Quadratically parametrized maps from a real projective space to a complex projective space are constructed as projections of the Veronese embedding. A classification theorem relates equivalence classes of projections to real congruence classes of complex symmetric matrix pencils. The images of some low-dimensional cases include certain quartic curves in the Riemann sphere, models of the real projective plane in complex projective 4-space, and some normal form varieties for real submanifolds of complex space with CR singularities.

This expository article considers non-circular ellipses in the Riemann sphere, and the action of the group of Mobius transformations. In particular, we find which Mobius transformations are symmetries of an ellipse, and which take one ellipse to another. We also survey some of the ``special plane curves'' which appear as inversive images of the ellipse.

This expository article considers non-circular ellipses in the Riemann sphere, and the action of the group of Mobius transformations. In particular, we find which Mobius transformations are symmetries of an ellipse, and which take one ellipse to another. We also survey some of the ``special plane curves'' which appear as inversive images of the ellipse.