# A question on using Cases [closed]

I am working on Wagon’s FindRoots2D section.

Clear[f];
f[x_, y_] := x – y^2 Cos[y];
g[x_, y_] := -y + x Sin[x];
fcnVec[{x_, y_}] := {f[x, y], g[x, y]};

I am starting (still a rookie) to understand that one can get the data of any object in Mathematica. For example, a contour plot.

cp = ContourPlot[f[x, y] == 0, {x, -10, 10}, {y, -10, 10}]

I’ve learned that if I look at FullForm[cp], I’ll see that the data is using the GraphicsComplex idea, which I have a beginner’s understanding of. If I look at FullForm[Normal[cp]], then I see the Line command using the actual data points instead of indices to the GraphicsComplex data. Now, Stan Wagon uses the Cases command to get those points. I try:

Cases[Normal[cp], Line[z_] :> z]

And I get an empty set. Then I try Stan’s:

Cases[Normal[cp], Line[z_] :> z, âˆž]

And I get all of the points. This is where I am stuck. Why does the Infinity symbol make a difference here?

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The explanation is readily found in the documentation of Cases.

Cases uses standard level specifications
The default value for levelspec in Cases is {1}

The Line expressions are not at level 1, and in fact may be at more than one level. Using âˆž as the level spec tells Cases to look at all levels.

Thanks for a nice explanation. Very, very helpful.
– David
Jul 19 ’15 at 18:07