plot::PointList2d – finite lists of 2D points

plot::PointList3d – finite lists of 3D points

plot::PointList2d and plot::PointList3d hold lists of points in 2D or 3D.

→ Examples

Calls:

plot::PointList2d(pts, <a = amin .. amax>, Options)

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

plot::PointList2d( $`M_{2d}`$ , <a = amin .. amax>, Options)

plot::PointList3d( $`M_{3d}`$ , <a = amin .. amax>, Options)

Parameters:

pts:	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.
$`M_{2d}`$ :	An array or a matrix with 2 columns. Each row provides the coordinates of one point. $`M_{2d}`$ is equivalent to the attribute Points2d.
$`M_{3d}`$ :	An array or a matrix with 3 columns. Each row provides the coordinates of one point. $`M_{3d}`$ is equivalent to the attribute Points3d.

Details:

These types are containers for a (large) finite number of points. They let you avoid constructing large numbers of objects of type plot::Point2d and 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 attributes Points2d and Points3d are 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.

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]],
PointSize=5))

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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])):

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

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

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

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

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):
plot(plot::PointList2d(data))

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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)

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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: expecting a list of lists of 2 expressions and an optional color value for attribute 'Points2d' in PointList2d object [plot]

Example 4

(Feigenbaum's period doubling route to chaos)

We consider the iteration where is the “logistic map” with a parameter . The iteration map maps the interval to itself for . For small values of , the sequence has a finite number of accumulation points that are visited cyclically. Increasing , the accumulation points split into 2 separate accumulation points for certain critical values of (“period doubling”). For , there are infinitely many accumulation points and the sequence behaves chaotically.

We wish to visualize the accumulation points as functions of (“Feigenbaum diagram”).

For closely spaced values of , we construct the sequence starting with . We ignore the first values, expecting that the next 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):
    end_for:

     // 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.
           break;
        else
           xSequence[i]:= x;
        end_if;
    end_for:
    plotdata:= plotdata . [[p, rhs(x)] $ x in xSequence];
end_for:

plot(plot::PointList2d(plotdata,
                       PointColor = RGB::Black,
                       PointSize = 0.5*unit::mm)):

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

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