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

\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra newpoint.spad}
\author{The Axiom Team}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{category PTCAT PointCategory}
<<category PTCAT PointCategory>>=
)abbrev category PTCAT PointCategory
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Operations: point, elt, setelt, copy, dimension, minIndex, maxIndex,
++ convert
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: 
++ References:
++ Description: PointCategory is the category of points in space which
++ may be plotted via the graphics facilities.  Functions are provided for
++ defining points and handling elements of points.
 
PointCategory(R:Ring) : Category == VectorCategory(R) with
  point: List R -> %
    ++ point(l) returns a point category defined by a list l of elements from 
    ++ the domain R.
  dimension: % -> PositiveInteger
    ++ dimension(s) returns the dimension of the point category s.
  convert: List R -> %
    ++ convert(l) takes a list of elements, l, from the domain Ring and 
    ++ returns the form of point category.
  cross: (%,%) -> %
      ++ cross(p,q) computes the cross product of the two points \spad{p}
      ++ and \spad{q}. Error if the p and q are not 3 dimensional
  extend : (%,List R) -> %
	++ extend(x,l,r) \undocumented

@
\section{domain POINT Point}
<<domain POINT Point>>=
)abbrev domain POINT Point
++ Description:
++ This domain implements points in coordinate space
 
Point(R:Ring) : Exports == Implementation where
  -- Domains for points, subspaces and properties of components in
  -- a subspace
 
  Exports ==> PointCategory(R)
 
  Implementation ==> Vector (R) add
    PI   ==> PositiveInteger

    point(l:List R):% ==
      pt := new(#l,R)
      for x in l for i in minIndex(pt).. repeat
        pt.i := x
      pt
    dimension p == (# p)::PI  -- Vector returns NonNegativeInteger...?
    convert(l:List R):% == point(l)
    cross(p0, p1) ==
      #p0 ~=3 or #p1~=3 => error "Arguments to cross must be three dimensional"      
      point [p0.2 * p1.3 - p1.2 * p0.3, _
             p1.1 * p0.3 - p0.1 * p1.3, _
             p0.1 * p1.2 - p1.1 * p0.2]
    extend(p,l) == concat(p,point l)

@
\section{domain COMPPROP SubSpaceComponentProperty}
<<domain COMPPROP SubSpaceComponentProperty>>=
)abbrev domain COMPPROP SubSpaceComponentProperty
++ Description:
++ This domain implements some global properties of subspaces.
 
SubSpaceComponentProperty() : Exports == Implementation where
 
  O ==> OutputForm
  I    ==> Integer
  PI   ==> PositiveInteger
  NNI  ==> NonNegativeInteger
  L    ==> List
  B    ==> Boolean
 
  Exports ==> SetCategory with
    new     : () -> %
	++ new() \undocumented
    closed? : % -> B
	++ closed?(x) \undocumented
    solid?  : % -> B
	++ solid?(x) \undocumented
    close   : (%,B) -> B
	++ close(x,b) \undocumented
    solid   : (%,B) -> B
	++ solid(x,b) \undocumented
    copy    : % -> %
	++ copy(x) \undocumented
 
  Implementation ==> add
    import B
    Rep := Record(closed:B, solid:B)
    closed? p == p.closed
    solid? p == p.solid
    close(p,b) == p.closed := b
    solid(p,b) == p.solid := b
    new() == [false,false]
    copy p ==
      annuderOne := new()
      close(annuderOne,closed? p)
      solid(annuderOne,solid? p)
      annuderOne
    coerce p ==
      hconcat(["Component is "::O,
              (closed? p => ""::O; "not "::O),"closed, "::O, _
              (solid? p => ""::O; "not "::O),"solid"::O ])

@
\section{domain SUBSPACE SubSpace}
<<domain SUBSPACE SubSpace>>=
)abbrev domain SUBSPACE SubSpace
++ Description:
++ This domain \undocumented
SubSpace(n:PI,R:Ring) : Exports == Implementation where
  -- n is the dimension of the subSpace
  -- The SubSpace domain is implemented as a tree. The root of the tree
  -- is the only node in which the field dataList - which points to a
  -- list of points over the ring, R - is defined. The children of the
  -- root are the top level components of the SubSpace (in 2D, these
  -- would be separate curves; in 3D, these would be separate surfaces).
  -- The pt field is only defined in the leaves.
  -- By way of example, consider a three dimensional subspace with
  -- two components - a three by three grid and a sphere. The internal
  -- representation of this subspace is a tree with a depth of three.
  -- The root holds a list of all the points used in the subspace (so,
  -- if the grid and the sphere share points, the shared points would not
  -- be represented redundantly but would be referenced by index).
  -- The root, in this case, has two children - the first points to the
  -- grid component and the second to the sphere component. The grid child
  -- has four children of its own - a 3x3 grid has 4 endpoints - and each
  -- of these point to a list of four points. To see it another way, the 
  -- grid (child of the root) holds a list of line components which, when
  -- placed one above the next, forms a grid. Each of these line components
  -- is a list of points.
  -- Points could be explicitly added to subspaces at any level. A path
  -- could be given as an argument to the addPoint() function. It is a list
  -- of NonNegativeIntegers and refers, in order, to the n-th child of the
  -- current node. For example,
  --                     addPoint(s,[2,3],p)
  -- would add the point p to the subspace s by going to the second child of
  -- the root and then the third child of that node. If the path does extend
  -- to the full depth of the tree, nodes are automatically added so that 
  -- the tree is of constant depth down any path. By not specifying the full
  -- path, new components could be added - e.g. for s from SubSpace(3,Float)
  --                     addPoint(s,[],p)
  -- would create a new child to the root (a new component in N-space) and
  -- extend a path to a leaf of depth 3 that points to the data held in p.
  -- The subspace s would now have a new component which has one child
  -- which, in turn, has one child (the leaf). The new component is then a 
  -- point.
 
  I    ==> Integer
  PI   ==> PositiveInteger
  NNI  ==> NonNegativeInteger
  L    ==> List
  B    ==> Boolean
  POINT ==> Point(R)
  PROP ==> SubSpaceComponentProperty()
  S ==> String
  O ==> OutputForm
  empty ==> nil  -- macro to ease conversion to new aggcat.spad
 
  Exports ==> SetCategory with
    leaf?          : % -> B
	++ leaf?(x) \undocumented
    root?          : % -> B
	++ root?(x) \undocumented
    internal?      : % -> B
	++ internal?(x) \undocumented
    new            : () -> %
	++ new() \undocumented
    subspace       : () -> %
	++ subspace() \undocumented
    birth          : % -> %       -- returns a pointer to the baby
	++ birth(x) \undocumented
    child          : (%,NNI) -> %
	++ child(x,n) \undocumented
    children       : % -> List %
	++ children(x) \undocumented
    numberOfChildren: % -> NNI
	++ numberOfChildren(x) \undocumented
    shallowCopy    : % -> %
	++ shallowCopy(x) \undocumented
    deepCopy       : % -> %
	++ deepCopy(x) \undocumented
    merge          : (%,%) -> %
      ++ merge(s1,s2) the subspaces s1 and s2 into a single subspace.
    merge          : List % -> %
      ++ merge(ls) a list of subspaces, ls, into one subspace.
    separate       : % -> List %
      ++ separate(s) makes each of the components of the \spadtype{SubSpace},
      ++ s, into a list of separate and distinct subspaces and returns
      ++ the list.
    addPoint       : (%,List NNI,POINT) -> %
      ++ addPoint(s,li,p) adds the 4 dimensional point, p, to the 3 
      ++ dimensional subspace, s. The list of non negative integers, li, 
      ++ dictates the path to follow, or, to look at it another way, 
      ++ points to the component in which the point is to be added.  It's 
      ++ length should range from 0 to \spad{n - 1} where n is the dimension 
      ++ of the subspace. If the length is \spad{n - 1}, then a specific 
      ++ lowest level component is being referenced.  If it is less than 
      ++ \spad{n - 1}, then some higher level component (0 indicates top 
      ++ level component) is being referenced and a component of that level 
      ++ with the desired point is created.  The subspace s is returned
      ++ with the additional point.
    addPoint2      : (%,POINT) -> %
      ++ addPoint2(s,p) adds the 4 dimensional point, p, to the 3 
      ++ dimensional subspace, s. 
      ++ The subspace s is returned with the additional point.
    addPointLast   : (%,%,POINT, NNI) -> %
      ++ addPointLast(s,s2,li,p) adds the 4 dimensional point, p, to the 3 
      ++ dimensional subspace, s. s2 point to the end of the subspace
      ++ s. n is the path in the s2 component.
      ++ The subspace s is returned with the additional point.
    modifyPoint    : (%,List NNI,POINT) -> %
      ++ modifyPoint(s,li,p) replaces an existing point in the 3 dimensional
      ++ subspace, s, with the 4 dimensional point, p.  The list of non 
      ++ negative integers, li, dictates the path to follow, or, to look at 
      ++ it another way, points to the component in which the existing point 
      ++ is to be modified.  An error message occurs if s is empty, otherwise
      ++ the subspace s is returned with the point modification.
    addPoint       : (%,List NNI,NNI) -> %
      ++ addPoint(s,li,i) adds the 4 dimensional point indicated by the
      ++ index location, i, to the 3 dimensional subspace, s. The list of 
      ++ non negative integers, li, dictates the path to follow, or, to 
      ++ look at it another way, points to the component in which the point 
      ++ is to be added.  It's length should range from 0 to \spad{n - 1} 
      ++ where n is the dimension of the subspace. If the length is 
      ++ \spad{n - 1}, then a specific lowest level component is being 
      ++ referenced.  If it is less than \spad{n - 1}, then some higher 
      ++ level component (0 indicates top level component) is being 
      ++ referenced and a component of that level with the desired point 
      ++ is created.  The subspace s is returned with the additional point.
    modifyPoint    : (%,List NNI,NNI) -> %
      ++ modifyPoint(s,li,i) replaces an existing point in the 3 dimensional
      ++ subspace, s, with the 4 dimensional point indicated by the index 
      ++ location, i.  The list of non negative integers, li, dictates 
      ++ the path to follow, or, to look at it another way, points to the 
      ++ component in which the existing point is to be modified.  An error
      ++ message occurs if s is empty, otherwise the subspace s is returned
      ++ with the point modification.
    addPoint       : (%,POINT) -> NNI
      ++ addPoint(s,p) adds the point, p, to the 3 dimensional subspace, s,
      ++ and returns the new total number of points in s.
    modifyPoint    : (%,NNI,POINT) -> %
      ++ modifyPoint(s,ind,p) modifies the point referenced by the index
      ++ location, ind, by replacing it with the point, p in the 3 dimensional
      ++ subspace, s.  An error message occurs if s is empty, otherwise the
      ++ subspace s is returned with the point modification.
 
    closeComponent : (%,List NNI,B)     -> %
      ++ closeComponent(s,li,b) sets the property of the component in the
      ++ 3 dimensional subspace, s, to be closed if b is true, or open if 
      ++ b is false.  The list of non negative integers, li, dictates the 
      ++ path to follow, or, to look at it another way, points to the 
      ++ component whose closed property is to be set.  The subspace, s, 
      ++ is returned with the component property modification.
    defineProperty : (%,List NNI,PROP)  -> %
      ++ defineProperty(s,li,p) defines the component property in the
      ++ 3 dimensional subspace, s, to be that of p, where p is of the
      ++ domain \spadtype{SubSpaceComponentProperty}.  The list of non 
      ++ negative integers, li, dictates the path to follow, or, to look 
      ++ at it another way, points to the component whose property is 
      ++ being defined.  The subspace, s, is returned with the component 
      ++ property definition.
    traverse       : (%,List NNI) -> %
      ++ traverse(s,li) follows the branch list of the 3 dimensional 
      ++ subspace, s, along the path dictated by the list of non negative
      ++ integers, li, which points to the component which has been
      ++ traversed to.  The subspace, s, is returned, where s is now
      ++ the subspace pointed to by li.
    extractPoint    : %  -> POINT
      ++ extractPoint(s) returns the point which is given by the current 
      ++ index location into the point data field of the 3 dimensional 
      ++ subspace s.
    extractIndex    : % -> NNI
      ++ extractIndex(s) returns a non negative integer which is the current 
      ++ index of the 3 dimensional subspace s.
    extractClosed   : %  -> B
      ++ extractClosed(s) returns the \spadtype{Boolean} value of the closed 
      ++ property for the indicated 3 dimensional subspace s.  If the 
      ++ property is closed, \spad{True} is returned, otherwise \spad{False}
      ++ is returned.
    extractProperty : %  -> PROP
      ++ extractProperty(s) returns the property of domain
      ++ \spadtype{SubSpaceComponentProperty} of the indicated 3 dimensional
      ++ subspace s.
    level           : % -> NNI
      ++ level(s) returns a non negative integer which is the current 
      ++ level field of the indicated 3 dimensional subspace s.
    parent          : % -> %
      ++ parent(s) returns the subspace which is the parent of the indicated
      ++ 3 dimensional subspace s.  If s is the top level subspace an error
      ++ message is returned.
    pointData       : % -> L POINT
      ++ pointData(s) returns the list of points from the point data field
      ++ of the 3 dimensional subspace s.
 
  Implementation ==> add
    import String()

    Rep := Record(pt:POINT, index:NNI, property:PROP, _
                   childrenField:List %, _
                   lastChild: List %, _ 
                   levelField:NNI, _
                   pointDataField:L POINT, _
                   lastPoint: L POINT, _
                   noPoints: NNI, _
                   noChildren: NNI, _
                   parentField:List %) -- needn't be list but...base case?

    TELLWATT : String := "Non-null list: Please inform Stephen Watt"
 
    leaf? space ==  empty? children space
    root? space == (space.levelField = 0$NNI)
    internal? space == not (root? space and leaf? space)
 
    new() ==
      [point(empty())$POINT,0,new()$PROP,empty(),empty(),0,_
                                     empty(),empty(),0,0,empty()]
    subspace() == new()
 
    birth momma ==
      baby := new()
      baby.levelField := momma.levelField+1
      baby.parentField := [momma]
      if not empty?(lastKid := momma.lastChild) then
        not empty? rest lastKid => error TELLWATT
      if empty? lastKid
        then
          momma.childrenField := [baby]
          momma.lastChild := momma.childrenField
          momma.noChildren := 1
        else
          setrest_!(lastKid,[baby])
          momma.lastChild := rest lastKid
          momma.noChildren := momma.noChildren + 1
      baby
 
    child(space,num) ==
      space.childrenField.num

    children space == space.childrenField
    numberOfChildren space == space.noChildren
 
    shallowCopy space ==
      node := new()
      node.pt         := space.pt
      node.index      := space.index
      node.property   := copy(space.property)
      node.levelField := space.levelField
      node.parentField := nil()
      if root? space then
        node.pointDataField := copy(space.pointDataField)
        node.lastPoint := tail(node.pointDataField)
        node.noPoints := space.noPoints
      node

    deepCopy space ==
      node := shallowCopy(space)
      leaf? space => node
      for c in children space repeat
        cc := deepCopy c
        cc.parentField := [node]
        node.childrenField := cons(cc,node.childrenField)
      node.childrenField := reverse_!(node.childrenField)
      node.lastChild := tail node.childrenField
      node

    merge(s1,s2) ==
       ------------------ need to worry about reindexing s2 & parentField
      n1 : Rep := deepCopy s1
      n2 : Rep := deepCopy s2
      n1.childrenField := append(children n1,children n2)
      n1

    merge listOfSpaces ==
       ------------------ need to worry about reindexing & parentField
      empty? listOfSpaces => error "empty list passed as argument to merge"
           -- notice that the properties of the first subspace on the
           -- list are the ones that are inherited...hmmmm...
      space := deepCopy first listOfSpaces
      for s in rest listOfSpaces repeat
           -- because of the initial deepCopy, above, everything is
           -- deepCopied to be consistent...more hmmm...
        space.childrenField := append(space.childrenField,[deepCopy c for c in s.childrenField])
      space

    separate space ==
       ------------------ need to worry about reindexing & parentField
      spaceList := empty()
      for s in space.childrenField repeat
        spc:=shallowCopy space
        spc.childrenField:=[deepCopy s]
        spaceList := cons(spc,spaceList)
      spaceList
 
    addPoint(space:%,path:List NNI,point:POINT) ==
      if not empty?(lastPt := space.lastPoint) then
        not empty? rest lastPt => error TELLWATT
      if empty? lastPt
        then
          space.pointDataField := [point]
          space.lastPoint := space.pointDataField
        else
          setrest_!(lastPt,[point])
          space.lastPoint := rest lastPt
      space.noPoints := space.noPoints + 1
      which := space.noPoints
      node := space
      depth : NNI := 0
      for i in path repeat
        node := child(node,i)
        depth := depth + 1
      for more in depth..(n-1) repeat
        node := birth node
      node.pt := point      -- will be obsolete field
      node.index := which
      space

    addPoint2(space:%,point:POINT) ==
      if not empty?(lastPt := space.lastPoint) then
        not empty? rest lastPt => error TELLWATT
      if empty? lastPt
        then
          space.pointDataField := [point]
          space.lastPoint := space.pointDataField
        else
          setrest_!(lastPt,[point])
          space.lastPoint := rest lastPt
      space.noPoints := space.noPoints + 1
      which := space.noPoints
      node := space
      depth : NNI := 0
      node := birth node
      first := node
      for more in 1..n-1 repeat
        node := birth node
      node.pt := point      -- will be obsolete field
      node.index := which
      first

    addPointLast(space:%,node:%, point:POINT, depth:NNI) ==
      if not empty?(lastPt := space.lastPoint) then
        not empty? rest lastPt => error TELLWATT
      if empty? lastPt
        then
          space.pointDataField := [point]
          space.lastPoint := space.pointDataField
        else
          setrest_!(lastPt,[point])
          space.lastPoint := rest lastPt
      space.noPoints := space.noPoints + 1
      which := space.noPoints
      if depth = 2 then node := child(node, 2)
      for more in depth..(n-1) repeat
        node := birth node
      node.pt := point      -- will be obsolete field
      node.index := which
      node -- space

    addPoint(space:%,path:List NNI,which:NNI) ==
      node := space
      depth : NNI := 0
      for i in path repeat
        node := child(node,i)
        depth := depth + 1
      for more in depth..(n-1) repeat
        node := birth node
      node.pt := space.pointDataField.which   -- will be obsolete field
      node.index := which
      space

    addPoint(space:%,point:POINT) ==
      root? space =>
        if not empty?(lastPt := space.lastPoint) then
          not empty? rest lastPt => error TELLWATT
        if empty? lastPt
          then
            space.pointDataField := [point]
            space.lastPoint := space.pointDataField
          else
            setrest_!(lastPt,[point])
            space.lastPoint := rest lastPt
        space.noPoints := space.noPoints + 1
      error "You need to pass a top level SubSpace (level should be zero)"
 
    modifyPoint(space:%,path:List NNI,point:POINT) ==
      if not empty?(lastPt := space.lastPoint) then
        not empty? rest lastPt => error TELLWATT
      if empty? lastPt
        then
          space.pointDataField := [point]
          space.lastPoint := space.pointDataField
        else
          setrest_!(lastPt,[point])
          space.lastPoint := rest lastPt
      space.noPoints := space.noPoints + 1
      which := space.noPoints
      node := space
      for i in path repeat
        node := child(node,i)
      node.pt := point       ---------- will be obsolete field
      node.index := which
      space

    modifyPoint(space:%,path:List NNI,which:NNI) ==
      node := space
      for i in path repeat
        node := child(node,i)
      node.pt := space.pointDataField.which       ---------- will be obsolete field
      node.index := which
      space

    modifyPoint(space:%,which:NNI,point:POINT) ==
      root? space =>
        space.pointDataField.which := point
        space
      error "You need to pass a top level SubSpace (level should be zero)"
 
    closeComponent(space,path,val) ==
      node := space
      for i in path repeat
        node := child(node,i)
      close(node.property,val)
      space

    defineProperty(space,path,prop) ==
      node := space
      for i in path repeat
        node := child(node,i)
      node.property := prop
      space
 
    traverse(space,path) ==
      for i in path repeat space := child(space,i)
      space

    extractPoint space ==
      node := space
      while not root? node repeat node := parent node
      (node.pointDataField).(space.index)
    extractIndex space == space.index
    extractClosed space == closed? space.property
    extractProperty space == space.property
 
    parent space ==
      empty? space.parentField => error "This is a top level SubSpace - it does not have a parent"
      first space.parentField
    pointData space == space.pointDataField
    level space == space.levelField
    s1 = s2 ==
        ------------ extra checks for list of point data
      (leaf? s1 and leaf? s2) =>
        (s1.pt = s2.pt) and (s1.property = s2.property) and (s1.levelField = s2.levelField)
      -- note that the ordering of children is important
      #s1.childrenField ~= #s2.childrenField => false
      and/[c1 = c2 for c1 in s1.childrenField for c2 in s2.childrenField]
       and (s1.property = s2.property) and (s1.levelField = s2.levelField)
    coerce(space:%):O ==
      hconcat([n::O,"-Space with depth of "::O,                     _
              (n - space.levelField)::O," and "::O,(s:=(#space.childrenField))::O,  _
              (s=1 => " component"::O;" components"::O)])

@
\section{package PTPACK PointPackage}
<<package PTPACK PointPackage>>=
)abbrev package PTPACK PointPackage
++ Description:
++ This package \undocumented
PointPackage(R:Ring):Exports == Implementation where
 
  POINT ==> Point(R)
  I    ==> Integer
  PI   ==> PositiveInteger
  NNI  ==> NonNegativeInteger
  L    ==> List
  B    ==> Boolean
 
  Exports == with
    xCoord       : POINT -> R
      ++ xCoord(pt) returns the first element of the point, pt,
      ++ although no assumptions are made as to the coordinate
      ++ system being used.  This function is defined for the
      ++ convenience of the user dealing with a Cartesian
      ++ coordinate system.
    yCoord       : POINT -> R
      ++ yCoord(pt) returns the second element of the point, pt,
      ++ although no assumptions are made as to the coordinate
      ++ system being used.  This function is defined for the
      ++ convenience of the user dealing with a Cartesian
      ++ coordinate system.
    zCoord       : POINT -> R
      ++ zCoord(pt) returns the third element of the point, pt,
      ++ although no assumptions are made as to the coordinate
      ++ system being used.  This function is defined for the
      ++ convenience of the user dealing with a Cartesian
      ++ or a cylindrical coordinate system.
    rCoord       : POINT -> R
      ++ rCoord(pt) returns the first element of the point, pt,
      ++ although no assumptions are made as to the coordinate
      ++ system being used.  This function is defined for the
      ++ convenience of the user dealing with a spherical
      ++ or a cylindrical coordinate system.
    thetaCoord   : POINT -> R
      ++ thetaCoord(pt) returns the second element of the point, pt,
      ++ although no assumptions are made as to the coordinate
      ++ system being used.  This function is defined for the
      ++ convenience of the user dealing with a spherical
      ++ or a cylindrical coordinate system.
    phiCoord     : POINT -> R
      ++ phiCoord(pt) returns the third element of the point, pt,
      ++ although no assumptions are made as to the coordinate
      ++ system being used.  This function is defined for the
      ++ convenience of the user dealing with a spherical
      ++ coordinate system.
    color        : POINT -> R
      ++ color(pt) returns the fourth element of the point, pt, 
      ++ although no assumptions are made with regards as to
      ++ how the components of higher dimensional points are
      ++ interpreted.  This function is defined for the
      ++ convenience of the user using specifically, color
      ++ to express a fourth dimension.
    hue : POINT -> R
      ++ hue(pt) returns the third element of the two dimensional point, pt,
      ++ although no assumptions are made with regards as to how the 
      ++ components of higher dimensional points are interpreted. This 
      ++ function is defined for the convenience of the user using 
      ++ specifically, hue to express a third dimension.
    shade : POINT -> R
      ++ shade(pt) returns the fourth element of the two dimensional 
      ++ point, pt, although no assumptions are made with regards as to 
      ++ how the components of higher dimensional points are interpreted.
      ++ This function is defined for the convenience of the user using 
      ++ specifically, shade to express a fourth dimension.
 
      -- 2D and 3D extraction of data
  Implementation ==> add
 
    xCoord p == elt(p,1)
    yCoord p == elt(p,2)
    zCoord p == elt(p,3)
    rCoord p == elt(p,1)
    thetaCoord p == elt(p,2)
    phiCoord p == elt(p,3)
    color p == 
      #p > 3 => p.4
      p.3
    hue p == elt(p,3)       -- 4D points in 2D using extra dimensions for palette information
    shade p == elt(p,4)     -- 4D points in 2D using extra dimensions for palette information

@
\section{package PTFUNC2 PointFunctions2}
<<package PTFUNC2 PointFunctions2>>=
)abbrev package PTFUNC2 PointFunctions2
++ Description:
++ This package \undocumented 
PointFunctions2(R1:Ring,R2:Ring):Exports == Implementation where
 
  Exports == with
    map : ((R1->R2),Point(R1)) -> Point(R2)
	++ map(f,p) \undocumented
 
  Implementation ==> add
    import Point(R1)
    import Point(R2)
 
    map(mapping,p) ==
      point([mapping p.(i::PositiveInteger) for i in minIndex(p)..maxIndex(p)])$Point(R2)

@
\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>>

<<category PTCAT PointCategory>>
<<domain POINT Point>>
<<domain COMPPROP SubSpaceComponentProperty>>
<<domain SUBSPACE SubSpace>>
<<package PTPACK PointPackage>>
<<package PTFUNC2 PointFunctions2>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}