Let me start with the context. This is definitely not a "research level" question, but I'm hoping that the research community will be able to settle for me whether or not a particular construction has been done before or not. I'm also hoping that the wider context will mean that this question doesn't get closed instantaneously.

I do an extension class with 10/11 year olds, not selected for their interest in mathematics (ie it's just a regular class). One of the activities I get them to do is to draw a cardioid by joining points on a circle with straight lines. One pair decided that instead of using straight lines, they would use curves instead. The resulting picture is similar to a cardioid, but is not the same.

My goal is to be able to say to them one of two things:

Either:

"Well done! You took the construction and played with it, seeing what else you could make. That's exactly what mathematicians do. What you've discovered is called the hyperlogical antiometric cardiobloid and it was first made in 1832 by de Grundie. It's linked to hyperlogical geometry which is a really key area of mathematics."

Or:

"Well done! You took the construction and played with it, seeing what else you could make. That's exactly what mathematicians do. What you've discovered is quite probably completely new! No-one I've asked has heard of it. That's brilliant! What would you like to call it?"

But to say either one, I need to know if this has been seen before. Note that it doesn't matter what the answer is, and also please note that I'm not claiming that the curve itself has any intrinsic value. Its value is purely extrinsic in that I can use it to show the students what being a real mathematician is like.

Without further waffle, here's a more precise construction.

On a circle, draw an arc between $\theta$ and $2 \theta$ so that the arc meets the circle at right-angles (in the manner of the Poincaré disc model of hyperbolic geometry - indeed, if this has been seen before then "curve stitching in hyperbolic space" seems a likely candidate). This draws out an envelope, just as the straight-line version draws a cardioid. In actual fact, one can draw the full circles - not just the arcs - and get another part of the curve outside the original circle.

Using the theory of envelopes, and sticking stuff into Sagemath, I found an equation of the curve:

$$ \begin{align} x^8 + y^8 &+ 16x^7 + 4(x^2 + 4x + 19)y^6 + 76x^6 + 48x^5 \\ &+ 6(x^4 + 8x^3 + 38x^2 + 8x - 47)y^4 - 282x^4 + 48x^3\\ &+ 4(x^6 + 12x^5 + 57x^4 + 24x^3 - 141x^2 + 12x + 19)y^2 + 76x^2 + 16x + 1 \end{align} $$

Using Geogebra, I drew a diagram showing the curve and its construction:

Any pointers will be gratefully received, (but please note that I no longer have access to paywalled articles (nor to MathSciNet) so if you supply references please try to find accessible ones).

Let me start with the context. This is definitely not a "research level" question, but I'm hoping that the research community will be able to settle for me whether or not a particular construction has been done before or not. I'm also hoping that the wider context will mean that this question doesn't get closed instantaneously.

I do an extension class with 10/11 year olds, not selected for their interest in mathematics (ie it's just a regular class). One of the activities I get them to do is to draw a cardioid by joining points on a circle with straight lines. One pair decided that instead of using straight lines, they would use curves instead. The resulting picture is similar to a cardioid, but is not the same.

My goal is to be able to say to them one of two things:

Either:

"Well done! You took the construction and played with it, seeing what else you could make. That's exactly what mathematicians do. What you've discovered is called the hyperlogical antiometric cardiobloid and it was first made in 1832 by de Grundie. It's linked to hyperlogical geometry which is a really key area of mathematics."

Or:

"Well done! You took the construction and played with it, seeing what else you could make. That's exactly what mathematicians do. What you've discovered is quite probably completely new! No-one I've asked has heard of it. That's brilliant! What would you like to call it?"

But to say either one, I need to know if this has been seen before. Note that it doesn't matter what the answer is, and also please note that I'm not claiming that the curve itself has any intrinsic value. Its value is purely extrinsic in that I can use it to show the students what being a real mathematician is like.

Without further waffle, here's a more precise construction.

On a circle, draw an arc between $\theta$ and $2 \theta$ so that the arc meets the circle at right-angles (in the manner of the Poincaré disc model of hyperbolic geometry - indeed, if this has been seen before then "curve stitching in hyperbolic space" seems a likely candidate). This draws out an envelope, just as the straight-line version draws a cardioid. In actual fact, one can draw the full circles - not just the arcs - and get another part of the curve outside the original circle.

Using the theory of envelopes, and sticking stuff into Sagemath, I found an equation of the curve:

$$ \begin{align} x^8 + y^8 &+ 16x^7 + 4(x^2 + 4x + 19)y^6 + 76x^6 + 48x^5 \\ &+ 6(x^4 + 8x^3 + 38x^2 + 8x - 47)y^4 - 282x^4 + 48x^3\\ &+ 4(x^6 + 12x^5 + 57x^4 + 24x^3 - 141x^2 + 12x + 19)y^2 + 76x^2 + 16x + 1 = 0 \end{align} $$

Using Geogebra, I drew a diagram showing the curve and its construction:

Any pointers will be gratefully received, (but please note that I no longer have access to paywalled articles (nor to MathSciNet) so if you supply references please try to find accessible ones).