Given 5 points that determine a conic, fix 4 of them and allow one to move. Identify the connected region around the initial location of the movable point within which a new conic constructed through the 4 fixed points and 1 moved point is qualitatively similar to the original conic. For example, a hyperbola stays a hyperbola or an ellipse stays an ellipse. We might also be interested in the ellipse/circle transition and where a hyperbola becomes two lines. Repeat, fixing a different 4 points. The resulting map of regions around each point might be very similar to the perturbation map of points in general position, as a conic constructed from points not in general position will probably be a conic on the boundary of two different conic types.
Similarly, explore perturbations of 9 points determining a cubic curve, identifying where one cubic curve turns into a qualitatively different one. Portions of cubic curves sometimes pinch off into loops disconnected from the rest of the curve, resulting in a qualitatively different curve.
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