documentclass{article}
usepackage{tikz}
usepgfmodule{nonlineartransformations}
makeatletter
defpolartransformation{% from the pgfmanual section 103.4.2
pgfmathsincos@{pgf@sys@tonumberpgf@x}%
pgf@x=pgfmathresultxpgf@y%
pgf@y=pgfmathresultypgf@y%
} % note that the problem is not specific to this transformation
makeatother
begin{document}
section*{Ordinary linear transformations}
It is well known that one can refer to coordinates inside a scope subject to
linear transformations without problems.

begin{tikzpicture}
begin{scope}[xshift=-4cm]
draw (-1,0) coordinate (l1) -- (3,1) coordinate (l2);
end{scope}
node[circle,draw] (A) at (-1,-2) {A};
node[circle,draw] (B) at (2,-2) {B};

draw[blue] (A) -- (l1);
draw[red] (B) -- (l2);
end{tikzpicture}

section*{Ordinary nonlinear transformations}
This is also true for ``ordinary'' nonlinear transformations.

begin{tikzpicture}
begin{scope}
pgftransformnonlinear{polartransformation}
draw (-1,0) coordinate (m1) -- (3,1) coordinate (m2); % no, m does not stand for marmot ;-)
end{scope}
node[circle,draw] (A) at (-1,-2) {A};
node[circle,draw] (B) at (2,-2) {B};
draw[blue] (A) -- (m1);
draw[red] (B) -- (m2);
end{tikzpicture}

section*{Nonlinear transformations with texttt{transform shape nonlinear=true}}
However, once one adds texttt{transform shape nonlinear=true}, this does no
longer work.


begin{tikzpicture}
begin{scope}[transform shape nonlinear=true] % p. 234 of the pgf manual
pgftransformnonlinear{polartransformation}
draw (-1,0) coordinate (r1) -- (3,1) coordinate (r2);
end{scope}
node[circle,draw] (A) at (-1,-2) {A};
node[circle,draw] (B) at (2,-2) {B};
draw[blue] (A) -- (r1);
draw[red] (B) -- (r2);
end{tikzpicture}

On the other hand, one might need these things, as only with this option
switched on the shape gets transformed.

begin{tikzpicture}[baseline=(A.base)]
begin{scope}
pgftransformnonlinear{polartransformation}
node[draw] (n1) at (2,2){hello!};
end{scope}
node[circle,draw] (A) at (-1,-2) {A};
node[circle,draw] (B) at (2,-2) {B};
draw[blue] (A) -- (n1.west);
draw[red] (B) -- (n1.east);
end{tikzpicture}quad vs.quad
begin{tikzpicture}[baseline=(A.base)]
begin{scope}[transform shape nonlinear=true]
pgftransformnonlinear{polartransformation}
node[draw] (n2) at (2,2){hello!};
end{scope}
node[circle,draw] (A) at (-1,-2) {A};
node[circle,draw] (B) at (2,-2) {B};
draw[blue] (A) -- (n2.west);
draw[red] (B) -- (n2.east);
end{tikzpicture}
end{document}

tikzset{
  transform shape nonlinear/.is choice,
  transform shape nonlinear/.default=true,
  transform shape nonlinear/true/.code=lettikz@nltrelax,
  transform shape nonlinear/false/.code=deftikz@nlt{pgfapproximatenonlineartranslation},
  transform shape nonlinear=false
}