Consider the following construction. Start with the standard envelope construction of a cardioid: on a circle, join each point $\theta$ to $2\theta$. Only, instead of joining with a line, join with an arc. Of course, there are many such arcs so choose the one that makes a right-angle where it intersects the original circle (ie at $\theta$ and $2\theta$).
The result looks like this:
(To get the outer curve, you draw the full circle, not just the arc inside the original circle.)
An implicit equation of this curve is:
$$ \begin{align} x^8 + y^8 &+ 16x^7 + 4(x^2 + 4x + 19)y^6 + 76x^6 + 48x^5 \\ &+ 6(x^4 + 8x^3 + 38x^2 + 8x - 47)y^4 - 282x^4 + 48x^3\\ &+ 4(x^6 + 12x^5 + 57x^4 + 24x^3 - 141x^2 + 12x + 19)y^2 + 76x^2 + 16x + 1 \end{align} $$
which is a bit tricky to search for!
My reason for asking is that this came up when some students were playing around with the envelope construction of the cardioid and replaced the lines with arcs as described above. I'd like to be able to tell them a bit about what they've come up with, but to do that I need to find out myself!