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With parametric form I mean a parametrization like $(\cos{t}, \sin{t})$ for a circle.
A conic section has such a parametrization, but suppose it degenerates in 2 lines (ranges of points), is there a parametrization that goes from $-\infty$ to $+\infty$ on both lines?

Then in case the 2 lines degenerate even further into one line of multiplicity 2, does the parametric form cover that doubly degenerated line 2 times?

Gerard
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1 Answers1

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The range of a continuous mapping defined on a connected set is itself connected. Hence you cannot find a continuous parameterization of a disconnected set on the real line.

But this is a problem also for non degenerate disconnected conics (namely: the hyperbola).

Emanuele Paolini
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