Efficient dynamic drawing

Question

So for example I have this code:

points = {}; k = 0;
Dynamic@k
Dynamic@Graphics[Point@points]

x = .1; y = .3; K = .9;
While[True,
  k++;
  {x, y} = {FractionalPart[x + K y], FractionalPart[x]};
  AppendTo[points, {x, y}];
]

The problem is that the list of points is growing and it slows down with time. What I want is to be able to add a point (or another primitive) directly on the dynamic graph without storing unnecessary chunks of data.

Answer 1

As it was mentioned in the comments: your loop is infinite, which eventually will cause the slowdown of any computation that accumulates data in the memory. The important rule of thumb for Dynamic updating is: only update when necessary and only update what is necessary. Accordingly, you can speed up the performance of the dynamic drawing by wrapping only points and k in Dynamic. By this way, only the list of points is updated (and the label) and Mathematica does not have to redraw the whole Graphics object again and again (which involves a lot of extra computation).

points = {};
Graphics[Point@Dynamic@points, PlotLabel -> Dynamic@k]

x = .1; y = .3; K = .9;
Do[
 {x, y} = {FractionalPart[x + K y], FractionalPart[x]};
 points = Append[points, {x, y}];
 , {k, 20000}]

Starting from @belisarius' comment, I came up with a more economic version (time scales linearly with k). If one does not have to keep all the points we can apply a reasonable resolution to bin the ranges and saving new datapoints in a matrix, overwriting previous data.

resolution = 256;  (* divide the (0,1) range into 256 bins *)
array = Array[0 &, {resolution, resolution}];

Dynamic@ArrayPlot[array, PlotLabel -> Dynamic@k]

x = .1; y = .3; K = .9;
Do[
 {x, y} = {FractionalPart[x + K y], FractionalPart[x]};
 array = ReplacePart[array, (Min[#, resolution] & /@ (Round[{x, y}*resolution] + 1)) -> 1],
 {k, 1000000}]