# Does this locus have a name?

This comes from variational analysis and in particular from the definition of the tangent cone of a set.

Say we have a circle centered at $$(0, 1/\tau)$$ with radius $$1/\tau$$, $$\tau > 0$$. All of these circles run through the origin. Take a point $$(w, 0)$$, $$w \neq 0$$ on the horizontal axis and let $$(x_\tau, y_\tau)$$ be the point where the line segment that connects $$(w, 0)$$ and $$(0,1/\tau)$$ intersects the circle.

As $$\tau$$ moves on $$(0, \infty)$$ it forms the locus shown in the figure above (blue line). The points of the locus can be written in the parametric form

\begin{align} x(t) {}={}& \frac{tw}{\sqrt{w^2 + t^2}},\\ y(t) {}={}& t - \frac{t^2}{\sqrt{w^2 + t^2}}, \end{align}

for $$t\in (0,\infty)$$. I wonder whether this locus has a name.

The parametric coordinates satisfy the Cartesian equation $$y^2 = \frac{x^2(w - x)}{ w + x}$$ which describes the "right strophoid". The full curve has reflective symmetry in the $$x$$-axis and includes asymptotic branches (with asymptote $$x=-w$$) corresponding to the locus of the "other" point of intersection of the line with the circle.