Real locus of a cone or a double cone
By Daniel Johnston
If we take the algebraic set of $X^2+Y^2-Z^2=0$, $V(X^2+Y^2-Z^2)$, why do we obtain a single cone, rather than the real locus of $X^2+Y^2-Z^2$ being a double cone? This seems absolutely wrong, since $(-z)^2=z^2$ obviously. Is there something more going on here?
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$\begingroup$I'm answering to get this out of the unanswered queue.
$X^2+Y^2-Z^2=0$ without any further restrictions does describe a double cone. People who just call it “cone” might be lazy, and might have established a convention where “cone” without qualification means “double cone”. People who call it a “single cone”, or depict it as such, are in my opinion just wrong.
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