path: root/src/algebra/draw.spad.pamphlet

\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra draw.spad}
\author{Clifton J. Williamson, Scott Morrison, Jon Steinbach, Mike Dewar}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package DRAWCFUN TopLevelDrawFunctionsForCompiledFunctions}
<<package DRAWCFUN TopLevelDrawFunctionsForCompiledFunctions>>=
)abbrev package DRAWCFUN TopLevelDrawFunctionsForCompiledFunctions
++ Author: Clifton J. Williamson
++ Date Created: 22 June 1990
++ Date Last Updated: January 1992 by Scott Morrison
++ Basic Operations: draw, recolor
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description: TopLevelDrawFunctionsForCompiledFunctions provides top level 
++ functions for drawing graphics of expressions.
TopLevelDrawFunctionsForCompiledFunctions():
 Exports == Implementation where
  ANY1 ==> AnyFunctions1
  B    ==> Boolean
  F    ==> Float
  L    ==> List
  SEG  ==> Segment Float
  SF   ==> DoubleFloat
  DROP ==> DrawOption
  PLOT ==> Plot
  PPC  ==> ParametricPlaneCurve(SF -> SF)
  PSC  ==> ParametricSpaceCurve(SF -> SF)
  PSF  ==> ParametricSurface((SF,SF) -> SF)
  Pt   ==> Point SF
  PSFUN ==> (SF, SF) -> Pt
  PCFUN ==> SF -> Pt
  SPACE3 ==> ThreeSpace(SF)
  VIEW2 ==> TwoDimensionalViewport
  VIEW3 ==> ThreeDimensionalViewport

  Exports ==> with

--% Two Dimensional Function Plots

    draw: (SF -> SF,SEG,L DROP) -> VIEW2
      ++ draw(f,a..b,l) draws the graph of \spad{y = f(x)} as x
      ++ ranges from \spad{min(a,b)} to \spad{max(a,b)}.
      ++ The options contained in the list l of
      ++ the domain \spad{DrawOption} are applied.
    draw: (SF -> SF,SEG) -> VIEW2
      ++ draw(f,a..b) draws the graph of \spad{y = f(x)} as x
      ++ ranges from \spad{min(a,b)} to \spad{max(a,b)}.

--% Parametric Plane Curves

    draw: (PPC,SEG,L DROP) -> VIEW2
      ++ draw(curve(f,g),a..b,l) draws the graph of the parametric
      ++ curve \spad{x = f(t), y = g(t)} as t ranges from \spad{min(a,b)} to 
      ++ \spad{max(a,b)}.
      ++ The options contained in the list l of the domain \spad{DrawOption}
      ++ are applied.
    draw: (PPC,SEG) -> VIEW2
      ++ draw(curve(f,g),a..b) draws the graph of the parametric
      ++ curve \spad{x = f(t), y = g(t)} as t ranges from \spad{min(a,b)} to 
      ++ \spad{max(a,b)}.

--% Parametric Space Curves

    draw: (PSC,SEG,L DROP) -> VIEW3
      ++ draw(curve(f,g,h),a..b,l) draws the graph of the parametric
      ++ curve \spad{x = f(t), y = g(t), z = h(t)} as t ranges from 
      ++ \spad{min(a,b)} to \spad{max(a,b)}.
      ++ The options contained in the list l of the domain
      ++ \spad{DrawOption} are applied.
    draw: (PSC,SEG) -> VIEW3
      ++ draw(curve(f,g,h),a..b,l) draws the graph of the parametric
      ++ curve \spad{x = f(t), y = g(t), z = h(t)} as t ranges from 
      ++ \spad{min(a,b)} to \spad{max(a,b)}.
    draw: (PCFUN,SEG,L DROP) -> VIEW3
      ++ draw(f,a..b,l) draws the graph of the parametric
      ++ curve \spad{f} as t ranges from 
      ++ \spad{min(a,b)} to \spad{max(a,b)}.
      ++ The options contained in the list l of the domain
      ++ \spad{DrawOption} are applied.
    draw: (PCFUN,SEG) -> VIEW3
      ++ draw(f,a..b,l) draws the graph of the parametric
      ++ curve \spad{f} as t ranges from 
      ++ \spad{min(a,b)} to \spad{max(a,b)}.

    makeObject: (PSC,SEG,L DROP) -> SPACE3
      ++ makeObject(curve(f,g,h),a..b,l) returns a space of the
      ++ domain \spadtype{ThreeSpace} which contains the graph of the
      ++ parametric curve \spad{x = f(t), y = g(t), z = h(t)} as t ranges from 
      ++ \spad{min(a,b)} to \spad{max(a,b)};
      ++ The options contained in the list l of the domain
      ++ \spad{DrawOption} are applied.
    makeObject: (PSC,SEG) -> SPACE3
      ++ makeObject(sp,curve(f,g,h),a..b) returns the space \spad{sp}
      ++ of the domain \spadtype{ThreeSpace} with the addition of the graph
      ++ of the parametric curve \spad{x = f(t), y = g(t), z = h(t)} as t
      ++ ranges from \spad{min(a,b)} to \spad{max(a,b)}.
    makeObject: (PCFUN,SEG,L DROP) -> SPACE3
      ++ makeObject(curve(f,g,h),a..b,l) returns a space of the
      ++ domain \spadtype{ThreeSpace} which contains the graph of the
      ++ parametric curve \spad{x = f(t), y = g(t), z = h(t)} as t ranges from 
      ++ \spad{min(a,b)} to \spad{max(a,b)}.
      ++ The options contained in the list l of the domain
      ++ \spad{DrawOption} are applied.
    makeObject: (PCFUN,SEG) -> SPACE3
      ++ makeObject(sp,curve(f,g,h),a..b) returns the space \spad{sp}
      ++ of the domain \spadtype{ThreeSpace} with the addition of the graph
      ++ of the parametric curve \spad{x = f(t), y = g(t), z = h(t)} as t
      ++ ranges from \spad{min(a,b)} to \spad{max(a,b)}.

--% Three Dimensional Function Plots

    draw: ((SF,SF) -> SF,SEG,SEG,L DROP) -> VIEW3
      ++ draw(f,a..b,c..d,l) draws the graph of \spad{z = f(x,y)}
      ++ as x ranges from \spad{min(a,b)} to \spad{max(a,b)} and y ranges from
      ++ \spad{min(c,d)} to \spad{max(c,d)}.
      ++ and the options contained in the list l of the domain
      ++ \spad{DrawOption} are applied.
    draw: ((SF,SF) -> SF,SEG,SEG) -> VIEW3
      ++ draw(f,a..b,c..d) draws the graph of \spad{z = f(x,y)}
      ++ as x ranges from \spad{min(a,b)} to \spad{max(a,b)} and y ranges from
      ++ \spad{min(c,d)} to \spad{max(c,d)}.
    makeObject: ((SF,SF) -> SF,SEG,SEG,L DROP) -> SPACE3
      ++ makeObject(f,a..b,c..d,l) returns a space of the domain
      ++ \spadtype{ThreeSpace} which contains the graph of \spad{z = f(x,y)}
      ++ as x ranges from \spad{min(a,b)} to \spad{max(a,b)} and y ranges from
      ++ \spad{min(c,d)} to \spad{max(c,d)}, and the options contained in the
      ++ list l of the domain \spad{DrawOption} are applied.
    makeObject: ((SF,SF) -> SF,SEG,SEG) -> SPACE3
      ++ makeObject(f,a..b,c..d) returns a space of the domain
      ++ \spadtype{ThreeSpace} which contains the graph of \spad{z = f(x,y)}
      ++ as x ranges from \spad{min(a,b)} to \spad{max(a,b)} and y ranges from
      ++ \spad{min(c,d)} to \spad{max(c,d)}.

--% Parametric Surfaces

    draw: (PSFUN, SEG, SEG, L DROP) -> VIEW3
      ++ draw(f,a..b,c..d) draws the
      ++ graph of the parametric surface \spad{f(u,v)}
      ++ as u ranges from \spad{min(a,b)} to \spad{max(a,b)}
      ++ and v ranges from \spad{min(c,d)} to \spad{max(c,d)}.
      ++ The options contained in the
      ++ list l of the domain \spad{DrawOption} are applied.
    draw: (PSFUN, SEG, SEG) -> VIEW3
      ++ draw(f,a..b,c..d) draws the
      ++ graph of the parametric surface \spad{f(u,v)}
      ++ as u ranges from \spad{min(a,b)} to \spad{max(a,b)}
      ++ and v ranges from \spad{min(c,d)} to \spad{max(c,d)}
      ++ The options contained in the list
      ++ l of the domain \spad{DrawOption} are applied.
    makeObject: (PSFUN, SEG, SEG, L DROP) -> SPACE3
      ++ makeObject(f,a..b,c..d,l) returns a
      ++ space of the domain \spadtype{ThreeSpace} which contains the
      ++ graph of the parametric surface \spad{f(u,v)}
      ++ as u ranges from \spad{min(a,b)} to
      ++ \spad{max(a,b)} and v ranges from \spad{min(c,d)} to \spad{max(c,d)};
      ++ The options contained in the
      ++ list l of the domain \spad{DrawOption} are applied.
    makeObject: (PSFUN, SEG, SEG) -> SPACE3
      ++ makeObject(f,a..b,c..d,l) returns a
      ++ space of the domain \spadtype{ThreeSpace} which contains the
      ++ graph of the parametric surface \spad{f(u,v)}
      ++ as u ranges from \spad{min(a,b)} to
      ++ \spad{max(a,b)} and v ranges from \spad{min(c,d)} to \spad{max(c,d)}.
    draw: (PSF,SEG,SEG,L DROP) -> VIEW3
      ++ draw(surface(f,g,h),a..b,c..d) draws the
      ++ graph of the parametric surface \spad{x = f(u,v)}, \spad{y = g(u,v)},
      ++ \spad{z = h(u,v)} as u ranges from \spad{min(a,b)} to \spad{max(a,b)}
      ++ and v ranges from \spad{min(c,d)} to \spad{max(c,d)};
      ++ The options contained in the
      ++ list l of the domain \spad{DrawOption} are applied.
    draw: (PSF,SEG,SEG) -> VIEW3
      ++ draw(surface(f,g,h),a..b,c..d) draws the
      ++ graph of the parametric surface \spad{x = f(u,v)}, \spad{y = g(u,v)},
      ++ \spad{z = h(u,v)} as u ranges from \spad{min(a,b)} to \spad{max(a,b)}
      ++ and v ranges from \spad{min(c,d)} to \spad{max(c,d)};
    makeObject: (PSF,SEG,SEG,L DROP) -> SPACE3
      ++ makeObject(surface(f,g,h),a..b,c..d,l) returns a
      ++ space of the domain \spadtype{ThreeSpace} which contains the
      ++ graph of the parametric surface \spad{x = f(u,v)}, \spad{y = g(u,v)},
      ++ \spad{z = h(u,v)} as u ranges from \spad{min(a,b)} to
      ++ \spad{max(a,b)} and v ranges from \spad{min(c,d)} to \spad{max(c,d)}.
      ++ The options contained in the
      ++ list l of the domain \spad{DrawOption} are applied.
    makeObject: (PSF,SEG,SEG) -> SPACE3
      ++ makeObject(surface(f,g,h),a..b,c..d,l) returns a
      ++ space of the domain \spadtype{ThreeSpace} which contains the
      ++ graph of the parametric surface \spad{x = f(u,v)}, \spad{y = g(u,v)},
      ++ \spad{z = h(u,v)} as u ranges from \spad{min(a,b)} to
      ++ \spad{max(a,b)} and v ranges from \spad{min(c,d)} to \spad{max(c,d)}.
    recolor: ((SF,SF) -> Pt,(SF,SF,SF) -> SF) -> ((SF,SF) -> Pt)
      ++ recolor(), uninteresting to top level user; exported in order to 
      ++ compile package.

  Implementation ==> add
    --!!  I have had to work my way around the following bug in the compiler:
    --!!  When a local variable is given a mapping as a value, e.g.
    --!!  foo : SF -> SF := makeFloatFunction(f,t),
    --!!  the compiler cannot distinguish that local variable from a local
    --!!  function defined elsewhere in the package.  Thus, when 'foo' is
    --!!  passed to a function, e.g.
    --!!  bird := fcn(foo),
    --!!  foo will often be compiled as |DRAW;foo| rather than |foo|. This,
    --!!  of course, causes a run-time error.
    --!!  To avoid this problem, local variables are not given mappings as
    --!!  values, but rather (singleton) lists of mappings.  The first element
    --!!  of the list can always be extracted and everything goes through
    --!!  as before.  There is no major loss in efficiency, as the computation
    --!!  of points will always dominate the computation time.
    --!!                                     - cjw,  22 June MCMXC

    import PLOT
    import TwoDimensionalPlotClipping
    import GraphicsDefaults
    import ViewportPackage
    import ThreeDimensionalViewport
    import DrawOptionFunctions0
    import MakeFloatCompiledFunction(Ex)
    import MeshCreationRoutinesForThreeDimensions
    import SegmentFunctions2(SF,Float)
    import ViewDefaultsPackage
    import AnyFunctions1(Pt -> Pt)
    import AnyFunctions1((SF,SF,SF) -> SF)
    import DrawOptionFunctions0
    import SPACE3
    import DROP

    EXTOVARERROR : String := _
      "draw: when specifying function, left hand side must be a variable"
    SMALLRANGEERROR : String := _
      "draw: range is in interval with only one point"
    DEPVARERROR : String := _
      "draw: independent variable appears on lhs of function definition"

------------------------------------------------------------------------
--                     2D - draw's  
------------------------------------------------------------------------

    drawToScaleRanges: (Segment SF,Segment SF) -> L SEG
    drawToScaleRanges(xVals,yVals) ==
      -- warning: assumes window is square
      xHi := convert(hi xVals)@Float; xLo := convert(lo xVals)@Float
      yHi := convert(hi yVals)@Float; yLo := convert(lo yVals)@Float
      xDiff := xHi - xLo; yDiff := yHi - yLo
      pad := abs(yDiff - xDiff)/2
      yDiff > xDiff =>
        [segment(xLo - pad,xHi + pad),map(convert(#1)@Float,yVals)]
      [map(convert(#1)@Float,xVals),segment(yLo - pad,yHi + pad)]

    drawPlot: (PLOT,L DROP) -> VIEW2
    drawPlot(plot,l) ==
      branches := listBranches plot
      xRange := xRange plot; yRange := yRange plot
      -- process clipping information
      if (cl := option(l,"clipSegment" :: Symbol)) case "failed" then
        if clipBoolean(l,clipPointsDefault()) then
          clipInfo :=
            parametric? plot => clipParametric plot
            clip plot
          branches := clipInfo.brans
          xRange := clipInfo.xValues; yRange := clipInfo.yValues
        else
          "No explicit user-specified clipping"
      else
        segList := retract(cl :: Any)$ANY1(L SEG)
        empty? segList =>
          error "draw: you may specify at least 1 segment for 2D clipping"
        #segList > 2 =>
          error "draw: you may specify at most 2 segments for 2D clipping"
        xLo : SF := 0; xHi : SF := 0; yLo : SF := 0; yHi : SF := 0
        if empty? rest segList then
          xLo := lo xRange; xHi := hi xRange
          yRangeF := first segList
          yLo := convert(lo yRangeF)@SF; yHi := convert(hi yRangeF)@SF
        else
          xRangeF := first segList
          xLo := convert(lo xRangeF)@SF; xHi := convert(hi xRangeF)@SF
          yRangeF := second segList
          yLo := convert(lo yRangeF)@SF; yHi := convert(hi yRangeF)@SF
        clipInfo := clipWithRanges(branches,xLo,xHi,yLo,yHi)
        branches := clipInfo.brans
        xRange := clipInfo.xValues; yRange := clipInfo.yValues
      -- process scaling information
      if toScale(l,drawToScale()) then
        scaledRanges := drawToScaleRanges(xRange,yRange)
        -- add scaled ranges to list of options
        l := concat(ranges scaledRanges,l)
      else
        xRangeFloat : SEG := map(convert(#1)@Float,xRange)
        yRangeFloat : SEG := map(convert(#1)@Float,yRange)
        -- add ranges to list of options
        l := concat(ranges(ll : L SEG := [xRangeFloat,yRangeFloat]),l)
      -- process color information
      ptCol := pointColorPalette(l,pointColorDefault())
      crCol := curveColorPalette(l,lineColorDefault())
      -- draw
      drawCurves(branches,ptCol,crCol,pointSizeDefault(),l)

    normalize: SEG -> Segment SF
    normalize seg ==
      -- normalize [a,b]:
      -- error if a = b, returns [a,b] if a < b, returns [b,a] if b > a
      a := convert(lo seg)@SF; b := convert(hi seg)@SF
      a = b => error SMALLRANGEERROR
      a < b => segment(a,b)
      segment(b,a)

--% functions for creation of maps SF -> Point SF (two dimensional)

    myTrap1: (SF-> SF, SF) -> SF
    myTrap1(ff:SF-> SF, f:SF):SF ==
      s := trapNumericErrors(ff(f))$Lisp :: Union(SF, "failed")
      s case "failed" => quietDoubleNaN()$Foreign(Builtin)
      r:=s::SF
      r >max()$SF or r < min()$SF => quietDoubleNaN()$Foreign(Builtin)
      r

    makePt2: (SF,SF) -> Point SF
    makePt2(x,y) == point(l : List SF := [x,y])

--% Two Dimensional Function Plots
 
    draw(f:SF -> SF,seg:SEG,l:L DROP) ==
      -- set adaptive plotting off or on
      oldAdaptive := adaptive?()$PLOT
      setAdaptive(adaptive(l,oldAdaptive))$PLOT
      -- create function SF -> Point SF
      ff : L(SF -> Point SF) := [makePt2(myTrap1(f,#1),#1)]
      -- process change of coordinates
      if (c := option(l,"coordinates" :: Symbol)) case "failed" then
        -- default coordinate transformation
        ff := [makePt2(#1,myTrap1(f,#1))]
      else
        cc : L(Pt -> Pt) := [retract(c :: Any)$ANY1(Pt -> Pt)]
        ff := [(first cc)((first ff)(#1))]
      -- create PLOT
      pl := pointPlot(first ff,normalize seg)
      -- reset adaptive plotting
      setAdaptive(oldAdaptive)$PLOT
      -- draw
      drawPlot(pl,l)
 
    draw(f:SF -> SF,seg:SEG) == draw(f,seg,nil())
 
--% Parametric Plane Curves

    draw(ppc:PPC,seg:SEG,l:L DROP) ==
      -- set adaptive plotting off or on
      oldAdaptive := adaptive?()$PLOT
      setAdaptive(adaptive(l,oldAdaptive))$PLOT
      -- create function SF -> Point SF
      f := coordinate(ppc,1); g := coordinate(ppc,2)
      fcn : L(SF -> Pt) := [makePt2(myTrap1(f,#1),myTrap1(g,#1))]
      -- process change of coordinates
      if not (c := option(l,"coordinates" :: Symbol)) case "failed" then
        cc : L(Pt -> Pt) := [retract(c :: Any)$ANY1(Pt -> Pt)]
        fcn := [(first cc)((first fcn)(#1))]
      -- create PLOT
      pl := pointPlot(first fcn,normalize seg)
      -- reset adaptive plotting
      setAdaptive(oldAdaptive)$PLOT
      -- draw
      drawPlot(pl,l)
 
    draw(ppc:PPC,seg:SEG) == draw(ppc,seg,nil())

------------------------------------------------------------------------
--                     3D - Curves  
------------------------------------------------------------------------

--% functions for creation of maps SF -> Point SF (three dimensional)

    makePt4: (SF,SF,SF,SF) -> Point SF
    makePt4(x,y,z,c) == point(l : List SF := [x,y,z,c])

--% Parametric Space Curves

    id: SF -> SF
    id x == x

    zCoord: (SF,SF,SF) -> SF
    zCoord(x,y,z) == z

    colorPoints: (List List Pt,(SF,SF,SF) -> SF) -> List List Pt
    colorPoints(llp,func) ==
      for lp in llp repeat for p in lp repeat
        p.4 := func(p.1,p.2,p.3)
      llp

    makeObject(psc:PSC,seg:SEG,l:L DROP) ==
      sp := space l
      -- obtain dependent variable and coordinate functions
      f := coordinate(psc,1); g := coordinate(psc,2); h := coordinate(psc,3)
      -- create function SF -> Point SF with default or user-specified
      -- color function
      fcn : L(SF -> Pt) := [makePt4(myTrap1(f,#1),myTrap1(g,#1),myTrap1(h,#1),_
                            myTrap1(id,#1))]
      pointsColored? : Boolean := false
      if not (c1 := option(l,"colorFunction1" :: Symbol)) case "failed" then
        pointsColored? := true
        fcn := [makePt4(myTrap1(f,#1),myTrap1(g,#1),myTrap1(h,#1),_
                retract(c1 :: Any)$ANY1(SF -> SF)(#1))]
      -- process change of coordinates
      if not (c := option(l,"coordinates" :: Symbol)) case "failed" then
        cc : L(Pt -> Pt) := [retract(c :: Any)$ANY1(Pt -> Pt)]
        fcn := [(first cc)((first fcn)(#1))]
      -- create PLOT
      pl := pointPlot(first fcn,normalize seg)$Plot3D
      -- create ThreeSpace
      s := sp
      -- draw Tube
--      print(pl::OutputForm)
      option?(l,"tubeRadius" :: Symbol) =>
        pts := tubePoints(l,8)
        rad := convert(tubeRadius(l,0.25))@DoubleFloat
        tub := tube(pl,rad,pts)$NumericTubePlot(Plot3D)
        loops := listLoops tub
        -- color points if this has not been done already
        if not pointsColored? then
          if (c3 := option(l,"colorFunction3" :: Symbol)) case "failed"
            then colorPoints(loops,zCoord)  -- default color function
            else colorPoints(loops,retract(c3 :: Any)$ANY1((SF,SF,SF) -> SF))
        mesh(s,loops,false,false)
        s
      -- draw curve
      br := listBranches pl
      for b in br repeat curve(s,b)
      s

    makeObject(psc:PCFUN,seg:SEG,l:L DROP) ==
      sp := space l
      -- create function SF -> Point SF with default or user-specified
      -- color function
      fcn : L(SF -> Pt) := [psc]
      pointsColored? : Boolean := false
      if not (c1 := option(l,"colorFunction1" :: Symbol)) case "failed" then
        pointsColored? := true
        fcn := [concat(psc(#1), retract(c1 :: Any)$ANY1(SF -> SF)(#1))]
      -- process change of coordinates
      if not (c := option(l,"coordinates" :: Symbol)) case "failed" then
        cc : L(Pt -> Pt) := [retract(c :: Any)$ANY1(Pt -> Pt)]
        fcn := [(first cc)((first fcn)(#1))]
      -- create PLOT
      pl := pointPlot(first fcn,normalize seg)$Plot3D
      -- create ThreeSpace
      s := sp
      -- draw Tube
      option?(l,"tubeRadius" :: Symbol) =>
        pts := tubePoints(l,8)
        rad := convert(tubeRadius(l,0.25))@DoubleFloat
        tub := tube(pl,rad,pts)$NumericTubePlot(Plot3D)
        loops := listLoops tub
        -- color points if this has not been done already
        mesh(s,loops,false,false)
        s
      -- draw curve
      br := listBranches pl
      for b in br repeat curve(s,b)
      s

    makeObject(psc:PSC,seg:SEG) ==
      makeObject(psc,seg,nil())

    makeObject(psc:PCFUN,seg:SEG) ==
      makeObject(psc,seg,nil())

    draw(psc:PSC,seg:SEG,l:L DROP) ==
      sp := makeObject(psc,seg,l)
      makeViewport3D(sp, l)

    draw(psc:PSC,seg:SEG) ==
      draw(psc,seg,nil())

    draw(psc:PCFUN,seg:SEG,l:L DROP) ==
      sp := makeObject(psc,seg,l)
      makeViewport3D(sp, l)

    draw(psc:PCFUN,seg:SEG) ==
      draw(psc,seg,nil())

------------------------------------------------------------------------
--                     3D - Surfaces  
------------------------------------------------------------------------

    myTrap2: ((SF, SF) -> SF, SF, SF) -> SF
    myTrap2(ff:(SF, SF) -> SF, u:SF, v:SF):SF ==
      s := trapNumericErrors(ff(u, v))$Lisp :: Union(SF, "failed")
      s case "failed" =>  quietDoubleNaN()$Foreign(Builtin)
      r:SF := s::SF
      r >max()$SF or r < min()$SF => quietDoubleNaN()$Foreign(Builtin)
      r

    recolor(ptFunc,colFunc) ==
      pt := ptFunc(#1,#2)
      pt.4 := colFunc(pt.1,pt.2,pt.3)
      pt

    xCoord: (SF,SF) -> SF
    xCoord(x,y) == x

--% Three Dimensional Function Plots

    makeObject(f:(SF,SF) -> SF,xSeg:SEG,ySeg:SEG,l:L DROP) ==
      sp := space l
      -- process color function of two variables
      col2 : L((SF,SF) -> SF) := [xCoord]     -- dummy color function
      pointsColored? : Boolean := false
      if not (c2 := option(l,"colorFunction2" :: Symbol)) case "failed" then
        pointsColored? := true
        col2 := [retract(c2 :: Any)$ANY1((SF,SF) -> SF)]
      fcn : L((SF,SF) -> Pt) :=
        [makePt4(myTrap2(f,#1,#2),#1,#2,(first col2)(#1,#2))]
      -- process change of coordinates
      if (c := option(l,"coordinates" :: Symbol)) case "failed" then
        -- default coordinate transformation
        fcn := [makePt4(#1,#2,myTrap2(f,#1,#2),(first col2)(#1,#2))]
      else
        cc : L(Pt -> Pt) := [retract(c :: Any)$ANY1(Pt -> Pt)]
        fcn := [(first cc)((first fcn)(#1,#2))]
      -- process color function of three variables, if there was no
      -- color function of two variables
      if not pointsColored? then
        c := option(l,"colorFunction3" :: Symbol)
        fcn := 
          c case "failed" => [recolor((first fcn),zCoord)]
          [recolor((first fcn),retract(c :: Any)$ANY1((SF,SF,SF) -> SF))]
      -- create mesh
      mesh := meshPar2Var(sp,first fcn,normalize xSeg,normalize ySeg,l)
      mesh

    makeObject(f:(SF,SF) -> SF,xSeg:SEG,ySeg:SEG) ==
      makeObject(f,xSeg,ySeg,nil())

    draw(f:(SF,SF) -> SF,xSeg:SEG,ySeg:SEG,l:L DROP) ==
      sp := makeObject(f, xSeg, ySeg, l)
      makeViewport3D(sp, l)

    draw(f:(SF,SF) -> SF,xSeg:SEG,ySeg:SEG) ==
      draw(f,xSeg,ySeg,nil())

--% parametric surface

    makeObject(s:PSF,uSeg:SEG,vSeg:SEG,l:L DROP) ==
      sp := space l
      -- create functions from expressions
      f : L((SF,SF) -> SF) := [coordinate(s,1)]
      g : L((SF,SF) -> SF) := [coordinate(s,2)]
      h : L((SF,SF) -> SF) := [coordinate(s,3)]
      -- process color function of two variables
      col2 : L((SF,SF) -> SF) := [xCoord]     -- dummy color function
      pointsColored? : Boolean := false
      if not (c2 := option(l,"colorFunction2" :: Symbol)) case "failed" then
        pointsColored? := true
        col2 := [retract(c2 :: Any)$ANY1((SF,SF) -> SF)]
      fcn : L((SF,SF) -> Pt) := 
        [makePt4(myTrap2((first f),#1,#2),myTrap2((first g),#1,#2),myTrap2((first h),#1,#2),_
                 myTrap2((first col2),#1,#2))]
      -- process change of coordinates
      if not (c := option(l,"coordinates" :: Symbol)) case "failed" then
        cc : L(Pt -> Pt) := [retract(c :: Any)$ANY1(Pt -> Pt)]
        fcn := [(first cc)((first fcn)(#1,#2))]
      -- process color function of three variables, if there was no
      -- color function of two variables
      if not pointsColored? then
        col3 : L((SF,SF,SF) -> SF) := [zCoord]  -- default color function
        if not (c := option(l,"colorFunction3" :: Symbol)) case "failed" then 
          col3 := [retract(c :: Any)$ANY1((SF,SF,SF) -> SF)]
        fcn := [recolor((first fcn),(first col3))]
      -- create mesh
      mesh := meshPar2Var(sp,first fcn,normalize uSeg,normalize vSeg,l)
      mesh

    makeObject(s:PSFUN,uSeg:SEG,vSeg:SEG,l:L DROP) ==
      sp := space l
      -- process color function of two variables
      col2 : L((SF,SF) -> SF) := [xCoord]     -- dummy color function
      pointsColored? : Boolean := false
      if not (c2 := option(l,"colorFunction2" :: Symbol)) case "failed" then
        pointsColored? := true
        col2 := [retract(c2 :: Any)$ANY1((SF,SF) -> SF)]
      fcn : L((SF,SF) -> Pt) := 
        pointsColored? => [concat(s(#1, #2), (first col2)(#1, #2))]
        [s]
      -- process change of coordinates
      if not (c := option(l,"coordinates" :: Symbol)) case "failed" then
        cc : L(Pt -> Pt) := [retract(c :: Any)$ANY1(Pt -> Pt)]
        fcn := [(first cc)((first fcn)(#1,#2))]
      -- create mesh
      mesh := meshPar2Var(sp,first fcn,normalize uSeg,normalize vSeg,l)
      mesh

    makeObject(s:PSF,uSeg:SEG,vSeg:SEG) ==
      makeObject(s,uSeg,vSeg,nil())

    draw(s:PSF,uSeg:SEG,vSeg:SEG,l:L DROP) ==
      -- draw
      mesh := makeObject(s,uSeg,vSeg,l)
      makeViewport3D(mesh,l)

    draw(s:PSF,uSeg:SEG,vSeg:SEG) ==
      draw(s,uSeg,vSeg,nil())
 
    makeObject(s:PSFUN,uSeg:SEG,vSeg:SEG) ==
      makeObject(s,uSeg,vSeg,nil())

    draw(s:PSFUN,uSeg:SEG,vSeg:SEG,l:L DROP) ==
      -- draw
      mesh := makeObject(s,uSeg,vSeg,l)
      makeViewport3D(mesh,l)

    draw(s:PSFUN,uSeg:SEG,vSeg:SEG) ==
      draw(s,uSeg,vSeg,nil())
 
@
\section{package DRAW TopLevelDrawFunctions}
<<package DRAW TopLevelDrawFunctions>>=
)abbrev package DRAW TopLevelDrawFunctions
++ Author: Clifton J. Williamson
++ Date Created: 23 January 1990
++ Date Last Updated: October 1991 by Jon Steinbach
++ Basic Operations: draw
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description: TopLevelDrawFunctions provides top level functions for 
++ drawing graphics of expressions.
TopLevelDrawFunctions(Ex:Join(ConvertibleTo InputForm,SetCategory)):
 Exports == Implementation where
  B    ==> Boolean
  BIND ==> SegmentBinding Float
  L    ==> List
  SF   ==> DoubleFloat
  DROP ==> DrawOption

  PPC  ==> ParametricPlaneCurve Ex
  PPCF ==> ParametricPlaneCurve(SF -> SF)
  PSC  ==> ParametricSpaceCurve Ex
  PSCF ==> ParametricSpaceCurve(SF -> SF)
  PSF  ==> ParametricSurface Ex
  PSFF ==> ParametricSurface((SF,SF) -> SF)
  SPACE3 ==> ThreeSpace(SF)
  VIEW2 ==> TwoDimensionalViewport
  VIEW3 ==> ThreeDimensionalViewport

  Exports ==> with

--% Two Dimensional Function Plots

    draw: (Ex,BIND,L DROP) -> VIEW2
      ++ draw(f(x),x = a..b,l) draws the graph of \spad{y = f(x)} as x
      ++ ranges from \spad{min(a,b)} to \spad{max(a,b)}; \spad{f(x)} is the 
      ++ default title, and the options contained in the list l of
      ++ the domain \spad{DrawOption} are applied.
    draw: (Ex,BIND) -> VIEW2
      ++ draw(f(x),x = a..b) draws the graph of \spad{y = f(x)} as x
      ++ ranges from \spad{min(a,b)} to \spad{max(a,b)}; \spad{f(x)} appears 
      ++ in the title bar.

--% Parametric Plane Curves

    draw: (PPC,BIND,L DROP) -> VIEW2
      ++ draw(curve(f(t),g(t)),t = a..b,l) draws the graph of the parametric
      ++ curve \spad{x = f(t), y = g(t)} as t ranges from \spad{min(a,b)} to 
      ++ \spad{max(a,b)}; \spad{(f(t),g(t))} is the default title, and the
      ++ options contained in the list l of the domain \spad{DrawOption}
      ++ are applied.
    draw: (PPC,BIND) -> VIEW2
      ++ draw(curve(f(t),g(t)),t = a..b) draws the graph of the parametric
      ++ curve \spad{x = f(t), y = g(t)} as t ranges from \spad{min(a,b)} to 
      ++ \spad{max(a,b)}; \spad{(f(t),g(t))} appears in the title bar.

--% Parametric Space Curves

    draw: (PSC,BIND,L DROP) -> VIEW3
      ++ draw(curve(f(t),g(t),h(t)),t = a..b,l) draws the graph of the
      ++ parametric curve \spad{x = f(t)}, \spad{y = g(t)}, \spad{z = h(t)}
      ++ as t ranges from \spad{min(a,b)} to \spad{max(a,b)}; \spad{h(t)}
      ++ is the default title, and the options contained in the list l of
      ++ the domain \spad{DrawOption} are applied.
    draw: (PSC,BIND) -> VIEW3
      ++ draw(curve(f(t),g(t),h(t)),t = a..b) draws the graph of the parametric
      ++ curve \spad{x = f(t)}, \spad{y = g(t)}, \spad{z = h(t)} as t ranges
      ++ from \spad{min(a,b)} to \spad{max(a,b)}; \spad{h(t)} is the default
      ++ title.
    makeObject: (PSC,BIND,L DROP) -> SPACE3
      ++ makeObject(curve(f(t),g(t),h(t)),t = a..b,l) returns a space of
      ++ the domain \spadtype{ThreeSpace} which contains the graph of the
      ++ parametric curve \spad{x = f(t)}, \spad{y = g(t)}, \spad{z = h(t)}
      ++ as t ranges from \spad{min(a,b)} to \spad{max(a,b)}; \spad{h(t)}
      ++ is the default title, and the options contained in the list l of
      ++ the domain \spad{DrawOption} are applied.
    makeObject: (PSC,BIND) -> SPACE3
      ++ makeObject(curve(f(t),g(t),h(t)),t = a..b) returns a space of the
      ++ domain \spadtype{ThreeSpace} which contains the graph of the
      ++ parametric curve \spad{x = f(t)}, \spad{y = g(t)}, \spad{z = h(t)}
      ++ as t ranges from \spad{min(a,b)} to \spad{max(a,b)}; \spad{h(t)} is
      ++ the default title.

--% Three Dimensional Function Plots

    draw: (Ex,BIND,BIND,L DROP) -> VIEW3
      ++ draw(f(x,y),x = a..b,y = c..d,l) draws the graph of \spad{z = f(x,y)}
      ++ as x ranges from \spad{min(a,b)} to \spad{max(a,b)} and y ranges from
      ++ \spad{min(c,d)} to \spad{max(c,d)}; \spad{f(x,y)} is the default
      ++ title, and the options contained in the list l of the domain
      ++ \spad{DrawOption} are applied.
    draw: (Ex,BIND,BIND) -> VIEW3
      ++ draw(f(x,y),x = a..b,y = c..d) draws the graph of \spad{z = f(x,y)}
      ++ as x ranges from \spad{min(a,b)} to \spad{max(a,b)} and y ranges from
      ++ \spad{min(c,d)} to \spad{max(c,d)}; \spad{f(x,y)} appears in the title bar.
    makeObject: (Ex,BIND,BIND,L DROP) -> SPACE3
      ++ makeObject(f(x,y),x = a..b,y = c..d,l) returns a space of the
      ++ domain \spadtype{ThreeSpace} which contains the graph of
      ++ \spad{z = f(x,y)} as x ranges from \spad{min(a,b)} to \spad{max(a,b)}
      ++ and y ranges from \spad{min(c,d)} to \spad{max(c,d)}; \spad{f(x,y)}
      ++ is the default title, and the options contained in the list l of the
      ++ domain \spad{DrawOption} are applied.
    makeObject: (Ex,BIND,BIND) -> SPACE3
      ++ makeObject(f(x,y),x = a..b,y = c..d) returns a space of the domain
      ++ \spadtype{ThreeSpace} which contains the graph of \spad{z = f(x,y)}
      ++ as x ranges from \spad{min(a,b)} to \spad{max(a,b)} and y ranges from
      ++ \spad{min(c,d)} to \spad{max(c,d)}; \spad{f(x,y)} appears as the
      ++ default title.

--% Parametric Surfaces

    draw: (PSF,BIND,BIND,L DROP) -> VIEW3
      ++ draw(surface(f(u,v),g(u,v),h(u,v)),u = a..b,v = c..d,l) draws the
      ++ graph of the parametric surface \spad{x = f(u,v)}, \spad{y = g(u,v)},
      ++ \spad{z = h(u,v)} as u ranges from \spad{min(a,b)} to \spad{max(a,b)}
      ++ and v ranges from \spad{min(c,d)} to \spad{max(c,d)}; \spad{h(t)}
      ++ is the default title, and the options contained in the list l of
      ++ the domain \spad{DrawOption} are applied.
    draw: (PSF,BIND,BIND) -> VIEW3
      ++ draw(surface(f(u,v),g(u,v),h(u,v)),u = a..b,v = c..d) draws the
      ++ graph of the parametric surface \spad{x = f(u,v)}, \spad{y = g(u,v)},
      ++ \spad{z = h(u,v)} as u ranges from \spad{min(a,b)} to \spad{max(a,b)}
      ++ and v ranges from \spad{min(c,d)} to \spad{max(c,d)}; \spad{h(t)} is
      ++ the default title.
    makeObject: (PSF,BIND,BIND,L DROP) -> SPACE3
      ++ makeObject(surface(f(u,v),g(u,v),h(u,v)),u = a..b,v = c..d,l) returns
      ++ a space of the domain \spadtype{ThreeSpace} which contains the graph
      ++ of the parametric surface \spad{x = f(u,v)}, \spad{y = g(u,v)},
      ++ \spad{z = h(u,v)} as u ranges from \spad{min(a,b)} to \spad{max(a,b)}
      ++ and v ranges from \spad{min(c,d)} to \spad{max(c,d)}; \spad{h(t)} is
      ++ the default title, and the options contained in the list l of
      ++ the domain \spad{DrawOption} are applied.
    makeObject: (PSF,BIND,BIND) -> SPACE3
      ++ makeObject(surface(f(u,v),g(u,v),h(u,v)),u = a..b,v = c..d) returns
      ++ a space of the domain \spadtype{ThreeSpace} which contains the
      ++ graph of the parametric surface \spad{x = f(u,v)}, \spad{y = g(u,v)},
      ++ \spad{z = h(u,v)} as u ranges from \spad{min(a,b)} to \spad{max(a,b)}
      ++ and v ranges from \spad{min(c,d)} to \spad{max(c,d)}; \spad{h(t)} is
      ++ the default title.

  Implementation ==> add
    import TopLevelDrawFunctionsForCompiledFunctions
    import MakeFloatCompiledFunction(Ex)
    import ParametricPlaneCurve(SF -> SF)
    import ParametricSpaceCurve(SF -> SF)
    import ParametricSurface((SF,SF) -> SF)
    import ThreeSpace(SF)

------------------------------------------------------------------------
--                     2D - draw's  (given by formulae)
------------------------------------------------------------------------

--% Two Dimensional Function Plots
 
    draw(f:Ex,bind:BIND,l:L DROP) ==
      import DROP
      -- create title if necessary
      if not option?(l,"title" :: Symbol) then
        s:String := unparse(convert(f)@InputForm)
        if sayLength(s)$DisplayPackage > 50 then
          l := concat(title "AXIOM2D",l)
        else l := concat(title s,l)
      -- call 'draw'
      draw(makeFloatFunction(f,variable bind),segment bind,l)
 
    draw(f:Ex,bind:BIND) == draw(f,bind,nil())
 
--% Parametric Plane Curves

    draw(ppc:PPC,bind:BIND,l:L DROP) ==
      import DROP
      f := coordinate(ppc,1); g := coordinate(ppc,2)
      -- create title if necessary
      if not option?(l,"title" :: Symbol) then
        s:String := unparse(convert(f)@InputForm)
        if sayLength(s)$DisplayPackage > 50 then
          l := concat(title "AXIOM2D",l)
        else l := concat(title s,l)
      -- create curve with functions as coordinates
      curve : PPCF := curve(makeFloatFunction(f,variable bind),_
                            makeFloatFunction(g,variable bind))$PPCF
      -- call 'draw'
      draw(curve,segment bind,l)
 
    draw(ppc:PPC,bind:BIND) == draw(ppc,bind,nil())

------------------------------------------------------------------------
--                     3D - Curves  (given by formulas)
------------------------------------------------------------------------

    makeObject(psc:PSC,tBind:BIND,l:L DROP) ==
      import DROP
      -- obtain dependent variable and coordinate functions
      t := variable tBind; tSeg := segment tBind
      f := coordinate(psc,1); g := coordinate(psc,2); h := coordinate(psc,3)
      -- create title if necessary
      if not option?(l,"title" :: Symbol) then
        s:String := unparse(convert(f)@InputForm)
        if sayLength(s)$DisplayPackage > 50 then
          l := concat(title "AXIOM3D",l)
        else l := concat(title s,l)
      -- indicate draw style if necessary
      if not option?(l,"style" :: Symbol) then
        l := concat(style unparse(convert(f)@InputForm),l)
      -- create curve with functions as coordinates
      curve : PSCF := curve(makeFloatFunction(f,t),_
                            makeFloatFunction(g,t),_
                            makeFloatFunction(h,t))
      -- call 'draw'
      makeObject(curve,tSeg,l)

    makeObject(psc:PSC,tBind:BIND) ==
      makeObject(psc,tBind,nil())

    draw(psc:PSC,tBind:BIND,l:L DROP) ==
      import DROP
      -- obtain dependent variable and coordinate functions
      t := variable tBind; tSeg := segment tBind
      f := coordinate(psc,1); g := coordinate(psc,2); h := coordinate(psc,3)
      -- create title if necessary
      if not option?(l,"title" :: Symbol) then
        s:String := unparse(convert(f)@InputForm)
        if sayLength(s)$DisplayPackage > 50 then
          l := concat(title "AXIOM3D",l)
        else l := concat(title s,l)
      -- indicate draw style if necessary
      if not option?(l,"style" :: Symbol) then
        l := concat(style unparse(convert(f)@InputForm),l)
      -- create curve with functions as coordinates
      curve : PSCF := curve(makeFloatFunction(f,t),_
                            makeFloatFunction(g,t),_
                            makeFloatFunction(h,t))
      -- call 'draw'
      draw(curve,tSeg,l)

    draw(psc:PSC,tBind:BIND) ==
      draw(psc,tBind,nil())

------------------------------------------------------------------------
--                     3D - Surfaces  (given by formulas)
------------------------------------------------------------------------

--% Three Dimensional Function Plots

    makeObject(f:Ex,xBind:BIND,yBind:BIND,l:L DROP) ==
      import DROP
      -- create title if necessary
      if not option?(l,"title" :: Symbol) then
        s:String := unparse(convert(f)@InputForm)
        if sayLength(s)$DisplayPackage > 50 then
          l := concat(title "AXIOM3D",l)
        else l := concat(title s,l)
      -- indicate draw style if necessary
      if not option?(l,"style" :: Symbol) then
        l := concat(style unparse(convert(f)@InputForm),l)
      -- obtain dependent variables and their ranges
      x := variable xBind; xSeg := segment xBind
      y := variable yBind; ySeg := segment yBind
      -- call 'draw'
      makeObject(makeFloatFunction(f,x,y),xSeg,ySeg,l)

    makeObject(f:Ex,xBind:BIND,yBind:BIND) ==
      makeObject(f,xBind,yBind,nil())

    draw(f:Ex,xBind:BIND,yBind:BIND,l:L DROP) ==
      import DROP
      -- create title if necessary
      if not option?(l,"title" :: Symbol) then
        s:String := unparse(convert(f)@InputForm)
        if sayLength(s)$DisplayPackage > 50 then
          l := concat(title "AXIOM3D",l)
        else l := concat(title s,l)
      -- indicate draw style if necessary
      if not option?(l,"style" :: Symbol) then
        l := concat(style unparse(convert(f)@InputForm),l)
      -- obtain dependent variables and their ranges
      x := variable xBind; xSeg := segment xBind
      y := variable yBind; ySeg := segment yBind
      -- call 'draw'
      draw(makeFloatFunction(f,x,y),xSeg,ySeg,l)

    draw(f:Ex,xBind:BIND,yBind:BIND) ==
      draw(f,xBind,yBind,nil())

--% parametric surface

    makeObject(s:PSF,uBind:BIND,vBind:BIND,l:L DROP) ==
      import DROP
      f := coordinate(s,1); g := coordinate(s,2); h := coordinate(s,3)
      if not option?(l,"title" :: Symbol) then
        s:String := unparse(convert(f)@InputForm)
        if sayLength(s)$DisplayPackage > 50 then
          l := concat(title "AXIOM3D",l)
        else l := concat(title s,l)
      if not option?(l,"style" :: Symbol) then
        l := concat(style unparse(convert(f)@InputForm),l)
      u := variable uBind; uSeg := segment uBind
      v := variable vBind; vSeg := segment vBind
      surf : PSFF := surface(makeFloatFunction(f,u,v),_
                             makeFloatFunction(g,u,v),_
                             makeFloatFunction(h,u,v))
      makeObject(surf,uSeg,vSeg,l)

    makeObject(s:PSF,uBind:BIND,vBind:BIND) ==
      makeObject(s,uBind,vBind,nil())

    draw(s:PSF,uBind:BIND,vBind:BIND,l:L DROP) ==
      import DROP
      f := coordinate(s,1); g := coordinate(s,2); h := coordinate(s,3)
      -- create title if necessary
      if not option?(l,"title" :: Symbol) then
        s:String := unparse(convert(f)@InputForm)
        if sayLength(s)$DisplayPackage > 50 then
          l := concat(title "AXIOM3D",l)
        else l := concat(title s,l)
      -- indicate draw style if necessary
      if not option?(l,"style" :: Symbol) then
        l := concat(style unparse(convert(f)@InputForm),l)
      -- obtain dependent variables and their ranges
      u := variable uBind; uSeg := segment uBind
      v := variable vBind; vSeg := segment vBind
      -- create surface with functions as coordinates
      surf : PSFF := surface(makeFloatFunction(f,u,v),_
                             makeFloatFunction(g,u,v),_
                             makeFloatFunction(h,u,v))
      -- call 'draw'
      draw(surf,uSeg,vSeg,l)

    draw(s:PSF,uBind:BIND,vBind:BIND) ==
      draw(s,uBind,vBind,nil())

@
\section{package DRAWCURV TopLevelDrawFunctionsForAlgebraicCurves}
<<package DRAWCURV TopLevelDrawFunctionsForAlgebraicCurves>>=
)abbrev package DRAWCURV TopLevelDrawFunctionsForAlgebraicCurves
++ Author: Clifton J. Williamson
++ Date Created: 26 June 1990
++ Date Last Updated:  October 1991 by Jon Steinbach
++ Basic Operations: draw
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description: TopLevelDrawFunctionsForAlgebraicCurves provides top level 
++ functions for drawing non-singular algebraic curves.

TopLevelDrawFunctionsForAlgebraicCurves(R,Ex): Exports == Implementation where
  R  : Join(IntegralDomain, OrderedSet, RetractableTo Integer)
  Ex : FunctionSpace(R)

  ANY1  ==> AnyFunctions1
  DROP  ==> DrawOption
  EQ    ==> Equation
  F     ==> Float
  FRAC  ==> Fraction
  I     ==> Integer
  L     ==> List
  P     ==> Polynomial
  RN    ==> Fraction Integer
  SEG   ==> Segment
  SY    ==> Symbol
  VIEW2 ==> TwoDimensionalViewport

  Exports ==> with

    draw: (EQ Ex,SY,SY,L DROP) -> VIEW2
      ++ draw(f(x,y) = g(x,y),x,y,l) draws the graph of a polynomial
      ++ equation.  The list l of draw options must specify a region
      ++ in the plane in which the curve is to sketched.

  Implementation ==> add
    import ViewportPackage
    import PlaneAlgebraicCurvePlot
    import ViewDefaultsPackage
    import GraphicsDefaults
    import DrawOptionFunctions0
    import SegmentFunctions2(RN,F)
    import SegmentFunctions2(F,RN)
    import AnyFunctions1(L SEG RN)
    import DROP

    drawToScaleRanges: (SEG F,SEG F) -> L SEG F
    drawToScaleRanges(xVals,yVals) ==
      -- warning: assumes window is square
      xHi := hi xVals; xLo := lo xVals
      yHi := hi yVals; yLo := lo yVals
      xDiff := xHi - xLo; yDiff := yHi - yLo
      pad := abs(yDiff - xDiff)/2
      yDiff > xDiff =>
        [segment(xLo - pad,xHi + pad),yVals]
      [xVals,segment(yLo - pad,yHi + pad)]

    intConvert: R -> I
    intConvert r ==
      (nn := retractIfCan(r)@Union(I,"failed")) case "failed" =>
        error "draw: polynomial must have rational coefficients"
      nn :: I

    polyEquation: EQ Ex -> P I
    polyEquation eq ==
      ff := lhs(eq) - rhs(eq)
      (r := retractIfCan(ff)@Union(FRAC P R,"failed")) case "failed" =>
        error "draw: not a polynomial equation"
      rat := r :: FRAC P R
      retractIfCan(denom rat)@Union(R,"failed") case "failed" =>
        error "draw: non-constant denominator"
      map(intConvert,numer rat)$PolynomialFunctions2(R,I)

    draw(eq,x,y,l) ==
      -- obtain polynomial equation
      p := polyEquation eq
      -- extract ranges from option list
      floatRange := option(l,"rangeFloat" :: Symbol)
      ratRange := option(l,"rangeRat" :: Symbol)
      (floatRange case "failed") and (ratRange case "failed") =>
        error "draw: you must specify ranges for an implicit plot"
      ranges : L SEG RN := nil()             -- dummy value
      floatRanges : L SEG F := nil()         -- dummy value
      xRange : SEG RN := segment(0,0)        -- dummy value
      yRange : SEG RN := segment(0,0)        -- dummy value
      xRangeFloat : SEG F := segment(0,0)    -- dummy value
      yRangeFloat : SEG F := segment(0,0)    -- dummy value
      if not ratRange case "failed" then
        ranges := retract(ratRange :: Any)$ANY1(L SEG RN)
        #ranges ~= 2 => error "draw: you must specify two ranges"
        xRange := first ranges; yRange := second ranges
        xRangeFloat := map(convert(#1)@Float,xRange)@(SEG F)
        yRangeFloat := map(convert(#1)@Float,yRange)@(SEG F)
        floatRanges := [xRangeFloat,yRangeFloat]
      else
        floatRanges := retract(floatRange :: Any)$ANY1(L SEG F)
        #floatRanges ~= 2 =>
          error "draw: you must specify two ranges"
        xRangeFloat := first floatRanges
        yRangeFloat := second floatRanges
        xRange := map(retract(#1)@RN,xRangeFloat)@(SEG RN)
        yRange := map(retract(#1)@RN,yRangeFloat)@(SEG RN)
        ranges := [xRange,yRange]
      -- create curve plot
      acplot := makeSketch(p,x,y,xRange,yRange)
      -- process scaling information
      if toScale(l,drawToScale()) then
        scaledRanges := drawToScaleRanges(xRangeFloat,yRangeFloat)
        -- add scaled ranges to list of options
        l := concat(ranges scaledRanges,l)
      else
        -- add ranges to list of options
        l := concat(ranges floatRanges,l)
      -- process color information
      ptCol := pointColorPalette(l,pointColorDefault())
      crCol := curveColorPalette(l,lineColorDefault())
      -- draw
      drawCurves(listBranches acplot,ptCol,crCol,pointSizeDefault(),l)

@
\section{package DRAWPT TopLevelDrawFunctionsForPoints}
<<package DRAWPT TopLevelDrawFunctionsForPoints>>=
)abbrev package DRAWPT TopLevelDrawFunctionsForPoints
++ Author: Mike Dewar
++ Date Created: 24 May 1995
++ Date Last Updated: 25 November 1996
++ Basic Operations: draw
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description: TopLevelDrawFunctionsForPoints provides top level functions for 
++ drawing curves and surfaces described by sets of points.
 
TopLevelDrawFunctionsForPoints(): Exports == Implementation where

  DROP  ==> DrawOption
  L     ==> List
  SF    ==> DoubleFloat
  Pt    ==> Point SF
  VIEW2 ==> TwoDimensionalViewport
  VIEW3 ==> ThreeDimensionalViewport

  Exports ==> with
    draw: (L SF,L SF) -> VIEW2
      ++ draw(lx,ly) plots the curve constructed of points (x,y) for x
      ++ in \spad{lx} for y in \spad{ly}.
    draw: (L SF,L SF,L DROP) -> VIEW2
      ++ draw(lx,ly,l) plots the curve constructed of points (x,y) for x
      ++ in \spad{lx} for y in \spad{ly}.
      ++ The options contained in the list l of
      ++ the domain \spad{DrawOption} are applied.
    draw: (L Pt) -> VIEW2
      ++ draw(lp) plots the curve constructed from the list of points lp.
    draw: (L Pt,L DROP) -> VIEW2
      ++ draw(lp,l) plots the curve constructed from the list of points lp.
      ++ The options contained in the list l of the domain \spad{DrawOption}
      ++ are applied.
    draw: (L SF, L SF, L SF) -> VIEW3
      ++ draw(lx,ly,lz) draws the surface constructed by projecting the values
      ++ in the \axiom{lz} list onto the rectangular grid formed by the 
      ++ \axiom{lx X ly}.
    draw: (L SF, L SF, L SF, L DROP) -> VIEW3
      ++ draw(lx,ly,lz,l) draws the surface constructed by projecting the values
      ++ in the \axiom{lz} list onto the rectangular grid formed by the 
      ++ The options contained in the list l of the domain \spad{DrawOption}
      ++ are applied.

  Implementation ==> add

    draw(lp:L Pt,l:L DROP):VIEW2 ==
      makeViewport2D(makeGraphImage([lp])$GraphImage,l)$VIEW2

    draw(lp:L Pt):VIEW2 == draw(lp,[])

    draw(lx: L SF, ly: L SF, l:L DROP):VIEW2 ==
      draw([point([x,y])$Pt for x in lx for y in ly],l)

    draw(lx: L SF, ly: L SF):VIEW2 == draw(lx,ly,[])

    draw(x:L SF,y:L SF,z:L SF):VIEW3 == draw(x,y,z,[])

    draw(x:L SF,y:L SF,z:L SF,l:L DROP):VIEW3 ==
      m  : Integer := #x
      zero? m => error "No X values"
      n  : Integer := #y
      zero? n => error "No Y values"
      zLen : Integer := #z
      zLen ~= (m*n) => 
        zLen > (m*n) => error "Too many Z-values to fit grid"
        error "Not enough Z-values to fit grid"
      points : L L Pt := []
      for j in n..1 by -1 repeat
        row : L Pt := []
        for i in m..1 by -1 repeat
          zval := (j-1)*m+i
          row := cons(point([x.i,y.j,z.zval,z.zval]),row)
        points := cons(row,points)
      makeViewport3D(mesh points,l)

@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
--    - Redistributions of source code must retain the above copyright
--      notice, this list of conditions and the following disclaimer.
--
--    - Redistributions in binary form must reproduce the above copyright
--      notice, this list of conditions and the following disclaimer in
--      the documentation and/or other materials provided with the
--      distribution.
--
--    - Neither the name of The Numerical ALgorithms Group Ltd. nor the
--      names of its contributors may be used to endorse or promote products
--      derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>

<<package DRAWCFUN TopLevelDrawFunctionsForCompiledFunctions>>
<<package DRAW TopLevelDrawFunctions>>
<<package DRAWCURV TopLevelDrawFunctionsForAlgebraicCurves>>
<<package DRAWPT TopLevelDrawFunctionsForPoints>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}