By picking orthogonal coordinates in the given plane you can make an angle preserving projective map $mathbb{P}^2tomathbb{P}^3$ whose image is the given plane. Composing with your camera mapping, you now have a mapping $Gcolonmathbb{P}^2tomathbb{P}^2$ that does not preserve angles. Let $H=G^{-1}$. The composition $HG=I$ clearly preserves angles; hence so does $HP$ when restricted to the given plane.
(Reverse the order of composition if you follow the usual computer graphics convention of letting matrices act on the right.)
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