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Consider the zero level set of $f : \mathbb{R}^3 \to \mathbb{R}$, where $0$ is a regular value. Consider also the space of planes passing through the origin, i.e. $\mathbb{RP}^2$. For a fixed plane $P \in \mathbb{RP}^2$, consider the orthogonal projection of $f^{-1}(0)$ onto $P$. The apparent contour, say $A_P \subset P$, is the critical value set of the projection of $f^{-1}(0)$ onto $P$.

Now consider a fixed set of morphisms of $\mathbb{R}^2$, say $M$, e.g. diffeomorphisms, homeomorphisms, isotopy equivalence, homotopy equivalence. I'm interested in the following space:

$\mathbb{RP}^2 / \sim$, where $P \sim P'$ if and only if there exists $\mu \in M$ such that $\mu(A_P) = A_{P'}$.

I'm thinking of all of the apparent contours as "living" in the same plane, and getting a two-parameter family of plane curves parametrised by the the choice of $P \in \mathbb{RP}^2$.

What is known, and what references can be given regarding this set-up. Especially when $M$ is the set of diffeomorphisms and homeomorphisms?

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what is $\mu A_P$? – Ryan Budney Aug 1 '12 at 6:27
The $A_P$ should be in brackets, i.e. $\mu(A_P)$. It's the image of an apparent contour. Two planes, say $P and P'$, are equivalent if and only if the apparent contour, i.e. the critical value set, of the orthogonal projection of $f^{-1}(0)$ onto $P$ is diffeo/homeo-morphic to the apparent contour of the orthogonal projection of $f^{-1}(0)$ onto $P'$. – Fly by Night Aug 1 '12 at 15:03

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