Pattern to match only “children” of certain elements

Question

I would like to be able to have a pattern that matches only expressions that are (alternately: are not) children of certain other elements.

For example, a pattern to

Answer 1

You might write a recursive function that descends an expression tree and acts on the types of expression you want only if inside the right type of sub-expression, while leaving everything else alone. Patterns would be used heavily in the definition of the function.

Consider, for example, the following expression.

test = {{1, 2}, Graphics[{
  Point[{{-1, 0}, {1, 0}}],
  Line[{{-1, 0}, {1, 0}}]},
 Frame -> True, 
 PlotRange -> {{-1, 1}, {-0.5, 0.5}}]};

Let's suppose that we want to rotate every ordered pair that we see in the first argument of Graphics about the origin through the angle Pi/4, while leaving other points alone. The following function does this.

Clear[f];
f[{x_?NumericQ, y_?NumericQ}] := If[flag === True,
  RotationMatrix[Pi/4].{x, y}, {x, y}];
f[Graphics[primitives_, rest___]] := Block[{flag = True},
  Graphics[f[primitives], rest]];
f[x_?AtomQ] := x;
f[x_] := f /@ x;

Now we check

f[test]