A toric problem

Starting innocently enough with the book at reference 1, I found myself at reference 2 and then at reference 3, where I came across a line drawing of a torus along the lines suggested above. The point of interest being the pair of horns in the middle. An oddity, if for no other reason, for being lines that just end, in mid-air, as it were, despite the originating torus itself being nice and smooth all round.

To get the line drawing, reference 3 explains that you start with a projection of sensible solid objects suspended in three dimensional space onto a two dimensional plane. We suppose that each point on that plane has two values. First, a scalar giving the distance from the eye to the object in question. Second, a unit vector which is orthogonal to the tangent plane at the point in question. Any point on the plane where either of these quantities is not continuous is a point on our line drawing. No doubt it can be shown that the line drawings which result from this proceeding are not very complicated. An isolated sphere, for example, gives a circle.

A cube, however, is more tricky, with the line drawing that results usually being ambiguous: the point in the middle in the figure above could go either way, and the burden of reference 3 is doing something about that ambiguity in its various guises. 

The present interest however is the horns of the torus, which one might think a simpler matter. 

Now the existence of the horns is very plausible when one looks at the figure above: they seem to be clearly visible right and left. But can we prove that they exist? Can we say a bit more about them?

A story

First, I offer an unsubstantiated story, what might be called a heuristic. Starting at the bottom in the figure above, we have the inner circle of our torus viewed from vertically above actually being a circle. With the angles α and β both being right angles, π/2 radians.

Note that none of the points or lines used here are to be found on the torus itself, which is a smooth, closed sheet without any vertices or edges. All these lines arise from having a point of view. From which I associate to colour being in the eye of the beholder, there being nothing of the sort out there in the real world.

Then as soon as we start rotating the torus, top away from us in the figure above, this inner circle starts to flatten and stops being a circle. Angles α and β have both decreased slightly, meeting at what is now a vertex. We also have the start of the two horns.

As we move up, the top line gradually shrinks and the bottom line straightens out, the whole ending as the straight line CD and vanishing as the rotation reaches the full quarter circle, π/2 radians (again). I assert that length CD is greater than length AB. Going further, that the width of the lower line (mostly less than the length), grows monotonically from AB to CD – a conjecture which is not captured in the figure.

Would it be enough to demonstrate that the width of the lower line is bounded below by AB, while that of the upper line is bounded above? Then the difference must be the horns?

Analysis

I then wondered about describing the torus analytically, thinking here of the equation for a circle in the plane, that is to say: (x-a)^2 + (y-b)^2 = c^2, where (a, b) are the coordinates of the centre and c > 0 is the radius.

My first offer was:

For some a > c; for coordinates θ, x, y; for all angles θ; for all values of x and y representing a point on the surface of the torus we have [(x-a)^2 + y^2 = c^2] ˅ [(x+a)^2 + y^2 = c^2]

Which does not help at all. Disjunctions might be informative, but do not help much with analysis.

Wikipedia does rather better at reference 4, with one way of doing it snapped above and another, a quartic, reproduced below.

            (x^2 + y^2 + z^2 + R^2- r^2)^2 = 4R^2(x^2 + y^2)

While reference 7 does even better.

The catch being that I am no longer competent to work with this sort of thing! So I go back to basic geometry. Maybe there is enough there.

Lemma 

Suppose we are looking down at the torus, the circle right being the outer ring, and have a vertical plane pivoting on point P left. What can we say about the length of AB as the angle θ increases? Intuitively, it decreases as the angle increases, but can we be sure?

Looking at the figure above, both α and β decrease as θ increases to its maximum.

Given the boxed identity, γ increases when α and β decrease. Then, given that we have radii of a circle and isosceles triangles, length AB varies as cos(γ).

Therefore, length AB decreases as angle θ increases, ending at zero, as it should. Monotonically but not linearly.

Adding the inside ring to the outside ring, the second length AB decreases monotonically too. We are interested in the way that the shape of the section through the torus cut by our vertical plane changes with increasing θ. It starts as two circles when θ = 0 and ends as a point, but what happens in-between?

Lemma

How ever we draw the line AE across the torus, centre C, length AB will always equal length DE. This is demonstrated by consideration of the triangles ABC and CDE.

Some labels

We start with the eye in the plane of the torus (the dotted line across the middle), then moving up and over. In the beginning, what the eye gets is the elongated sausage, illustrated top right. When the eye is right over the torus, it gets the image bottom right. Note that the boundaries there are slightly above the actual equator of the torus – to get that one would have to be at an infinite distance.

On this trajectory, we identify three zones. Zone O where region A of the torus completely masks region B to its right. Zone P where the upper edge of region B has appeared above the upper edge of region A. Zone Q (red spot) where the lower edge of region B has appeared above the upper edge of region A. Remembering that this is along the centre line: what happens as we move outwards is still to play for.

While at the blue spot, the lower edge of region A coincides with the upper edge of region B. Remembering that ‘upper’ and ‘lower’ switch round as one moves from left to right around the blue trajectory.

The horned zone is suggested top left, with the present task being to demonstrate its existence and size. Our problem being that one only gets tractable circles along certain lines of sight. Elsewhere, the shapes might be nicely rounded, but are otherwise uncertain.

A loss of symmetry

We are looking directly at the torus, from a position slightly above its central plane. The boundary of what we see is suggested top right in red, below the equator left, above the equator right, probably crossing over slightly to the left of the central vertical. As the eye moves up and over to the right, the red lines both move closer to the outer circle, ending by coinciding with it.

The loss of symmetry lies, along this centre line at least, in the dashed red line left being further from the inner circle than the line right, that is to say L(θ) > L(γ), although this does not come out in the sketch.

A simulation

At this point, I thought it might be an idea to ask Bing about the sections of the torus, which turned up references 5 thru 9. It turns out that people have been pondering about toric sections since well before the birth of Christ. And while in these earlier times, the apparatus of today was not available, they did turn up all kinds of curious examples, such as the Hippopedes of Proclus.

While a bit later on we had Villarceau's circles, Cassini's ovals and Bernoulli's Lemniscates. These last being related to the ellipse in the sense that rather than the sum of the distances to two focal points being constant, the product is.

All these and more are to be found at reference 7. Then at reference 9 we can produce our own sections. Most of which are neither circles nor ellipses – despite often being quite like circles or ellipses - and the simple certainties of schoolboy geometry are not applicable.

The torus, as seen here from slightly above its plane, has two visible boundaries, an outer boundary (OB) and an inner boundary (IB).

We are also shown the inner (IE) and outer equators (OE), a mixture of continuous and dashed lines, partly depending on their position relative to the vertical interesting plane.

On the page, OB lies outside OE. Above it at the back, below it at the front.

IB lies inside IE. Below it at the back, above it at the front, thus giving rise to the acute angles at the left and right extremities.

In both cases, the two cross over points occur at the extreme points, left and right, slightly adrift from the red arrow.

As we look at the extreme points of IB, the section is something like (allowing for the different angle), the red section above. Plenty of room there for the lower part of the inner boundary to acquire its horns.

As the preceding figure, but with the addition of the horizontal line AB, at right angles to the point of view, intersecting the torus near but not at the crossover points of both equators, the change over points from upper to lower parts of boundaries.

Not quite, because as is suggested in the figure above, from any finite point E, one sees slightly less than half the torus (or sphere for that matter), as suggested by the blue rectangle. But one does get closer and closer to the full half as θ goes towards zero, as the point E backs off to the left, off the page (or screen).

Maybe the previous figure has been constructed on the basis that θ does equal zero?

Which starts me wondering about what exactly do you have to put on the flat page in order to get a realistic image of a real, three dimensional torus on the retina? In the mind’s eye of reference 1, where I started? But it is a bit early in the morning for that one.

Shapes

Which brings me back to a figure which I had prepared earlier. And there, I think, the matter of horns is going to have to rest. I am satisfied that they exist and am content to have failed to produce a proper proof.

Conclusions

I have been impressed by the way the torus has introduced me to a world where the circle with its simple certainties no longer does the trick – this despite the torus being generated by the rotation of one circle around another. A quite different world to that obtained by rotating one line around another, the world of conics. But, for the intrepid, reference 7 does go on to make a connection between the two.

I have also been impressed by the help provided by reference 9. All so much harder when all one had was brain, paper and pencil!

And the whole exercise has reminded me of the importance of point of view. Without someone to look at it, nothing is of much interest!

References

Reference 1: The Mind's Eye – Oliver Sacks – 2010. Book_213

Reference 2: Character complexity and redundancy in writing systems over human history – Mark Changizi, Shinsuke Shimojo – 2005.

Reference 3: Interpreting Line Drawings of Curved Objects – Jitendra Malik – 1987.

Reference 4: https://en.wikipedia.org/wiki/Torus. 

Reference 5: https://en.wikipedia.org/wiki/Toric_section.

Reference 6: https://www.lucamoroni.it/simulations/intersection-torus-plane-simulation/.

Reference 7: The toric sections: a simple introduction – Luca Moroni – 2017. 

Reference 8: https://www.lucamoroni.it/. 

Reference 9: https://www.geogebra.org/m/MmTVuXYk. The program here is quite slow, but I believe there is a download option which would probably be faster.