© Jaspower

This all originated from a post in reddit's r/mildlyinteresting by user Jaspower. Looking at his cup the redditor noticed that the reflection of the saucer lined up with the saucer itself when viewed from a specific angle.

Being interested in maths and geometry this got me thinking:
*Is there a shape that would make this work from any angle?* I soon
came up with an idea and
posted it on the math subreddit.

This post contains a brief explanation of my idea and finally
*includes a proof*.

(Note that I tried to keep the proof simple, so the phrasing may sometimes be
inexact)

I soon discovered that *if* there was a cup shape with the desired
property it would have to be a hyperboloid - which is some sort of "3d
hyperbola".

A Hyperbola has an interesting property: any ray cast from one of its focal
points will be reflected in such a way that it appears to originate from the
other focal point.

In the image below this means an observer (in the top right corner) looking in
the direction of F_{2} would actually always see the mirror image of
F_{1}. The reason behind this is that the tangent in any point of a
hyperbola will always be the angle bisector of the two lines leading towards
the focal points.

(based on this image (CC-BY-SA))

As this works in 2 dimensions I hoped it would work in 3 dimensions as well and I came up with the following

Conjecture: To fulfill the desired property the cup has to be a hyperboloid and the rim of the saucer has to be the locus of all focal points of hyperbolas on the hyperboloid's surface.

The locus of all focal points is a circle which we will label *f*.

Does this conjecture hold up against a quick check? A visualization in Blender shows it does:

Soon I also received visualizations from other people: codepen user kzf created a great interactive WebGL visualization (doesn't work on all browsers).

Screenshot from kzf's interactive visualization

Through our arguments in the 2D case we're already very close to a proof.

(based on this image (CC-BY-SA))

We already know:

If we mirror the line PF_{1} along the tangent plane T we will get the
line PF_{2}.

Now we want to know:

If we mirror a random line PX along the tangent plane - will we get a line
that hits the circle f as well?

Let's imagine X moves along the cirlce f. Which shape would the connecting
line PX sweep out?

The answer is: an elliptic cone (image) - note that I'm referring to the infinite surface, not the solid.

An elliptic cone always has two planes of symmetry. In our case, one of these
planes of symmetry *must* be the tangent plane T - as T is the plane of
symmetry between PF_{1} and PF_{2}.

Therefore we know: Mirroring the elliptic cone along T will transform it into
itself.

Thus, mirroring any line on the elliptic cone will result in another line on
the cone. So:

A ray that's directed at a point of the circle f and gets reflected will always end up hitting another point on f.

If we reverse the direction of the ray in our minds we're able to confirm our conjecture.

I hope you enjoyed the read. If you have comments, message me on reddit or email me!