picture (the commands
    Copy Res->Def and Plot are executed in succession). When repeating
    this feature several times the picture can become "animated".
    Sometimes this can look interesting, and it helps finding fix points
    of the funtion (Fix points are points which aren´t changed by the
    function. Often some areas of the picture move closer and closer
    to such fix points - that´s called convergency).

    So the items of the menu bar are quickly explained:

     1. Commands:

     a) Plot (Amiga-P)
        As in the other two windows, the calculation of the picture
        will be started.

     b) Copy Res->Def (Amiga-K)
        The picture of the result area is copied into the definition
        area (the old drawing will be overwritten). Then the definition
        window is activated.

     c) Copy&Plot (Amiga-R)
        This feature does the commands Copy Res->Def and Plot in
        succession (without switching the windows).

     d) Quit program
        This command is reserved for all who are sick of this program.


     2. Windows:

     a) Parameters (Amiga-E)
        With this item you get back to the main parameters windows
        where you can edit the function term and the window borders.

     b) Definition area (Amiga-D)
        To get back to the definition editor you should select this.

     c) Zoom complete (Amiga-A)
        With this you can adjust the window borders to the picture
        borders. This command is called automatically if the automatic
        mode is selected in the main parameters window.

     d) Zoom out (Amiga-Z)
        With this menu item the window borders are pushed outwards
        so that the picture will become smaller. This is good to
        take back zoomings into parts of the picture.



    3. Descriptions of some complex functions

    Of course you want to know what the complex functions do to your
    drawings. I only can describe some basic mechanisms, because the
    whole theory actually fills books. So with some complicated
    functions you should "learn by trying".


    3.1 Translations (moving the drawing to another place)

    Moving the drawing around is simple: the addition of two complex
    numbers is like the addition of vectors, so you have to add a
    constant complex number which represents the translation. For
    example to move the drawing 2 measures to the right and 1 measure
    up you can use the function:

          f(z) = z + 2+i



    3.2 Making the drawing smaller/bigger

    A scalar multiplication with real numbers accomplishes this.
    For example the function

          f(z) = 2*x + .5i*y

    doubles the breadth and halves the height.



    3.3 Simple mirrorings/rotations

    To mirror the drawing at the real axis you can use the function

          f(z) = con(z)

    which produces the complex conjugated numbers (the imaginary
    part is being negated).

    If you negate the complex numbers the drawing will be mirrored
    at the origin:

          f(z) = -z

    If you want to mirror the drawing at the 45 degree line you can do
    this by swapping real and imaginary part:

          f(z) = swap(z)

    To rotate the figure 90 degree counterclockwise you can multiply
    the points with i:

          f(z) = i*z



    3.4 Other rotations

    Rotations with all kinds of angles kann be accomplished in different
    ways. The first is to multiply the numbers with a constant complex
    number. When multiplicating two complex numbers the angles (to the
    real axis) will be added and the lengths (the distance of the point
    to the origin = the absolute value) multiplied. For example to
    rotate the drawing 45 degree counterclockwise you have to multiply
    the points with that complex number which has the angle of 45
    degree and the length of 1, and that is sqrt(.5)+sqrt(.5)*i:

          f(z) = (sqrt(.5)+sqrt(.5)*i)*z

    Another possibility is to use the polar coordinate writing of
    the complex numbers and directly add the rotation angle:

          f(z) = abs(z)*exp(i*(arg(z)+pi/4))

    This function rotates with pi/4, which equals 45 degree counter-
    clockwise.

    And you can use the matrices of the special orthogonal group
    which are related to the linear algebra. Here the function for
    the 45 degree rotation:


        f(z) = x*cos(pi/4)+y*sin(pi/4)+i*(y*cos(pi/4)-x*sin(pi/4))

    Of course all three methods are mathematically equivalent.


    3.5 The parabola z^2

    To demonstrate the complex parabola use can use the following
    fishing net:

       Borders: Left: -3  Right: 3  Top: 2  Bottom: 1
       Sections: Horizontal: 25  Vertical:1

    This strip will be bend around the origin.


    3.6 The hyperbola 1/z

    To show some aspects of this function you can first use a
    spiders web with the center in the origin. This drawing will
    be pictured to a web-like thing where the circles are getting
    more crowded at the mid. This is so, because the function isn´t
    defined for z=0 and has a polar singularity there.


    3.7 The trigonometric functions (sin, cos, etc.)

    The trigonometric functions have several effects. Usually they
    have the likeness to catapult points which have some distance
    to the origin even farther away. So you should use drawings
    near the origin, not farther away than ca. 4 measures.

    If you for example use the sinus function the drawing will
    be bend and enlarged in several directions.

    The trigonometric and hyperbolic functions are related very
    closely to the exponential function.


    3.8 The exponential function und the logarithm

    The exponential function rotates according to the imaginary
    part of the number and enlarges exponentially according to
    the real part of the number. This results in bending in direction
    of the real axis and rotating in direction of the imaginary axis.

    The logarithm puts everything into a horizontal strip with the
    height of 2*pi, because im(log(z))=arg(z), and arg(z) being the
    angle of the complex number can only have values between -pi and
    pi.



    4. Examples and ideas


    4.1 Drawings already on disk

    In the directoy "Plotter/DefPics" you find some example drawings
    which can be loaded into the definition editor.

    The drawings "Quadrants", "QuadrantsII" and "Bar" should demonstrate
    that the generators for fishing net and spiders web can even be
    used to produce some more complex drawings. They are suited for
    directly studying complex functions.

    The drawings "Tree", "Lemming" and "Binky" show that it is possible
    to draw some nice pictures with the mouse. Especially the drawing
    "Putzi" can demonstrate funny effects of most basic functions.
    I advice to test this drawing with exp(z), log(z), sqrt(z), sin(z),
    cos(z), tan(z/2), sinh(z), tanh(z), arsinh(z) u.a.

    Now, isn´t that funny ?


    4.2 Emulation of other plotter-types

    Because the real numbers are just a part of the complex numbers,
    it is possible to emulate some other types of function plotters.
    This enhances the usefulness of this program even more.


    4.2.1 "Classic" function plotter (Real --> Real)

    If you want to display a real function g(x) in the interval
    [a;b] with the usual method, you can do this in the following
    way:

    First you go to the definition editor and define this fishing
    net:

       Borders: Left: a  Right: b  Top: 0  Bottom: -1
       Sections: Horizontal: 640  Vertical: 0

    A horizontal line appears. Now you go to the main parameter
    window and type in the function:

          f(z) = x+i*g(x)

    Of course you have to replace g(x), a and b with appropriate
    terms (e.g. sin(x), -2*pi and 2*pi). And now you can plot the
    function by selecting "Plot" from the commands menu.

    If you use a smaller number of horizontal sections the computation
    time will be shorter. The result is often good enough if you don´t
    make this number smaller than 100.


    4.2.2 "3D-" function plotter (Real x Real --> Real)

    This kind of plotters produces those nice net graphics, in which
    the x- and y- coordinates are to be interpreted as function
    arguments, and the z-coordinate is to be interpreted as the
    function value. This can be emulated, too.

    For that purpose you must have a net in the definition area. A
    simple fishernet will do most times. Spiders webs with their
    center in the origin are good when using functions with rotation
    symmetry. Or you can use every other drawing.

    Then you define the following function:

          f(z) = x+y/2+i*(y/2+g(x,y))

    For g(x,y) you have to write the actual 3D-function, e.g.

          sin(x+y)
          1/(.5+x^2+y^2)
          exp(x+y)

    The only disadvantage is that the "hidden" lines will also be drawn.


    4.2.3 Function plotter Real --> Real x Real

    Relatively seldom are plotters which produce out of one real
    variable a two-dimensional coordinate. Most prominent examples
    are the so-called Lissajouz pictures which appear when the
    function has periodicy.

    If you want to display such functions with the ComplexPlotter
    you can use the following path:

    As in 4.2.1 you define a horizontal line. Then you input the
    function:

          f(z) = gx(x)+i*gy(x)

    For example produces

          f(z) = sin(x)+i*cos(1.2*x)

    a closed Lissajouz picture (when the line of the definition
    area lies in the right interval). Of couse you can also use
    non-periodical functions.




    I hope you have fun with my program or can use it for some
    other nice purpose. Enjoy it !



    Michael Gentner