Syntractrix

A syntractrix is a curve of the form

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

The syntractrix for ${\displaystyle a=0.5}$ and ${\displaystyle b=1.}$

The syntractrix for ${\displaystyle a=1.5}$ and ${\displaystyle b=1.}$

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]