Locus of intersection of two perpendicular normals to an ellipse

Question

I came up with the following question while playing around with Geogebra, and rather than do it myself I figured I'd offer it here.

For a given ellipse, the director circle is defined to be the set of all points where two perpendicular tangents to the ellipse cross each other. Suppose we instead consider the locus of points where two perpendicular normals to the ellipse cross. Using Geogebra, I can use it to trace the curve out: It looks like a four-leaf clover that is symmetric about the ellipse's axes. But what curve is this?

Note: To construct this in Geogebra, I first constructed the director circle as the circumcircle of the minimum bounding box of the ellipse. I then picked a point on the circle, used this to find two points of tangency, and found the intersection of the corresponding normals at these points. The curve could then be traced out by varying the choice of point on the director circle.

Answer 1

I do not have even a sniff of a clue as to the stated question (the name of the curve), but I did find the formula for it.

Starting with an ellipse at origin, with axis-aligned semi-major axes $r_x$ and $r_y$, $$\begin{align} x_E(\varphi) &= r_x \cos \varphi \\ y_E(\varphi) &= r_y \sin \varphi \end{align}$$ the normal (defined as tangent rotated 90° counterclockwise in a right-handed coordinate system) is $$\begin{align} x_N(\varphi) = \frac{d y_E(\varphi)}{d \varphi} &= r_y \cos \varphi \\ y_N(\varphi) = - \frac{d x_E(\varphi)}{d \varphi} &= r_x \sin \varphi \end{align}$$ If the normals corresponding to $\varphi=\varphi_1$ and $\varphi=\varphi_2$ are perpendicular, then their dot product is zero, $$x_N(\varphi_1) x_N(\varphi_2) + y_N(\varphi_1) y_N(\varphi_2) = 0$$ i.e. $$r_y^2 \cos(\varphi_1) \cos(\varphi_2) + rx^2 \sin(\varphi_1) \sin(\varphi_2) = 0$$ Solving for $\varphi_2$ we get $$\varphi_2 = - \arctan\left(\frac{r_y^2 \cos\varphi_1}{r_x^2 \sin\varphi_1}\right)$$ I shall use that to define a function $$\theta(\varphi) = - \arctan\left(\frac{r_y^2 \cos\varphi_1}{r_x^2 \sin\varphi_1}\right)$$

The equation for the perpendicular line at $\varphi$, with $t$ as the line parameter is $$\begin{cases} x_1(\varphi, t_1) = x_E(\varphi) + t_1 x_P(\varphi) \\ y_1(\varphi, t_1) = y_E(\varphi) + t_1 y_P(\varphi) \end{cases}$$ Using the function $\theta(\varphi)$ defined before, we know that the above is also perpendicular to $$\begin{cases} x_2(\varphi, t_2) = x_E(\theta(\varphi)) + t_2 x_P(\theta(\varphi)) \\ y_2(\varphi, t_2) = y_E(\theta(\varphi)) + t_2 y_P(\theta(\varphi)) \end{cases}$$ Their intersection occurs at $$\begin{cases} x_1(\varphi, t_1) = x_2(\varphi, t_2) \\ y_1(\varphi, t_1) = y_2(\varphi, t_2) \end{cases}$$ which we can solve for $t_1$ and $t_2$. I shall omit $t_2$ here for brevity. $t_1 = t(\varphi)$: $$t(\varphi) = r_x r_y \frac{ (r_x^2 - r_y^2) \cos(\varphi) - \sqrt{r_x^4 + \frac{r_y^4 \cos(\varphi)^2}{\sin(\varphi)^2}}}{(r_y^2 \cos(\varphi)^2 + r_x^2 \sin(\varphi)^2)\sqrt{r_x^4 + \frac{r_y^4 \cos(\varphi)^2}{\sin(\varphi)^2}}}$$ Note that $t(\varphi)$ is not defined at integer multiples of $\pi$ (including zero), but it approaches the value $$-rx/ry$$.

The curve is then $$\begin{cases} x'(\varphi) = x_E(\varphi) + t(\varphi) x_P(\varphi) \\ y'(\varphi) = y_E(\varphi) + t(\varphi) y_P(\varphi) \end{cases}$$ Because of how we defined the normal, the actual curve is composed of two parts, $$\begin{cases} x(\varphi) = x'(\varphi) \\ y(\varphi) = y'(\varphi) \end{cases}, \text{ and } \begin{cases} x(\varphi) = -x'(\varphi) \\ y(\varphi) = -y'(\varphi) \end{cases}, \; \; \varphi = (0, \pi)$$ In pseudocode (with ^ denoting exponentiation),

tmp1 = sqrt( rx^4 + ry^4 / sin(phi)^2 )
tmp2 = ( ( rx^2 - ry^2 ) * cos(phi) - tmp1 ) / ( tmp1 * ( rx^2 * sin(phi)^2 + ry^2 * cos(phi)^2 ) )
x = rx * ( 1 + tmp2 * ry^2 ) * cos(phi)
y = ry * ( 1 + tmp2 * rx^2 ) * sin(phi)

Each of the four lobes is $C^2$-continuous, as at $\varphi \to 0$ and $\varphi \to \pi$ the curve parts approach the origin at the same slopes.