Rolling Not Slipping – Story of a Twirling Ellipsoid

I can’t help it – I use the first title that pops into my mind!

“Rolling, not slipping” is the punchline of a mathematical argument used in classical mechanics, in the theory of the motion of a gyroscope. But I am not going to do it justice in this blog post (apart from a math appendix you can skip). I just want to tell the story of the evolution of my related drawing!

This is again going to be a tribute to Louis Poinsot’s formulation of the theory of the gyroscope. But this time the ellipsoid is moving – and it is a different ellipsoid from the one featuring in my 2025 Poinsot ellipsoid series.

The energy ellipsoid is rolling without slipping along the plane defined by constant angular momentum. It is an abstract ellipsoid that is “virtually connected” to an actual physical spinning top. But you can think of this ellipsoid as made from wood: Fixed in its center, rolling along a desktop.

My plan is to show the evolution of the ellipsoid in three stages, colored in yellow, red, and blue. As with my Lissajous Figures in three dimensions, I am constructing the allegedly simple view first – when you see the scenario “straight-on” from three direction.

The first version has flaws, as I over-simplified:

The view in the upper right corner is correct: The yellow ellipsoid’s main axis is seen in its true length, then it rolls and turns by 90° (red) which makes it appear more stocky. Then it turns to the left by another 90° and becomes blue.

The yellow one is easy to depict when seen from the other directions – it is either an ellipse or a circle. The blue one is tilted in the other views, but still rather symmetrical. But the red ellipsoid is tilted twice in different directions, so I’ve shown it as too symmetrical.

As the ellipsoid is rather “fat” / “sphere-like” the error is not so egregious that I would spot it intuitively.

I know, I have to draw it again, but I enjoy messing up what I have, going over the red outline over and over to introduce the slight tilts – and develop the geometrical method to construct the needed line segments.

I am also using this version to think about the “3D” version in the center of the drawing – looking at the ellipsoid(s) from a diagonal direction.

Time for version 2! From version 1 I’ve also learned that the ellipsoid should better be more Zeppelin-/cigar-like – so that its apparent shapes really changes when it turns around.

Very cautiously, I am doing the construction. It dawns on me that the red ellipsoid is already looking pretty three-dimensional in the bottom and the left view. I need some helper constructions to figure out the lengths of all the foreshortened line segments.

I am building up each ellipsoid from two or three of its “main ellipses” – the intersection of the ellipsoids with a planes that contain a principle axis. The ellipsoid has to embrace these ellipses, and it has to stay within the pairs of tangents defined by the projection rays.

The hard part is now nearly done. But I will have to tweak the red “scaffolding ellipses” again and again! They looks so weird in the 3D view that it is hard to keep track of which ellipse projects onto what.

But finally, I can start adding color!

Next challenge though: This is technical drawing paper, not suitable at all for watercolors. I tried, in the past – I nearly ruined and then salvaged this Lissajous Figures drawing: I attempted to flatten the paper buckled from the water by generously watering it on the backside and weigh it down – a method that works well with thick watercolor paper. But with the thin paper, the watercolor marker dot got activated and smudgy as the water had reached the other side.

I am using watercolor pencils, but only a bit of water. It looks like not watered at all:

Decision: This will be my first drawing where I used watercolor pencils just like colored pencils. No more water.

I am adding color timidly at first (and I am gradually fixing the red ellipses in the left and the bottom view).

It starts to remind me of my “stained glass” watercolor pencils drawings. Will it be possible to achieve such an effect with colored pencils?

Then I notice I need to enhance the invariable plane.

… but then I also need to show the curves the points of contact on the ellipsoid and the sphere are tracing out! Both are circles as my chosen ellipsoid has two axes of the same length.

The more color I add, the more do the construction lines get covered. But the point of my drawings is to celebrate the construction: An analog calculation done by a human with the most ancient of tools. Predating AI, predating software and computers.

So, I need to re-trace them! Fortunately, without any more water, I do not need to care about the water solubility of any ink I might choose. I pick a thinner (0,3mm) fine liner from a set of pens for technical drawing.

I am going over every single line again, using a ruler.

But then I am longing for randomness, and I feel like crafting “fake ink blotches”!

I am mustering all my courage and finally add formulas and annotations. They should be considered “technical calligraphy” and not be too distracting.

And I am layering the colored pencils as if my life depended on it.

I have also cross-checked all the ellipses several times, so the mathematical part is over.

Enter: my subconscious!

More color! More lines! More ink blotches!

I notice, I have switched from colored ellipsoids on light background to light colored ellipsoids on darker background.

It seems done – but something about the balance of colors feels off. I wanted it to be yellow – red – blue without too much green. But it looks pale to me?

So: More color! Unusual colors! Dark colors! Greenish blues!

And suddenly – it feels complete! Now I just need to spend a while playing with lighting in my very makeshift “photo studio”. I am determined to retire my scanner.

~

Math / Physics Appendix

The relevant math is included in the drawing. Both energy and angular momentum can be calculated from the instantaneous angular velocity and the inertia tensor. This tensor defines the inertia ellipsoid and the energy ellipsoid similar to it: A surface defined by all possible vectors of angular velocity.

If you form the gradient of the definition of energy with respect to angular velocity, you obtain the vector of angular momentum. The gradient is perpendicular to the tangent plane to the surface defined by the energy condition. At every moment, the point of contact between the ellipsoid and the plane is the current angular velocity.

The spinning top is rotating about its axis, and the axis of rotation is changing slowly – rotating about the vector of angular momentum. Special case: Both are parallel – then the gyroscope is just rotating about a fixed axis.

The ellipsoid as an abstract concept can be thought of as attached to the rotating top – rotating with it. The instantaneous angular velocity has to remain the point of contact; so it must not slide along the plane. If it changes, this is achieved by lifting the ellipsoid off its current point of contact.

If the ellipsoid is symmetrical two of its axes are the same: Two of its intersections with planes containing the axes are ellipses, and the remaining one is a circle. For symmetry reasons both the Polhode inscribed in the ellipsoid as well as its trace on the invariable plane – the Herpolhode – have to be circles. Just as the rim of a wheel and its trace on the ground are tangential to each other, Polhode and Herpolhode are, too.

For an ellipsoid with three different axes, the Polhode is a closed curve (depicted in the Poinsot’s Paper Spaceship series – this page has also more links to articles with physics background), but the Herpolhode is an open, rather random-looking curve.

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Edit, March 10, 2026: Canvas prints are now available here in my store.