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.


1 Answer 1


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.

enter image description here

Image from Wikipedia. Attribution: Kmhkmh, CC BY 4.0, via Wikimedia Commons


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