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Cramér-Euler Paradox

A curve of order $n$ is generally determined by $n(n+3)/2$ points. So a Conic Section is determined by five points and a Cubic Curve should require nine. But the Maclaurin-Bezout Theorem says that two curves of degree $n$ intersect in $n^2$ points, so two Cubics intersect in nine points. This means that $n(n+3)/2$ points do not always uniquely determine a single curve of order $n$. The paradox was publicized by Stirling, and explained by Plücker.

See also Cubic Curve, Maclaurin-Bezout Theorem

© 1996-9 Eric W. Weisstein