How to replace data in a list (tree) in a functional Mathematica way?

Question

I have list containing 2D points in various groupings, i.e. points, lists of points (figures), lists of lists of points (patterns) and finally a list of patterns. The list also contains other data, i.e. Colors and other decorators.

For (a simple!) example:

fig1 = {{0,0}, {1,0}, {0,1}}
fig2 = {{2,0}, {3,0}, {2,1}}
fig3 = {{4,0}, {5,0}, {4,1}}
li = {{fig1, Red}, {{fig2, fig3}, Blue}}

Now I would like to replace the points in the list with the result of a function applied to that point, for example a matrix multiplication. For the example we can use a simple function. f[p:pnt2D] := 2p or multiplication with IdentityMatrix[2] will do to illustrate.

I am looking for a typical functional Mathematica way to attack this problem. Extract all the points from the list, apply the function and replace with the outcome. In such a way that it works for all sorts of (huge) lists.

How to replace data in a list (tree) in a typical Mathematica way, preferably using clear code?

Answer 1

ReplaceAll or /. is the function you're looking for:

li /. p : {_?NumericQ, _?NumericQ} :> f@p

By restricting the 2 element list to only numeric entries, we avoid matching things like {Red, Blue}.

Here's a version using MapAll or //@:

g[p : {_?NumericQ, _?NumericQ}] := f@p
g[x_] := x

g //@ li

Answer 2

I'm going to suggest a slightly different form from what R.M proposed (which I voted for), in an attempt to make it a bit more robust. A list of two numeric values might occur in other parts of your expression besides your points. For example, Hue[{0.3, 0.6}] is a valid color. If you will always have a list of points we can use that to further restrict the pattern, then map the function over these lists.

Using:

li = {{fig1, Red}, {{fig2, fig3}, Hue[{0.3, 0.6}]}}

Here are two options that preserve the Hue directive:

li /. x : {{_?NumericQ, _?NumericQ} ..} :> f /@ x

li /. x_ /; ArrayQ[x, 2, NumericQ] && Dimensions[x][[2]] == 2 :> f /@ x

{{{f[{0, 0}], f[{1, 0}], f[{0, 1}]}, 
  RGBColor[1, 0, 0]}, {{{f[{2, 0}], f[{3, 0}], f[{2, 1}]}, {f[{4, 0}], f[{5, 0}],
  f[{4, 1}]}}, Hue[{0.3, 0.6}]}}

You should compare their performance in your actual application.