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