Syntractrix

A syntractrix is a curve of the form

$x+\sqrt{b^2-y^2}= a \ln \frac{b+\sqrt{b^2-y^2}}{y}.$^[1]

The syntractrix for $a=0.5$ and $b=1.$

The syntractrix for $a=1.5$ and $b=1.$

It is the locus of a point on the tangent of a tractrix at a constant distance from the point of tangency, as the point of tangency is moved along the curve.^[2]

References

1. George Salmon (1879). A Treatise on the Higher Plane Curves: Intended as a Sequel to A Treatise on Conic Sections. Published by Hodges, Foster, and Figgis. Page 290.
2. Dionysius Lardner, A system of algebraic geometry 1823, p. 261–263