Nope, no need. Try this, use your own crank and pedals. Scribe the circle of your actual cranks. Now figure out what your actual pedal angle is at say every 10 to 20 degrees when you pedal (the more data points the better) now place dots where your pedal tops end up at those positions (you can connect those dots as smoothly as possible if you'd like, but not really required). Now find any point inside the scribed circle of your crank that is equidistant from all those points.

*sigh*

If the pedal spindle travels in a circle, another point at a fixed distance in a fixed direction from the pedal spindle can only travel in a circle. It's path is DEFINED BY the path of the pedal spindle.

That's correct, but that's not the problem under debate. You commented on a statement that was not directed at you, but at the person that has repeatedly posted a diagram that shows pedals that are directly above the pedal spindle at all times. In that case, the pedal tops travel in a circle. I'm sure you'll agree with that.

Let me know when we get to algebra and statistics. I was only average in geometry and physics.

Ok good, you are starting to make my point. The pedal top IS a fixed distance away from the spindle, but, as it spins (or rocks) the magnitude of the contribution to the effective length of the crank changes, try it. lock the crank in one spot, and spin the pedal. Baby steps...

I don't. The only way a pedal top travels a circle is if the pedal spindle does not spin at all. and if you agree with what I posted above, I don't see how you don't see that. Try your own suggested graph paper, and it will be the same. I must admit, I cheated, I was fairly sure about the non circle thing, but I built a mock crank pedal jig thing with a popsicle stick and a pog, and thumb tacked it to my wife's project table with the grid thing, hopefully she won't notice the hole...

This is incorrect. The pedal top also travels in a circle if the orientation of the pedal (in space) is constant.

Another good example of this is a Ferris wheel. The cars of a Ferris wheel have a fixed orientation in space (because gravity causes them to hang directly below their mounting points located on the main wheel) and they trace a circular path through space as the Ferris wheel rotates. The circular path of the cars is not centered on the hub of the main wheel, but on a point that "hangs" several feet beneath the hub of the main wheel. The fact that the cars don't remain at a constant distance from the hub of the main wheel isn't relevant -- the path of the cars is still circular.

We were simplifying the example to make it clearer, based on the diagram that njkayaker, which claimed the pedal top didn't travel in a circle because the distance from the BB center changes. His diagram showed a pedal parallel to the ground at all crank angles, so I have been making the point that in this situation, the pedal top does in fact travel in a circle but that circle - and therefore the center of that circle - is offset from circle traveled by the pedal spindle by the stack height, i.e. if the pedal top is ALWAYS displaced from the pedal spindle at the same angle and distance, and the pedal spindle travels in a circle, the pedal top MUST travel in a circle.

You raise a DIFFERENT question, which is the degree to which ankling changes the path of the pedal top. On average (I found) riders' foot angle changes 24 degrees from minimum to maximum deflection. Look Keos have a stack height to the top of the pedal of about 11.5mm. You can determine how much this would move a point on the pedal top with some simple geometry that I can't be arsed to bother doing. Knock yourself out.

I hope this wasn't mentioned earlier, I skipped a page or two...
As an experiment we make a pedal with the stack height of 10 meters. With the pedal kept perfectly horizontal, you rotate the cranks and make a perfect circle 10 meters in the air above the BB center.
EDIT: What I was trying to emphasize, if stack height contributes to eccentricity, symmetrical or not, wouldn't a 10 meter stack amplify the effect? I've been wrong before.

A quick calculation shows this effect would result in a deviation from circularity of less than 1.5%.

And,... SCENE....

Wow.

Pedantry R Us.

Is it over yet….?🫣

We're only at 5 pages. That's nothing!

You know that Japanese soldier that was still fighting WWII on an Island until 1974? Like that.....

Spatially Challenged ᗺ Yous...

To summarize, if the pedal is horizontal (or does not change angle relative to the horizontal), your foot travels in a perfect circle.

If the pedal angle varies, it’s almost a perfect circle but not quite.

Reading this thread makes me feel
like I’m taking crazy pills. The pedal is directly connected to the spindle. The spindle travels in a circle ergo the pedal travels in a circle. The foot on top of said pedal travels in a circle. If I balanced an apple on a pedal and turned it, the apple would go in a circle! The pedal travels in a circle!

Did anyone here ever do this?
Pedal stroke is different with the shorter crank. That's the whole idea.
In practice, adopting cranks 10mm or 20mm shorter means putting the saddle up 2 to 5mm. For a change in crank length of 5mm or 10mm very possible no one bothers.
Some of us have actually done this rather than spinning tales about theories.

The other one that theorizers always use is shorter cranks mandate lower gears. No, does not happen. The old gears work just fine. Maybe better, as the power stroke is better.

When I went to a shorter crank, I kept my seat height the same because I was looking to both reduce the amount of extension at the bottom as well as limit the knee bend angle at the top of the stroke. This was to resolve some IT band issues. I figure the need to adjust depends on one's goal.