Discriminants of convex curves are homeomorphic

For a given real generic curve gamma : S-1 --> P-n let D-gamma denote the ruled hypersurface in P-n consisting of all osculating subspaces to gamma of codimension 2. In this note we show that for any two convex real projective curves gamma(1) : S-1 --> P-n and gamma(2) : S-1 --> P-n the pairs (P-n, D-gamma 1) and (P-n, D-gamma 2) are homeomorphic.