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In algebraic geometry, the lemniscate of Gerono, or lemniscate of Huygens, or figure-eight curve, is a plane algebraic curve of degree four and genus zero and is a lemniscate curve shaped like an \infty symbol, or figure eight. It has equation

\( x^4-x^2+y^2 = 0. \)

It was studied by Camille-Christophe Gerono.

Because the curve is of genus zero, it can be parametrized by rational functions; one means of doing that is

\( x = \frac{t^2-1}{t^2+1},\ y = \frac{2t(t^2-1)}{(t^2+1)^2}. \)

Another representation is

\( x = \cos \varphi,\ y = \sin\varphi\,\cos\varphi = \sin(2\varphi)/2 \)

which reveals that this lemniscate is a special case of a lissajous figure.

The dual curve (see Pl├╝cker formula), pictured below, has therefore a somewhat different character. Its equation is

\( (x^2-y^2)^3 + 8y^4+20x^2y^2-x^4-16y^2=0. \)

Dual to the lemniscate of Gerono


References

J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. p. 124. ISBN 0-486-60288-5.

Notes
External links

O'Connor, John J.; Robertson, Edmund F., "Figure Eight Curve", MacTutor History of Mathematics archive, University of St Andrews.

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

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