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RAMPHOID CURVE

Curve proposed by Euler in 1744 (Letter
to Cramer on the 20th of October 1744).
Ramphoid comes from the Greek ramphos "bird beak";
the name was given in 1809 to the cusps of second kind by Louis-Benjamin
Francoeur Se; he gaves the name keratoid forthe first kind.
e this text. |

Cartesian parametrization: .
Cartesian equation or . Polynomial quartic. |

This curve is probably the simplest curve that has a cuspidal
point of the second kind (i.e. such that the two parts of the curve are
on the same side of the tangent).

Euler presented it as an answer to Cramer who believed
that cuspidal points of this kind could not be found in algebraic curves.

Other curves with bird beak: the Joukowski
curve, the involutes
of curves with inflection points.

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© Robert FERRÉOL
2023