Finite lists of 3D points

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.


plot::PointList3d(pts, <a = amin .. amax>, options)
plot::PointList3d(M3d, <a = amin .. amax>, options)


plot::PointList3d holds lists of points in 3D.

These types are containers for a (large) finite number of points. They let you avoid constructing large numbers of objects of type plot::Point3d, for two reasons. First, the point types have non-negligible overhead and constructing and plotting a large number of them (say, five thousand) takes more time than plotting the same number of points in a single container object. Second, and this may be even more important, having five thousand points in the object browser takes a significant amount of memory and is not as lucid as having a single point list displayed there.

The attribute Points3d is displayed in the inspector in the user interface only for short lists.

plot::PointList2d, PointList3d internally use lists for storing the points. It is therefore not advisable to add a large number of points one-by-one. See Example 2 for a better method of collecting data.

If you specify the color of one point, you must specify the colors of all other points in the list. See Example 3.


AttributePurposeDefault Value
AffectViewingBoxinfluence of objects on the ViewingBox of a sceneTRUE
Colorthe main colorRGB::MidnightBlue
Framesthe number of frames in an animation50
Legendmakes a legend entry 
LegendTextshort explanatory text for legend 
LegendEntryadd this object to the legend?FALSE
Namethe name of a plot object (for browser and legend) 
ParameterEndend value of the animation parameter 
ParameterNamename of the animation parameter 
ParameterBegininitial value of the animation parameter 
ParameterRangerange of the animation parameter 
Points3dlist of 3D points 
PointSizethe size of points1.5
PointColorthe color of pointsRGB::MidnightBlue
PointStylethe presentation style of pointsFilledCircles
TimeEndend time of the animation10.0
TimeBeginstart time of the animation0.0
TimeRangethe real time span of an animation0.0 .. 10.0
Titleobject title 
TitleFontfont of object titles[" sans-serif ", 11]
TitlePositionposition of object titles 
TitleAlignmenthorizontal alignment of titles w.r.t. their coordinatesCenter
TitlePositionXposition of object titles, x component 
TitlePositionYposition of object titles, y component 
TitlePositionZposition of object titles, z component 
VisibleAfterobject visible after this time value 
VisibleBeforeobject visible until this time value 
VisibleFromToobject visible during this time range 
VisibleAfterEndobject visible after its animation time ended?TRUE
VisibleBeforeBeginobject visible before its animation time starts?TRUE


Example 1

plot::PointList3d provides a basic form of scatter plot:

plot(plot::PointList3d([[1,1,1], [1,2,2], [1,3,2], [1,3,4],
                        [2,1,1], [2,2,3], [2,3.5, 4]],

We can use this, for example, to get a visual test of random number generators:

r := frandom(0):
plot(plot::PointList3d([[r(), r(), r()] $ i=1..10000])):

r := random(10^10)*1e-10:
plot(plot::PointList3d([[r(), r(), r()] $ i=1..10000])):

frandom and random fill the cube nicely, without noticeable patterns. The following generator, however, should probably not be used:

randseed := 12345:
r := proc()
       randseed := (randseed * 17 + 8) mod 10^10:
       1e-10 * randseed;
plot(plot::PointList3d([[r(), r(), r()] $ i=1..10000])):

Example 2

The following iteration leads to the so-called Hénon attractor (from chaos theory):

c1 := 1.4:
c2 := 0.3:
henon_iter := (x, y) -> [c1*x^2+y-1, c2*x]:

We start at (0, 0), let hundred iteration cycles pass by (to only plot the attractor) and then collect the next three thousand points:

[x, y] := [0, 0]:
for i from 1 to 100 do
  [x, y] := henon_iter(x, y);
data := {}:
for i from 1 to 3000 do
  [x, y] := henon_iter(x, y);
  data := data union {[x, y]};

Note that we collected the data in a set, because adding elements to a set is a fast operation, unlike changing the length of a list, and we don't care for the order in which points were reached. To plot the data, we must convert it to a list first:

data := coerce(data, DOM_LIST):

We'd like to invite you to experiment with different values of c1 and c2 and see how they change the resulting image.

Example 3

plot::PointList2d and plot::PointList3d allow you to specify the colors of the points. For example, the following list contains two points. When you plot this list, the first point appears in red, and the second point appears in green:

Coords := [[3, 4, RGB::Red], [5, 5, RGB::Green]];
plotCoords := plot::PointList2d(Coords):
plot(plotCoords,  PointSize=5)

If you specify the color of one point, you must also specify the colors of all other points in the list:

Coords := [[3, 4, RGB::Red], [5, 5]];
plotCoords := plot::PointList2d(Coords)

Error: The attribute 'Points2d' in the 'PointList2d' object must be a list of lists of two expressions and an optional color value. [plot]

Example 4

(Feigenbaum's period doubling route to chaos)

We consider the iteration xn + 1 = fp(xn) where is the "logistic map" with a parameter p. The iteration map fp maps the interval [0, 1] to itself for 0 ≤ p ≤ 4. For small values of p, the sequence (xn) has a finite number of accumulation points that are visited cyclically. Increasing p, the accumulation points split into 2 separate accumulation points for certain critical values of p ("period doubling"). For , there are infinitely many accumulation points and the sequence (xn) behaves chaotically.

We wish to visualize the accumulation points as functions of p ("Feigenbaum diagram").

For P closely spaced values of p, we construct the sequence (xn) starting with x0 = 0.5. We ignore the first N values, expecting that the next M values cycle over the accumulation points. These points are added to a list plotdata that is finally fed into a PointList2d for plotting:

f:= (p, x) -> p*x*(1-x):

 P:= 500: // number of steps in p direction
N:= 200: // transitional steps before we are close to the cycle
M:= 300: // maximal number of points on the cycle

 pmin:= 2.8: // Consider p between
pmax:= 4.0: // pmin and pmax
plotdata:= [ ]:
for p in [pmin + i*(pmax - pmin)/P $ i = 0..P] do
    // First, do N iterations to drive the 
    // point x towards the limit cycle 
    x:= 0.5:
    for i from 1 to N do
      x:= f(p, x):

     // consider the next M iterates and use them as plot data:
    xSequence:= table():
    xSequence[1]:= x;
    for i from 2 to M do
        x:= f(p, x):
        if abs(x - xSequence[1]) < 10^(-5) then
           // We are back at the beginning of the cycle;
           // the points will repeat. Go to the next p.
           xSequence[i]:= x;
    plotdata:= plotdata . [[p, rhs(x)] $ x in xSequence];

                       PointColor = RGB::Black,
                       PointSize = 0.5*unit::mm)):

delete f, P, N, M, pmin, pmax, plotdata, x, xSequence, i;



A list of points. A point must not be of type plot::Point2d or plot::Point3d, respectively. In 2D, each point must be a list of two real-valued expressions (the coordinates) and an optional RGB color. In 3D, each point mut be a list of three expressions (the coordinates) and an optional RGB or RGBa color. The lists specifying the points and the colors must all have the same length.

pts is equivalent to the attributes Points2d, Points3d.


An array or a matrix with 3 columns. Each row provides the coordinates of one point.

M3d is equivalent to the attribute Points3d.


Animation parameter, specified as a = amin..amax, where amin is the initial parameter value, and amax is the final parameter value.

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