\%      *****************************
\%      *  1D Stairs/Prev-Next      *
\%      *  for orbits of maps only  *
\%      *****************************

\% ********** 1D Stairs/Prev-Next **********
\topic kh_1DSWin {1D window for maps}
{\bf 1D stairs/prev-next windows} are used to represent
orbits of iterated maps as one-dimensional {\bf staircase}
or {\bf previous-next values} diagrams.
These windows may be opened for iterated maps only
and are available for manipulation only when the 
\jump kh_CurveType {curve type} is set to {\bf orbit}.\par
Let us consider k-th coordinates of two consequent points
${x\,}^{n+1}$ and ${x\,}^n$ of an orbit.
Then the {\bf stairs form} of representation consists
of two intervals $({x\,}^n_k,{x\,}^n_k)-({x\,}^n_k,{x\,}^{n+1}_k)$ and
$({x\,}^n_k,{x\,}^{n+1}_k)-({x\,}^{n+1}_k,{x\,}^{n+1}_k).$
The {\bf previous-next diagram} contains one point
$({x\,}^n_k,{x\,}^{n+1}_k).$ You can choose desired form of
representation of orbits as well as other graphic
attributes via the \jump kh_1DSWinLayout {layout dialog box.}
\par
You open the 1D stairs windows with the
\jump kh_menuWindow {Window|Open|1D Stairs} command from the
\jump kh_MainWin {Main window.}\par
There are the following components in the window.\par
{\desc 27
 {\bf Component\>What it does or is used for}\par
 {\bf Compute}\>\include kh_ComputeSubmenu \par
 {\sb 0 \li 1 \bf Forward}\>\include kh_ForwardItem \par
 {\sb 0 \li 1 \bf Extend}\>\include kh_ExtendItem \par
 {\bf Window}\>this menu contains commands which manipulate
  the window.\par
 {\sb 0 \li 1 \bf Redraw\+diagram}\>\include kh_RedrawDiagramItem \par
 {\sb 0 \li 1 \bf Redraw\+curve}\>\include kh_RedrawCurveItem \par
 {\sb 0 \li 1 \bf Clear}\>\include kh_ClearItem \par
 {\sb 0 \li 1 \bf Fit\+diagram}\>changes the
  {\bf limits of visibility} in such a way that all curves
  from the current diagram become totally visible.
  The command doesn't affect other windows.\par
 {\sb 0 \li 1 \bf Fit\+curve}\>changes the
  {\bf limits of visibility} in such a way that the
  current curve becomes totally visible.
  The command doesn't affect other windows.\par
 {\sb 0 \li 1 \bf Undo\+scale}\>restores the limits of visibility
  (four numbers defining the visible rectangle)
  for this window to their previous values. {CONTENT}
  remembers as much as last ten limits so you can go back
  more than one step (after ten steps the current limits
  are restored). {CONTENT} records the limits
  when you set them with the mouse (see below in this topic),
  or press the {\bf Enter} key or
  clear the window after editing them.
  \par
 {\sb 0 \li 1 \bf Layout}\>displays the
  \jump kh_1DSWinLayout {layout dialog box} where
  you change various settings which control the layout
  of the window.\par
 {\sb 0 \li 1 \bf Hardcopy}\>displays the
  \jump kh_1DSWinHardcopy {hardcopy dialog box} where you
  specify various option for the hardcopy of this window.\par
 {\sb 0 \li 1 \bf Duplicate}\>opens another 1D stairs window
  which has exactly the same attributes as the current window.
  This may be useful if you want to do some minor changes
  of attributes without restoring them back.\par
 {\sb 0 \li 1 \bf Hide}\>\include kh_HideItem \par
 {\bf Attributes}\>this menu allows you to change attributes
  used to draw and redraw curves.\par
 {\sb 0 \li 1 \bf Current}\>displays the
  \jump kh_1DSWinCurrentAttr {current graphic attributes} dialog box
  where you specify attributes which will be used to draw
  the curves to be computed next.
  \par
 {\sb 0 \li 1 \bf Curve}\>displays the
  \jump kh_1DSWinCurveAttr {curves graphic attributes} dialog box
  where you change attributes which are used to redraw
  the \jump kh_CurCurve {current curve.}\par
 {\bf Mouse}\>this menu allows you to set mouse related options.
  All items in it are disabled by default.\par
 {\sb 0 \li 1 \bf Pick\+up\+coordinates}\>when this option is set
  and you click somewhere in the {\bf Graphic area,}
  window coordinate of the point along the abscissa is converted
  to mathematical coordinate and replaces appropriate coordinate of
  the \jump kh_InitialPoint {initial point.} You can see new value
  of the coordinate in the \jump kh_StarterWin {starter window.}\par
 {\sb 0 \li 1 \bf Immediate\+forward}\>when this option is set
  and you click somewhere in the {\bf Graphic area,}
  {CONTENT} replaces coordinates of the initial
  point as described above. In addition it starts computation
  of new curve in the forward direction.
  This option can only be set if the
  {\bf Pick\+up\+coordinates} option is set.\par
 {\bf Help}\>\include kh_HelpSubmenu \par
 {\sb 0 \li 1 \bf This\+window}\>displays
  the description of 1D stairs/prev-next windows.\par
\% {\sb 0 \li 1 \bf Index}\>displays
\%  the contents of Help information.\par
 {\sb 0 \li 1 \bf Search}\>\include kh_HelpSearchItem \par
 {\bf Limits\+of\+visibility}\>is the area where you may specify
  minimum and maximum values for axes as well as
  the tick size. To change any value in the area click it with the
  mouse and then edit it. {CONTENT} will update the
  {\bf Graphics area} when you press the {\bf Enter} key
  or perform the {Clear} command from the {Window} menu.
  Any value may be given by an expression
  constructed from numbers, parentheses, operators
  (+, -, *, /, and ^), and standard function calls
  (abs, acos, asin, atan, ceil, cos, exp, floor, log, log10,
  sin, sqrt, tan), for example sqrt(2)/2.
  As you type CONTENT tries to parse the expression
  and compute its value. If it succeeds the value
  is used. Otherwise, the prompt which is left to the input field
  is grayed to indicate that the
  text does not form correct expression.
  Use the \jump kh_1DSWinLayout {layout dialog box}
  to specify which parts of the limits are present in the window.\par
 {\bf Palette}\>is a horizontal bar which shows all colors
  available for drawing curves.
  To choose a color click on it with the mouse.
  A small square at the right of the bar displays the
  currently chosen color.
  The {\bf layout dialog box} allows
  you to show or hide the palette.\par
 {\bf Graphic area}\>is the part of the window where {CONTENT}
  draws computed orbits. In addition you can ask {CONTENT} to
  draw axes, grid, scales, labels, and RHS in the area. Use the
  \jump kh_1DSWinLayout {layout dialog box} to make the
  appropriate settings.
  \par
}
In addition to the traditional way of changing the limits
of visibility (by typing them in) you can do this with the mouse.
Place the pointer at the point in the Graphic area which you want
to map to the upper left or bottom right corner of the area
and press the left mouse button. Holding the button drag
the pointer to the point you want to be mapped to the opposite corner
of the area and then release
the button. A thin frame is drawn as you drag the pointer.
Note that you have to move the pointer holding the button.
Press and immediate release the button without dragging the pointer
will change appropriate coordinate of the initial point instead.
The rectangle in the mathematical space you have specified maps onto
the whole Graphic area. To cancel the operation release
the button elsewhere outside the area. You can restore
previous limits of visibility later by the {\bf Undo\+scale}
command from the {\bf Window} menu.
\par
To destroy the window use the {\bf Close} command from
the {\bf System} menu of the window.\par
\end	\% kh_1DSWin

\% ********** Hardcopy dialog box **********
\topic kh_1DSWinHardcopy {hardcopy dialog box for 1D stairs window}
The {\bf hardcopy dialog box} is where you
specify various options for a hardcopy of a 1D stairs window.\par
You display the dialog box with the
\jump kh_1DSWin {Window|Hardcopy} command from a 1D stairs window.
A hardcopy is a {\bf PostScript} program which produces a copy
of the window's Graphic area with all the
\jump kh_Curve {curves} from the \jump kh_CurDiagram {current diagram}
drawn.
The hardcopy is always written into a file
and may optionally be sent to a printer.\par
There are following components in the dialog box.\par
#ifdef _UNIX
{\desc 19
#else	_UNIX
{\desc 17
#endif	_UNIX
 {\bf Component\>What it does or is used for}\par
 {\bf File}\>is where you specify the name of a file
  to which {CONTENT} will write the hardcopy.
  The file may be sent to a printer later
  (by means of appropriate {\bf copy} command.\par
 {\bf Browse}\>displays the list of available files
  which allows you to choose a file without typing
  its name. The name of the file you choose is put
  into the input field.\par
 {\bf Picture}\>is where you specify dimensions of a picture.\par
 {\li 1 \sb 0 \bf Width}\>specifies the width in current units.\par
 {\li 1 \sb 0 \bf Height}\>specifies the height in current units.\par
 {\li 1 \sb 0 \bf Unit}\>specifies the units for width and height.\par
 {\bf Font}\>contains two lists which you use to choose
  font name and its size. The font is used for all
  characters in the picture.\par
{\bf Options}\>allows you to specify some characteristics
  of the picture and where to send it. The values of three
  scale factors depend on the resolution of your printer.
  You may need to experiment to choose appropriate values for them.
  \par
 {\li 1 \sb 0 \bf Curve\+width}\>specifies the value of the
  curve width factor. It is used for all curves.\par
 {\li 1 \sb 0 \bf Line\+width}\>specifies the value of the
  line width factor. It is used for axes, grid, and frame
  of the picture.\par
 {\li 1 \sb 0 \bf Marker\+scale}\>specifies the value of the
  marker width factor. It is used to draw markers.\par
 {\li 1 \sb 0 \bf Preserve\+colors}\>when this box is checked
  {CONTENT} tries to produce a color picture by preserving
  colors set for Graphic area and curves.\par
#ifdef _UNIX
 {\li 1 \sb 0 \bf Show\+on\+screen}\>when this box is checked
  {CONTENT} shows the picture on the screen. The command
  for this is given by the value of the {\bf HCVIEWER} key which is
  in the {\bf [Visual]} section of the {\bf settings file.}\par
 {\li 1 \sb 0 \bf Send\+to\+printer}\>when this box is checked
  {CONTENT} sends the picture to a printer. The command
  for this is given by the value of the {\bf HCPRINTER} key which is
  in the {\bf [Visual]} section of the {\bf settings file.}\par
#endif _UNIX
 {\bf Buttons}\>\par
 {\li 1 \sb 0 \bf Ok}\>closes the dialog box and produces the
  hardcopy in accordance with the settings you have made.\par
 {\li 1 \sb 0 \bf Cancel}\>closes the dialog box without
  changing anything and producing the hardcopy.\par
 {\li 1 \sb 0 \bf Help}\>opens the \jump kh_HelpWin {help window} which
  displays the description of this dialog box.\par
}
\end	\% kh_1DSWinHardcopy

\% ********** Layout dialog box **********
\topic kh_1DSWinLayout {layout dialog box for 1D stairs window}
The {\bf layout dialog box} is where you specify
various settings which affects the layout of 1D stairs window.
You display the dialog box by the
\jump kh_1DSWin {Window|Layout} command in 1D stairs
window. Note that the settings affect only the window
from which you perform the command.\par
The following components are in the dialog box.\par
{\desc 23
 {\bf Component\>What it does or is used for}\par
 {\bf Window\+title}\>allows you to modify the title of the window.\par
 {\bf Limits\+of\+visibility}\>is where you specify limits
  of visibility for abscissa and ordinate.\par
 {\sb 0 \li 1 \bf Min}\>specifies minimum values.\par
 {\sb 0 \li 1 \bf Max}\>specifies maximum values.\par
 {\sb 0 \li 1 \bf Tick}\>specifies steps
  used for drawing scale along axes. 0 means you
  want {CONTENT} to calculate the value by itself
  using given minimum and maximum values.\par
 {\bf Show\+in\+window}\>contains check boxes
  which you use to specify which components will be shown
  in the 1D stairs window. Check appropriate boxes for
  the components you want to see in or to be able to change
  directly from the window\par
 {\sb 0 \li 1 \bf Min..Max}\>shows minimum and maximum values.\par
 {\sb 0 \li 1 \bf Tick}\>shows the tick values.\par
 {\sb 0 \li 1 \bf Scale}\>draws the scale along axes using
  the tick values.\par
 {\sb 0 \li 1 \bf Label}\>prints the name of a variable
  whose values are plotted. The labels are placed
  at upper left and bottom right corners of the Graphic area.\par
 {\sb 0 \li 1 \bf Grid}\>draws a grid, which is a set
  of horizontal and vertical dotted lines. The mathematical distances
  between lines are given by the tick values.\par
 {\sb 0 \li 1 \bf Axes}\>draws horizontal and vertical lines
  which correspond to the mathematical zero.
  Note that nothing will be drawn
  if mathematical zero is outside of the limits of visibility.\par
 {\sb 0 \li 1 \bf Palette}\>sets the mode in which
  a palette will be placed in the 1D stairs window to facilitate
  selection of colors for drawing curves.\par
 {\sb 0 \li 1 \bf Graph}\>draws the bisector.
  In addition if the {\bf Stairs} check box is checked
  (see the \jump kh_1DSWinCurrentAttr {current attributes}
  dialog box) {CONTENT} draws the graph of RHS with all values
  frozen except for the coordinate chosen for axes which
  varies from $Min$ to $Max.$\par
 {\bf Colors}\>is where you set options related to
  colors.\par
 {\sb 0 \li 1 \bf Foreground}\>opens the {\bf Standard Color dialog box}
  which you use to select the foreground color.
  The color will be used to draw axes, labels, the grid, the frame
  and the graph.\par
 {\sb 0 \li 1 \bf Background}\>opens the {\bf Standard Color dialog box}
  which you use to select the background color for the window.\par
 {\bf Sample\+of\+window}\>displays a sample of
  the window in accordance with the current settings.\par
 {\bf Buttons}\>\par
 {\sb 0 \li 1 \bf Ok}\>closes the dialog box and redraws
  the 1D stairs window using the settings you have made.\par
 {\sb 0 \li 1 \bf Cancel}\>closes the dialog box without
  changing the settings.\par
 {\sb 0 \li 1 \bf Help}\>opens the \jump kh_HelpWin {help window} which
  displays the description of this dialog box.\par
}
\end	\% kh_1DSWinLayout

\% ********** Current graphic attributes dialog box **********
\topic kh_1DSWinCurrentAttr {current attributes dialog box for 1D stairs window}
The {\bf current attributes dialog box} is where you
specify various graphic attributes of \jump kh_Curve {curves} which you
will compute next.\par
You display the dialog box by the
\jump kh_1DSWin {Attributes|Current} command in a 1D stairs window. 
Note that the settings affect only the window
from which you perform the command.\par
The following components are in the dialog box.\par
{\desc 17
{\bf Component\>What it does or is used for}\par
{\bf Axes}\>allows you to choose the coordinate
 whose values will be plotted in the window.\par
{\sb 0 \li 1 \bf List\+of\+names}\>contains the two-level list
 of names. The first level
 contains groups of names. Click the name of a group to
 see the names from the group. Click a name to choose it.
 To return to the first level of the list click the line which
 contains two dots.\par
{\sb 0 \li 1 \bf Name\+chosen}\>shows the currently chosen name.\par
{\bf Graphic\+attributes}\>allows you to set various
 graphic attributes.\par
{\sb 0 \li 1 \bf Set\+attributes\+of}\>allows you to choose
 graphic element for which you want to set attributes.
 Click one of the following: {\bf points} or
 \jump kh_Path {paths.}
 \par
{\sb 0 \li 1 \bf Join}\>allows you to choose the style of lines
 which join points. One of the following can be chosen:
 {\bf No,} {\bf Solid,} or {\bf Dotted.}\par
{\sb 0 \li 1 \bf Width}\>allows you to choose the width of points or lines.
 If this value is zero, the width is one pixel and fast but possibly
 not accurate device-dependent drawing algorithm will be used.
 \par
{\sb 0 \li 1 \bf Change\+Color}\>opens the {\bf Standard Color Dialog Box}
 which you use to select the color for points or lines.\par
{\bf Sample\+of\+window}\>displays a sample of
 a curve in accordance with the current settings.\par
{\bf Options}\>allows you to choose the form of representation
 of orbits.
 See the table at the end of this topic which describes
 how these options are combined with the {\bf Join} option.\par
{\sb 0 \li 1 \bf Stairs}\>tells {CONTENT} to represent orbits
 as 1D stairs.\par
{\sb 0 \li 1 \bf Prev-next}\>tells {CONTENT} to represent orbits
 as Previous-next values diagram.\par
{\bf Buttons}\>\par
{\sb 0 \li 1 \bf Ok}\>remembers the graphic attributes you have set
 and closes the dialog box.\par
{\sb 0 \li 1 \bf Cancel}\>closes the dialog box without
 changing the settings.\par
{\sb 0 \li 1 \bf Help}\>opens the \jump kh_HelpWin {help window} which
 displays the description of this dialog box.\par
}
The following table describes representation of orbits
depending on the form (Stairs or Prev-Next) and join style
(No or Solid/Dotted).
$x^n$ and $x^{n+1}$ are chosen coordinates of two consequent
points of an orbit.
$(a,b)$ denotes the point with coordinates $a$ and $b$,\
$(a,b)-(c,d)-\ldots$ denoted polyline joining $(a,b)$ with $(c,d)$ etc.
\par
{\li 3
$
\table 3 lll 1 1
 { }         {No}           {Solid/Dotted}
 {Stairs}    {({x\,}^n,{x\,}^n),({x\,}^{n+1},{x\,}^{n+1})} {({x\,}^n,{x\,}^{n+1})-({x\,}^n,{x\,}^{n+1})-({x\,}^{n+1},{x\,}^{n+1})}
 {Prev-next} {({x\,}^n,{x\,}^{n+1})} {({x\,}^n,{x\,}^{n+1})}
$
}
\end	\% kh_1DSWinCurrentAttr

\% ********** Curve's graphics attributes dialog box **********
\topic kh_1DSWinCurveAttr {curve attributes dialog box for 1D stairs window}
The {\bf curve's attributes dialog box} is where you
change various graphic attributes of the
\jump kh_CurCurve {current orbit.}\par
You display the dialog box by the
\jump kh_1DWin {Attributes|Curve} command in a 1D stairs
window. Note that the settings affect only the window
from which you perform the command.\par
The following are components in the dialog box.\par
{\desc 11
{\bf Component\>What it does or is used for}\par
{\bf Graphic\+attributes}\>allows you to change graphic attributes.
 \par
{\sb 0 \li 1 \bf Set\+attributes\+of}\>allows you to choose
 graphic element for which you want to change attributes.
 Click one of the following: {\bf points} or \jump kh_Path {paths.}
 \par
{\sb 0 \li 1 \bf Join}\>allows you to change the style of lines
 which join points. One of the following can be chosen:
 {\bf No,} {\bf Solid,} or {\bf Dotted.}\par
{\sb 0 \li 1 \bf Width}\>allows you to change the width of points or lines.
 If this value is zero, the width is one pixel and fast but possibly
 not accurate device-dependent drawing algorithm will be used.
 \par
{\sb 0 \li 1 \bf Change\+Color}\>opens the {\bf Standard Color Dialog Box}
 which you use to change the color of points or lines.\par
{\bf Sample\+of\+window}\>displays a sample of
 the current orbit in accordance with the settings.\par
{\bf Buttons}\>\par
{\sb 0 \li 1 \bf Ok}\>remembers the changes you have made
 and closes the dialog box.\par
{\sb 0 \li 1 \bf Cancel}\>closes the dialog box without
 changing the attributes.\par
{\sb 0 \li 1 \bf Help}\>opens the \jump kh_HelpWin {help window} which
 displays the description of this dialog box.\par
}
\end	\% kh_1DSWinCurveAttr