!  When z = 0,  the yellow line should vanish altogether from both
!  windows. 
!  
!  When the a slider is in use, a dark blue line should appear on
!  the left screen, the line  Im z = b - as it does. The image of
!  this line is a ray from the origin in the e^z plane, not a line
!  through the origin as drawn. This is my oversight, faithfully
!  reproduced by you and then Ashot.

! I keep wondering about what symbols or
! pictures are needed to actually understand what is going
! on in this one. It creates a mystery and doesn't explain
! itself.

! Maybe it takes a preliminary picture to show how the
! conversion to polar coordinates happens, and maybe another
! picture that shows an application.

! given a complex value a+ib and its exponential e^(a+ib)
!   e^(a+ib) = e^a * e^ib
! but Euler showed that e^ib = (cos(b) + i sin(b) so
!   e^(a+ib) = e^a * (cos(b) + i sin(b))
! now distribute e^a
!   e^(a+ib) = e^a * cos(b) + e^a * i sin(b)
! e^a * cos(b) + e^a * i sin(b) is a complex value so
! we can separate the real and imaginary parts of e^(a+ib)
!   re = e^a * cos(b)
!   im = e^a * i sin(b)
! but graphically, i just means "on the complex vertical axis" so
!   re = e^a * cos(b)    ! on x
!   im = e^a * sin(b)    ! on y
! so e^a is the radius, and b is the angle, for the
! point that represents e^(a+ib) on the complex plane

! we don't help students understand this in the pictures
! or the symbols. I was trying to find a way. I have a version
! with symbolic steps, but I know you don't like showing the
! symbols, so I was trying to find graphics to suggest that I
! am using
!    x = e^a * cos(b)
!    y = e^a * sin(b)
! behind the scenes. Will they automatically know how to use
! Euler's formula to understand how to get to the polar coordinates?
! I thought that if a remedial tool is actually needed at this
! level, then they might not know about Euler's Formula.


!! File: ComplexExponential.68k
!! Jume 27, 2003 Hubert Hohn

PUBLIC PCFlag,Mac5Flag,M68KFlag,UnixFlag,xmax,ymax
PUBLIC toolLft,toolRgt,toolBas,toolTop,toolhdr,toolHgt,toolWid  ! tool boundaries
PUBLIC winLft,winRgt,winBas,winTop,winHgt,winWid      ! window
PUBLIC workLft,workRgt,workBas,workTop,workMidx  ! work area
PUBLIC backclr,black,drkgry,drkmid,midgry,litmid,litgry,white
PUBLIC red,yellow,green,cyan,blue,magenta,pink,colorscheme
PUBLIC planeclr,gridclr,rimclr,axisclr,axislabelclr,titleclr,rightsclr
PUBLIC numberlineclr,slotdrkclr,slotlgtclr,slideclr
PUBLIC largefonts, title$, SLUmode

LET toolHgt= 560
LET toolWid= 780
LET window$= "The d'Arbeloff Interactive Math Project"
LET colorscheme= 0

LET title$ = "Complex Exponentials"

SUB ThisProgram
    CALL ComplexExponential
    CLEAR
END SUB


!! ---------------------------------------------------------
!! ------ Start TB4 Mac Header and Subs ------

!LET M68KFlag = 1
!LIBRARY "MacTools*", "HHLib.trc"
!ASK PIXELS winWid,winHgt
!LET winLft= 0
!LET winTop= 0
!LET winRgt= winWid-1
!LET winBas= winHgt-1
!SET WINDOW 0,winRgt,winBas,0
!CALL Palette
!CLEAR
!
!CALL ToolPanel
!CALL ThisProgram
!
!END
!EXTERNAL
!
!MODULE Mac4Parts
!    SUB SetTextFont(font,size,style$)
!        CALL MacTextFont(font)
!        CALL MacTextSize(size)
!        CALL MacTextFace(style$)
!    END SUB
!
!    SUB StringWidth(sw$,sl)
!        DECLARE DEF MacStringWidth
!        LET sl= MacStringWidth(sw$)
!    END SUB
!
!    SUB SetLineWeight(wgt)
!        CALL MacPenSize(wgt,wgt)
!    END SUB
!
!    SUB BoxDisk(Lft,Rgt,Bas,Top)
!        CALL MacPaintOval(Lft,Rgt,Bas,Top)
!    END SUB
!END MODULE

!! --- End TB4 Mac Header and Subs ---
!!---------------------------------------------------------
!!--- Start TB5 Cross-Platform header and subs ---

LIBRARY "c:\TB Gold 51a\TBLibs\TrueCtrl.trc"     ! windows
LIBRARY "c:\TB Gold 51a\TBLibs\HHLib.trc"

!LIBRARY ":TBLibs:TrueCtrl.trc"                   ! macintosh
!LIBRARY "HHLib.trc"

PUBLIC WinID
DECLARE PUBLIC OBJM_SET,OBJM_SYSINFO
LET winHgt= toolHgt
LET winWid= toolWid
DIM values(1)

CALL TC_Init

CALL Object(OBJM_SYSINFO,WinID,"MACHINE",system$,values())
IF system$="MAC" then
   LET Mac5Flag= 1
ELSE IF system$="WIN32" then
   LET PCFlag  = 1
END IF

CALL TC_SetUnitsToPixels                         ! 5.1 and up needs this
CALL TC_GetScreenSize(scrnLft,scrnRgt,scrnBas,scrnTop)
LET winLft= int((scrnRgt-scrnLft-winWid)/2)
LET winRgt= winLft+winWid-1
LET winTop= int((scrnBas-scrnTop-winHgt)/2) + 10
LET winBas= winTop+winHgt-1

CALL TC_Win_Create (WinID,"TITLE",winLft,winRgt,winBas,winTop)
LET values(1)= 2
CALL Object(OBJM_SET, WinID, "TYPE", "", values())

IF PCFlag=1 then                                 ! kill dithering
   LET  values(1)= 1
   CALL Object(OBJM_SET, WinID, "SOLID MIX", "", values())
END IF

LET  values(1)= 0
CALL TC_SetRect(WinID,winLft,winRgt,winBas,winTop)
CALL TC_Win_SetTitle(WinID,window$)
CALL TC_Show(WinID)

SET MODE "COLORSTANDARD"

ASK PIXELS winWid,winHgt                         ! must follow set mode
LET winLft= 0
LET winTop= 0
LET winRgt= winWid-1
LET winBas= winHgt-1
SET WINDOW 0,winRgt,winBas,0

CALL Palette

IF PCFlag=1 then
   LET  values(1)= 0                             ! now force solid colors
   CALL Object(OBJM_SET, WinID, "SOLID MIX", "", values())
   CALL TC_Win_RealizePalette(WinID)             ! some PCs need this

   CALL TC_Win_SetFont(WinID,"arial",9,"plain")
   CALL StringWidth("0",sw)
   IF sw>7 then LET largefonts=1 else LET largefonts=0
END IF

CALL TC_Win_Switch(WinID)

CALL ToolPanel
CALL ThisProgram

CALL SetTextFont(1,12,"bold")                    ! now shut down and clean up
LET quit$= "click the mouse or press a key to close..."
CALL PlotTextCJ(workmidx,(workbas+worktop)/2,quit$,yellow)
CALL TC_CleanUp

END
EXTERNAL

MODULE TB5Parts
    SUB StringWidth(sw$,sl)
        DECLARE PUBLIC WinID
        LET sl= StrWidth(WinID,sw$)
    END SUB

    SUB SetLineWeight(wgt)
        DECLARE PUBLIC OBJM_SET
        DECLARE PUBLIC WinID
        DIM values(1)
        LET values(1)= wgt
        CALL Object(OBJM_SET,WinID, "WIDTH", "", values())
    END SUB

    SUB SetTextFont(font,size,style$)
        DECLARE PUBLIC WinID,Mac5Flag,PCFlag,largefonts
        IF Mac5Flag=1 then
           SELECT CASE font
           CASE 4
                LET font$= "Courier"
           CASE 16
                LET font$= "Times"
           CASE else
                LET font$= "Geneva"
           END SELECT
        ELSE IF PCFlag=1 then
           IF largefonts=1 then
              IF size<12 then
                 LET size=  6
              ELSE IF size=14 then
                 LET size= 10
              ELSE IF size=18 then
                 LET size= 12
              ELSE IF size=24 then
                 LET size= 14
              ELSE IF size=12 then
                 LET size=  8
              ELSE
                 LET size= round(72/96 * size * .8)
              END IF
           ELSE
              IF size<12 then
                 LET size=  7
              ELSE IF size=14 then
                 LET size= 12
              ELSE IF size=12 then
                 LET size=  9
              ELSE IF size=18 then
                 LET size= 14
              ELSE IF size=24 then
                 LET size= 18
              ELSE
                 LET size= round(72/96 * size)
              END IF
           END IF
           SELECT CASE font
           CASE 4
                LET font$= "Courier New"
           CASE 16
                LET font$= "Times New Roman"
           CASE else
                LET font$= "Verdana"
           END SELECT
        END IF
        IF style$= "normal" then LET style$= "plain"
        CALL TC_Win_SetFont(WinID,font$,size,style$)
    END SUB

    SUB BoxDisk(Lft,Rgt,Bas,Top)
        BOX DISK Lft,Rgt,Bas,Top
    END SUB
END MODULE

!  --- End TB5 Cross-platform header and subs ---
!! ---------------------------------------------------------

!! --- Start Unix Header and Subs ---

!library "HHLib.unix"
!LET UnixFlag= 1
!ASK PIXELS winWid,winHgt
!LET winLft= 0
!LET winTop= 0
!LET winRgt= winWid-1
!LET winBas= winHgt-1
!SET WINDOW 0,winRgt,winBas,0
!CALL Palette
!
!CALL ToolPanel
!CALL ThisProgram
!
!END
!EXTERNAL
!
!MODULE UnixParts
!    SHARE CharWidth
!
!    SUB SetTextFont(font,size,style$)
!        LET font$= "-adobe-courier-"
!        IF style$= "normal" then
!           LET style$= "medium-r-normal--"
!        ELSE
!           LET style$= "bold-r-normal--"
!        END IF
!        IF size=9 then
!           LET size$= str$(10)
!        ELSE
!           LET size$= str$(size)
!        END IF
!        LET test= SetFont(font$&style$&size$&"*")
!
!        IF size=9 then
!           LET CharWidth=  6
!        ELSE IF size=12 then  ! numeric output - axis labels
!           LET CharWidth=  7
!        ELSE IF size=14 then  ! rare
!           LET CharWidth=  8
!        ELSE IF size=18 then  ! rare
!           LET CharWidth= 10
!        END IF
!    END SUB
!
!    SUB StringWidth(sw$,sl)                      ! string width in pixels
!        ! LET sl= StrWidth(sw$)
!        LET chars= len(sw$)
!        LET sl   = chars*CharWidth
!    END SUB
!
!    SUB SetLineWeight(wgt)
!        ! CALL PenSize(wgt,wgt)
!    END SUB
!
!    SUB BoxDisk(Lft,Rgt,Bas,Top)
!        CALL Fill_Circle(Lft,Rgt,Bas,Top)
!    END SUB
!END MODULE

!! ------ End of TB Unix Header and Subs ------
! -----------------------------------------------------------


! ***

SUB ComplexExponential
    DECLARE PUBLIC worklft,workrgt,workbas,worktop,workmid      ! work area
    DECLARE PUBLIC black,drkgry,drkmid,midgry,litmid,litgry,white
    DECLARE PUBLIC red,yellow,green,cyan,blue,magenta,colorscheme
    DECLARE PUBLIC slideclr,axislabelclr,true,false

    DECLARE DEF InfoWithin, QuitWithin

    ! --- help screen array ---

    DIM  info$(1:20)
    MAT READ info$

    DATA "Complex Exponentials"
    DATA ""
    DATA "This tool supports visual exploration of the relationships between complex values on the z plane and their corresponding complex exponentials on the e^[z] plane."
    DATA ""
    DATA "Given a complex value  a+ib  and its exponential  e^[a+ib]"

    DATA "-e^[a+ib] = e^[a] e^[ib]"
    DATA "Euler showed that  e^[ib] = cos(b) + i sin(b)  so"
    DATA "-e^[a+ib] = e^[a] (cos(b) + i sin(b))"
    DATA "-e^[a+ib] = e^[a] cos(b) + e^[a] i sin(b)"
    DATA "To plot the point, we separate the real and imaginary parts"

    DATA "-re = e^[a] cos(b)"
    DATA "-im = e^[a] i sin(b)"
    DATA "and this is a conversion from polar to cartesian coordinates, where"
    DATA "e^[a] is the radius and b is the angle."
    DATA "When graphing, the i refers to the imaginary vertical axis, so we plot"

    DATA "-re = e^[a] cos(b)  on the horizontal axis"
    DATA "-im = e^[a] sin(b)  on the vertical axis"
    DATA ""
    DATA "To set the input value, click or drag the mouse in the z plane, or use the a and b sliders."
    DATA "Drag the t slider to see corresponding points on the two planes."


    ! ---------- Utility functions ---

    DECLARE DEF clamp,roundn,e


    ! ---- Functions ----

    DEF cs(a,b,t) = e^(a*t) * cos(b*t)
    DEF sn(a,b,t) = e^(a*t) * sin(b*t)

    SUB ComplexExpVal(a,b,t,x,y)
        LET rad= e^(a*t)
        LET bt = b*t
        LET x  = rad * cos(bt)
        LET y  = rad * sin(bt)
    END SUB


    ! ---------- Graphing plane parameters and methods ----------

    LET wsize = 288

    ! --- w1 plane data ---

    DECLARE PUBLIC w1Lft,w1Rgt,w1Bas,w1Top,w1Midx,w1Midy
    DECLARE PUBLIC w1fLft,w1fRgt,w1fBas,w1fTop,w1x0,w1y0
    DECLARE PUBLIC w1xFirst, w1xSTik, w1xLTik, w1xLabel, w1xGridstep
    DECLARE PUBLIC w1yFirst, w1ySTik, w1yLTik, w1yLabel, w1yGridstep
    DECLARE PUBLIC w1wWid,w1wHgt,w1fWid,w1fHgt
    DECLARE PUBLIC w1fxRatio,w1fyRatio,w1wxRatio,w1wyRatio,w1Aspect
    DECLARE PUBLIC w1xPiFlag, w1xMult, w1yPiFlag, w1yMult

    LET w1Flag   = 1
    LET w1xPiFlag= 0
    LET w1xMult  = 1
    LET w1yPiFlag= 0
    LET w1yMult  = 1

    LET w1Lft = workrgt - 340                    ! pixel bounds
    LET w1Rgt = w1Lft + wsize
    LET w1Top = workTop + 55
    LET w1Bas = w1Top + wsize

    LET fsize =  3
    LET w1fLft= -fsize                           ! function bounds
    LET w1fRgt=  fsize
    LET w1fTop=  fsize
    LET w1fBas= -fsize

    LET w1xAx$= "Re"                             ! axis labels
    LET w1yAx$= "Im"

    LET w1xGridstep= 1                           ! horizontal grid intervals
    LET w1yGridstep= 1                           ! vertical grid intervals

    LET w1xSTik = 0                              ! horizontal axis Tik marks
    LET w1xLTik = 1
    LET w1xLabel= 1
    LET w1xFirst= w1fLft

    LET w1ySTik = 0                              ! vertical axis Tik marks
    LET w1yLTik = 1
    LET w1yLabel= 1
    LET w1yFirst= w1fBas

    ! --- w1 Plane methods ---

    DECLARE DEF w1fncx,w1fncy,w1wndx,w1wndy      ! window/function transforms
    DECLARE DEF w1wWithin,w1Within               !,w1fWithin

    CALL w1Variables
    LET w1x1 = w1Wndx(1)
    LET w1pix= w1wxratio

    SUB w1Init
        CALL w1DrawPlane(1,1,1)                  ! grid, axes, zeroaxes
        CALL SetTextFont(1,12,"bold")
        CALL PlotTextLJ(w1Rgt+8,w1y0+3,w1xAx$,axislabelclr)     ! axis labels
        CALL PlotTextCJ(w1x0,w1Top-10,w1yAx$,axislabelclr)
        CALL PlotDiamondClr(w1x1,w1y0,red)
        CALL w1KeepGridLayer
        CALL SuperSubScriptCJ(w1Midx,w1Top-30,"e^[z] plane",white)
    END SUB


    ! --- w2 plane data ---

    DECLARE PUBLIC w2Lft,w2Rgt,w2Bas,w2Top,w2Midx,w2Midy
    DECLARE PUBLIC w2fLft,w2fRgt,w2fBas,w2fTop,w2x0,w2y0
    DECLARE PUBLIC w2xFirst, w2xSTik, w2xLTik, w2xLabel, w2xGridstep
    DECLARE PUBLIC w2yFirst, w2ySTik, w2yLTik, w2yLabel, w2yGridstep
    DECLARE PUBLIC w2wWid,w2wHgt,w2fWid,w2fHgt
    DECLARE PUBLIC w2fxRatio,w2fyRatio,w2wxRatio,w2wyRatio,w2Aspect
    DECLARE PUBLIC w2xPiFlag, w2xMult, w2yPiFlag, w2yMult

    LET w2Flag   = 1
    LET w2xPiFlag= 0
    LET w2xMult  = 1
    LET w2yPiFlag= 0
    LET w2yMult  = 1

    LET w2Lft = worklft + 50                     ! pixel bounds
    LET w2Rgt = w2Lft + wsize
    LET w2Top = w1Top
    LET w2Bas = w2Top + wsize

    LET w2fLft= -1                               ! function bounds
    LET w2fRgt=  1
    LET w2fTop=  8
    LET w2fBas= -8

    LET w2xAx$= "a"                              ! axis labels
    LET w2yAx$= "b"

    LET w2xGridstep= 0.5                         ! grid line intervals
    LET w2yGridstep= 4

    LET w2xSTik = 0                              ! axis Tik marks
    LET w2xLTik = 0.5
    LET w2xLabel= 0.5
    LET w2xFirst= w2fLft

    LET w2ySTik = 0
    LET w2yLTik = 4
    LET w2yLabel= 4
    LET w2yFirst= w2fBas

    ! --- w2 plane methods ---

    DECLARE DEF w2fncx,w2fncy,w2wndx,w2wndy      ! window/function transforms
    DECLARE DEF w2wWithin,w2Within

    CALL w2Variables

    SUB w2Init
        CALL w2DrawPlane(1,1,1)                  ! grid, axes, zeroaxes
        CALL SetTextFont(1,12,"bold")
        CALL PlotTextLJ(w2Rgt+8,w2y0+3,w2xAx$,axislabelclr)     ! axis labels
        CALL PlotTextCJ(w2x0,w2Top-10,w2yAx$,axislabelclr)
        CALL PlotTextCJ(w2Midx,w2Top-30,"z plane",white)
        CALL PlotDiamondClr(w2x0,w2y0,red)
        CALL w2KeepGridLayer
    END SUB

    ! --- horizontal sliders ---

    ! --- h1 slider - a value ---

    DECLARE PUBLIC h1axis,h1wLft,h1wRgt,h1wBas,h1wTop,h1fLft,h1fRgt
    DECLARE PUBLIC h1name$,h1form$,h1clr,h1First,h1STik,h1LTik,h1Label
    DECLARE PUBLIC h1PiAxis,h1Mult,h1fMin,h1fMax

    LET h1PiAxis= 0
    LET h1Mult  = 1

    LET h1clr   = slideclr
    LET h1name$ = "a"
    LET h1form$ = "-%.##"
    LET h1Places= 2
    LET h1Click = 0.1

    LET h1axis  = workbas - 75
    LET h1wLft  = w2Lft
    LET h1wRgt  = h1wLft + wsize

    LET h1fLft  = -1
    LET h1fRgt  =  1

    LET h1STik  = .1
    LET h1LTik  = .5
    LET h1Label = .5
    LET h1First = h1fLft

    CALL h1SliderVariables

    ! --- h1 slider methods ---

    DECLARE DEF h1Within

    SUB h1Init
        CALL h1DrawSlider(h1name$,a)
    END SUB


    ! --- h2 slider - b value ---

    DECLARE PUBLIC h2axis,h2wLft,h2wRgt,h2wBas,h2wTop,h2fLft,h2fRgt
    DECLARE PUBLIC h2name$,h2form$,h2clr,h2First,h2STik,h2LTik,h2Label
    DECLARE PUBLIC h2PiAxis,h2Mult,h2fMin,h2fMax

    LET h2PiAxis= 0
    LET h2Mult  = 1

    LET h2clr   = slideclr
    LET h2name$ = "b"
    LET h2form$ = "-%.##"
    LET h2Places=  2
    LET h2Click =  pi/4

    LET h2axis  = h1axis + 45
    LET h2wLft  = h1wLft
    LET h2wRgt  = h1wRgt

    LET h2fLft  = -8
    LET h2fRgt  =  8

    LET h2STik  =  1
    LET h2LTik  =  4
    LET h2Label =  4
    LET h2First = h2fLft

    CALL h2SliderVariables

    DECLARE DEF h2Within                         ! window/function transforms

    SUB h2Init
        CALL h2DrawSlider(h2name$,b)
    END SUB

    ! --- h3 slider - t parameter ---

    DECLARE PUBLIC h3axis,h3wLft,h3wRgt,h3wBas,h3wTop,h3fLft,h3fRgt
    DECLARE PUBLIC h3name$,h3form$,h3clr,h3First,h3STik,h3LTik,h3Label
    DECLARE PUBLIC h3PiAxis,h3Mult,h3fMin,h3fMax

    LET h3PiAxis= 0
    LET h3Mult  = 1

    LET h3clr   = yellow                         !slideclr
    LET h3name$ = "t"
    LET h3form$ = "-%.##"
    LET h3Places= 2
    LET h3Click = 0.1

    LET h3axis  = h1axis
    LET h3wLft  = w1Midx
    LET h3wRgt  = h3wLft + 100

    LET h3fLft  = 0
    LET h3fRgt  = 1

    LET h3STik  = 0.1
    LET h3LTik  = 0.5
    LET h3Label = 0.5
    LET h3First = h3fLft

    CALL h3SliderVariables

    DECLARE DEF h3Within                         ! window/function transforms

    SUB h3Init
        CALL h3DrawSlider(h3name$,t)
    END SUB

    ! ---------- Text Output Rects ----------

    ! --- text rectangle 1 - complex expressions ---

    LET t1BasLn1= w1Bas + 35
    LET t1BasLn2= t1BasLn1 + 25
    LET t1Lft   = w1Lft
    LET t1Rgt   = w1Rgt - 80
    LET t1Bas   = t1BasLn1 +  5
    LET t1Top   = t1BasLn1 - 15

    LET t1Label$= "e^[a + bi] = "

    CALL SetTextFont(1,12,"bold")
    CALL StringWidth("e = ",sw1)
    CALL SetTextFont(1, 9,"bold")
    CALL StringWidth("a + bi",sw2)
    LET t1Eqx= t1Lft + sw1 + sw2

    SUB t1Label
        CALL SuperSubScriptRJ(t1Eqx,t1BasLn1,t1Label$,cyan)
        CALL SuperSubScriptLJ(t1Lft,t1BasLn2,"e^[(a + bi)t]",yellow)
    END SUB

    SUB t1Value(x,y)
        CALL t1Clear
        CALL CmplxStringTrim(cmplx$,x,y)         ! Numeric to text
        CALL SuperSubScriptLJ(t1Eqx,t1BasLn1,cmplx$,cyan)
    END SUB

    SUB t1Clear
        BOX CLEAR t1Eqx,t1Rgt,t1Bas,t1Top
    END SUB

    SUB t1Init
        CALL t1Label
    END SUB


    ! --- text rectangle 2 - complex time expressions ---

    LET t2BasLn1= t1BasLn1
    LET t2BasLn2= t1BasLn2
    LET t2Lft   = w2Lft
    LET t2Rgt   = w2Rgt - 80
    LET t2Bas   = t2BasLn1 +  5
    LET t2Top   = t2BasLn1 - 15
    LET t2Label$= "a + bi = "
    CALL SetTextFont(1,12,"bold")
    CALL StringWidth(t2Label$,sw)
    LET t2eqx   = t2Lft + sw

    SUB t2Label
        CALL PlotTextRJ(t2Eqx,t2BasLn1,t2Label$,cyan)
        CALL PlotTextLJ(t2Lft,t1BasLn2,"(a + bi)t",yellow)
    END SUB

    SUB t2Value(x,y)
        CALL t2Clear
        CALL CmplxStringTrim(cmplx$,a,b)         ! Numeric to text
        CALL PlotTextLJ(t2eqx,t2BasLn1,cmplx$,cyan)
    END SUB

    SUB t2Clear
        BOX CLEAR t2Eqx,t2Rgt,t2Bas,t2Top
    END SUB

    SUB t2Init
        CALL t2Label
    END SUB


    ! --- t3 equation steps (not used in Mit version) ---

    LET t3BasLn = w1Bas + 90
    LET t3LnSpc = 18
    LET t3Lft   = w1Lft
    LET t3Rgt   = w1Rgt - 60
    LET t3Bas   = t3BasLn +  4*t3LnSpc + 5
    LET t3Top   = t3BasLn - 15
    LET t3Label$= "e^[a + bi] = "
    LET t3Eqx   = t1Lft + 70

    LET t3bLft= w1Rgt-55
    LET t3bRgt= t3bLft+55
    LET t3bBas= t3Bas
    LET t3bTop= t3bBas-20

    SUB t3Clear
        BOX CLEAR t3lft,t3Rgt,t3Bas,t3Top
    END SUB

    SUB t3Button
        CALL SetTextFont(1,12,"bold")
        CALL DrawButton(t3bLft,t3bRgt,t3bBas,t3bTop,6,"Steps")
    END SUB

    SUB t3Set
        ! e^(a+bi) = e^a e^bi
        CALL SuperSubScriptRJ(t3Eqx,t3BasLn+0*t3LnSpc,t3Label$,cyan)
        CALL SuperSubScriptLJ(t3Eqx,t3BasLn+0*t3LnSpc,"e^[a] e^[ib]",cyan)

        ! e^(a+ib) = e^a (cos(b) + i sin(b))        ! euler's formula
        CALL SuperSubScriptRJ(t3Eqx,t3BasLn+1*t3LnSpc,"= ",cyan)
        CALL SuperSubScriptLJ(t3Eqx,t3BasLn+1*t3LnSpc,"e^[a] (cos(b) + i sin(b))",cyan)

        ! e^(a+ib) = e^a cos(b) + i e^a sin(b)
        CALL SuperSubScriptRJ(t3Eqx,t3BasLn+2*t3LnSpc,"= ",cyan)
        CALL SuperSubScriptLJ(t3Eqx,t3BasLn+2*t3LnSpc,"e^[a] cos(b) + i e^[a] sin(b)",cyan)

        !       re = e^a * cos(b)
        CALL SuperSubScriptRJ(t3Eqx,t3BasLn+3*t3LnSpc,"Re = ",cyan)
        CALL SuperSubScriptLJ(t3Eqx,t3BasLn+3*t3LnSpc,"e^[a] cos(b)",cyan)

        !       im = i e^a * i sin(b)
        CALL SuperSubScriptRJ(t3Eqx,t3BasLn+4*t3LnSpc,"Im = ",cyan)
        CALL SuperSubScriptLJ(t3Eqx,t3BasLn+4*t3LnSpc,"i e^[a] sin (b)",cyan)
    END SUB


    ! --- default parameters ---

    LET maxa   = w2fRgt
    LET maxb   = w2fTop
    LET a,olda = 0.5
    LET b,oldb = pi
    LET t,oldt = 0
    LET t3State= 0


    ! --- draw the screen ---

    CALL InitScreen

    SUB InitScreen
        BOX CLEAR worklft,workrgt,workbas,worktop

        CALL w1Init
        CALL w2Init

        CALL t1Init
        CALL t2Init

        CALL h1Init
        CALL h2Init
        CALL h3Init

        CALL DrawingVariables(a,b,x1,y1,x2,y2,wx1,wy1,wx2,wy2,w1c1,w1s1)
        CALL DrawGraphs(yellow)
    END SUB


    ! ----------------- Event manager -----------------

    DO
       DO
          GET MOUSE: mx,my,ms
       LOOP until ms=2

       IF w2Within(mx,my)=true then
          CALL h3ResetTime
          CALL w2MouseEvent(mx,my)

       ELSE IF h1Within(mx,my)=true then
          CALL h3ResetTime
          IF my<h1axis then
             CALL h1ClickEvent(ms)
          ELSE
             CALL h1DragEvent(ms)
          END IF
       ELSE IF h2Within(mx,my)=true then
          CALL h3ResetTime
          IF my<h2axis then
             CALL h2ClickEvent(ms)
          ELSE
             CALL h2DragEvent(ms)
          END IF
       ELSE IF h3Within(mx,my)=true then
          IF my<h3axis then
             CALL h3ClickEvent(ms)
          ELSE
             CALL h3DragEvent(ms)
          END IF

          !       not used in MIT version
          !       ELSE IF mx>t3bLft and mx<t3bRgt and my>t3bTop and my<t3bBas then
          !          CALL MouseButtonUp(t3bLft,t3bRgt,t3bBas,t3bTop,ms)
          !          LET t3State= 1 - t3State
          !          IF t3State=1 then
          !             CALL t3Set
          !          ELSE
          !             CALL t3Clear
          !          END IF

       ELSE IF infoWithin(mx,my,ms)=true then
          CALL infoButtonUp(ms)
          CALL InfoPage(Info$)
          CALL InitScreen
       ELSE IF quitWithin(mx,my,ms)=true then
          CALL quitButtonUp(ms)
          EXIT SUB
       ELSE
          CALL MouseUp(mx,my,ms)
       END IF
    LOOP

    ! --- Window Event Methods ---

    SUB w2MouseEvent(mx,my)
        LET olda,oldb= -999
        DO
           CALL w2GetDragValues(mx,my,ms,2,2,a,b)
           IF a<>olda or b<>oldb then
              CALL h1Mark(a)
              CALL h2Mark(b)
              CALL w2MouseAction
              CALL SetVars(a,b,olda,oldb)
           END IF
        LOOP until ms=3
        CALL DrawGraphs(yellow)
    END SUB

    SUB w2MouseAction
        LOCAL x,y
        ! sketch line
        CALL w2ShowGridLayer
        CALL DrawingVariables(a,b,x1,y1,x2,y2,wx1,wy1,wx2,wy2,w1c1,w1s1)
        CALL PlotLinePoint(w2x0,w2y0,w2Wa,w2Wb,yellow,cyan)
        ! sketch exponential spiral
        CALL w1ShowGridLayer
        CALL FastSpiral
    END SUB

    ! ---- Slider Event Methods ----
    ! ----- slider 1 - a value -----

    SUB h1ClickEvent(ms)
        CALL h1GetClickVal(ms,h1Click,a)
        CALL DrawGraphs(yellow)
        LET olda= a
    END SUB
    SUB h1DragEvent(ms)
        LET olda,oldb= -999
        CALL w1ShowGridLayer
        CALL w2ShowGridLayer
        LET wy= w2Wndy(b)
        CALL PlotLine( w2Lft,wy, w2Rgt,wy, blue)

        CALL ComplexExpVal(a,b,1,x,y)
        LET rb= round(b,2)
        LET modb= round(mod(b,2*pi),2)
        IF rb= 6.28 then LET modb= 0
        IF x<>0 then
           LET slope= y/x                        ! find the slope
           IF abs(slope)<=1 then                 ! clip right or left
              LET x1= w1fRgt
              LET x2= w1fLft
              LET y1= slope*x1
              LET y2= slope*x2
              IF modb<pi/2 or modb>3*pi/2 then
                 LET x2= 0
                 LET y2= 0
              ELSE
                 LET x1= 0
                 LET y1= 0
              END IF
           ELSE                                  ! clip top or bottom
              LET y1= w1fTop
              LET y2= w1fBas
              LET x1= y1/slope
              LET x2= y2/slope
              IF modb>=0 and modb<pi then
                 LET x2= 0
                 LET y2= 0
              ELSE
                 LET x1= 0
                 LET y1= 0
              END IF
           END IF

           CALL PlotLine( w1Wndx(x1),w1Wndy(y1), w1Wndx(x2),w1Wndy(y2), blue)
        ELSE                                     ! vertical
           CALL PlotLine( w1x0,w1Top+1, w1x0,w1Bas-1, blue)
        END IF

        CALL w1KeepGraphLayer
        CALL w2KeepGraphLayer

        DO
           CALL h1GetDragVal(ms,h1Places,a)
           IF a<>olda then
              CALL DrawFastGraphs(drkmid)
              LET olda= a
           END IF
        LOOP until ms=3
        CALL DrawGraphs(yellow)
    END SUB

    ! --- slider 2 - b val ---

    SUB h2ClickEvent(ms)
        CALL h2GetClickVal(ms,h2Click,b)
        CALL DrawGraphs(yellow)
        LET oldb= b
    END SUB
    SUB h2DragEvent(ms)
        LET olda,oldb= -999
        CALL h2GetDragVal(ms,h2Places,b)
        LET wx= w2Wndx(a)
        CALL w2ShowGridLayer
        CALL PlotLine( wx,w2Top, wx,w2Bas, blue)

        CALL ComplexExpVal(a,b,1,x,y)
        LET r  = sqr(x*x + y*y)
        LET wr = r*w1pix
        CALL w1ShowGridLayer
        CALL BoxCircle (w1x0-wr,w1x0+wr,w1y0+wr,w1y0-wr,blue)
        CALL w1KeepGraphLayer
        CALL w2KeepGraphLayer

        DO
           CALL h2GetDragVal(ms,h2Places,b)
           IF b<>oldb then
              CALL DrawFastGraphs(drkmid)
              LET oldb= b
           END IF
        LOOP until ms=3
        CALL DrawGraphs(yellow)
    END SUB


    ! --- slider 3 - t value ---

    SUB h3ResetTime
        LET t= 0
        CALL h3Mark(t)
    END SUB

    SUB h3ClickEvent(ms)
        CALL h3GetClickVal(ms,h3Click,t)
        CALL h3Action
    END SUB
    SUB h3DragEvent(ms)
        CALL DrawGraphs(drkmid)
        DO
           CALL h3GetDragVal(ms,h3Places,t)
           CALL h3Action
        LOOP until ms=3
        CALL DrawGraphs(yellow)
        CALL PlotDiamondClr(wx,wy,yellow)
    END SUB

    SUB h3Action
        IF t<>oldt then
           CALL SetVars(a*t,b*t,x,y)
           CALL w2MathToPixels(x,y,wx,wy)
           CALL w2ShowGraphLayer
           CALL PlotDiamondClr(wx,wy,yellow)

           CALL ComplexExpVal(a,b,t,x,y)         ! e^(a+bi)
           CALL w1MathToPixels(x,y,wx,wy)
           SET COLOR yellow
           CALL w1ShowGraphLayer
           CALL PlotDiamondClr(wx,wy,yellow)
           LET oldt= t
        END IF
    END SUB

    ! --- Graph drawing methods ---

    SUB DrawFastGraphs(clr)                      ! real time response to sliders
        ! sketch linear input
        CALL FastLine(clr)
        ! sketch exponential spiral
        CALL w1ShowGraphLayer
        CALL FastSpiral
    END SUB

    SUB FastLine(clr)
        CALL DrawingVariables(a,b,x1,y1,x2,y2,wx1,wy1,wx2,wy2,w1c1,w1s1)
        CALL w2ShowGraphLayer
        CALL PlotLinePoint(w2x0,w2y0,w2Wa,w2Wb,yellow,cyan)
    END SUB

    SUB FastSpiral
        SET COLOR yellow
        FOR t1= 0 to 1 step 1/64
            CALL ComplexExpVal(a,b,t1,x,y)
            CALL w1MathtoPixels(x,y,w1r2,w1i2)
            PLOT w1r2,w1i2;
        NEXT t1
        PLOT

        CALL PlotDiamondClr(w1c1,w1s1,cyan)
    END SUB

    SUB DrawingVariables(a,b,x1,y1,x2,y2,wx1,wy1,wx2,wy2,w1c1,w1s1)
        ! left plane boundary intersects
        LOCAL slope,diag
        IF a<>0 then
           LET slope= b/a
           LET diag = maxb/maxa

           IF abs(slope)<=diag then              ! clip left and right
              LET maxt= maxa/abs(a)
              LET x1  = -maxa
              LET x2  =  maxa
              LET wx1 = w2Lft
              LET wx2 = w2Rgt

              LET y1  = x1*slope
              LET y2  = x2*slope
              LET wy1 = w2Wndy(y1)
              LET wy2 = w2Wndy(y2)
           ELSE IF abs(slope)>=diag then         ! clip top and bas
              LET maxt= maxb/abs(b)
              LET y1  =  maxb
              LET y2  = -maxb
              LET wy1 = w2Top
              LET wy2 = w2Bas

              LET x1  = y1/slope
              LET x2  = y2/slope
              LET wx1 = w2Wndx(x1)
              LET wx2 = w2Wndx(x2)
           END IF

        ELSE IF a=0 and b<>0 then                ! vertical
           LET maxt   = maxb/abs(b)
           LET x1,x2  = 0
           LET wx1,wx2= w2x0
           CALL SetVars(w2fBas,w2fTop,  y1, y2)
           CALL SetVars(w2Bas ,w2Top , wy1,wy2)

        ELSE IF a=0 and b=0 then                 ! origin
           LET maxt   = 1
           LET x1,x2  = 0
           LET y1,y2  = 0
           LET wx1,wx2= w2x0
           LET wy1,wy2= w2y0
        END IF

        CALL ComplexExpVal(a,b,1,c1,s1)          ! right complex values
        CALL w1MathtoPixels(c1,s1,w1c1,w1s1)

        CALL t1Value(c1,s1)                      ! left complex values
        CALL t2Value(a,b)
        CALL w2MathtoPixels(a,b,w2Wa,w2Wb)
    END SUB

    SUB DrawGraphs(clr)
        CALL DrawingVariables(a,b,x1,y1,x2,y2,wx1,wy1,wx2,wy2,w1c1,w1s1)

        CALL w2ShowGridLayer                     ! left line
        CALL PlotLine(wx1,wy1,wx2,wy2,clr)
        CALL PlotDiamondRimClr(w2Wa,w2Wb,cyan)   ! restore points
        CALL PlotDiamondRimClr(w2x0,w2y0,red)

        CALL w1ShowGridLayer                     ! right spiral
        CALL PlotDiamondClr(w1c1,w1s1,cyan)
        SET COLOR clr
        FOR t1= -maxt to maxt step maxt/128
            CALL ComplexExpVal(a,b,t1,x,y)       ! get x and y
            CALL w1MathToPixels(x,y,w1r2,w1i2)
            PLOT w1r2,w1i2;
        NEXT t1
        CALL ComplexExpVal(a,b,maxt,x,y)         ! complete the spiral
        CALL w1MathToPixels(x,y,w1r2,w1i2)
        PLOT w1r2,w1i2

        CALL PlotDiamondRimClr(w1x1,w1y0,red)
        CALL PlotDiamondRimClr(w1c1,w1s1,cyan)

        CALL w1KeepGraphLayer
        CALL w2KeepGraphLayer
    END SUB
END SUB

! --- end of complex exponential code ---