But you aren't translating the top of the pedal "the same amount in the same direction" relative to the center of the crank. The rotation axis would also have to move (which it doesn't do).

Again, you're measuring from the wrong point. The center of the circular path is not the crank spindle -- it's located above the spindle by an amount equal to the stack height.

The center of the crank is the right point. (Well, it's certainly not "wrong".) I was clear about what point I was using/considering. Other people were less clear.

Nope.

On edit: this is such a trivial math problem that it's really not worth spending anymore time on it.

That's a useless "argument".

The interaction is through the center of the crank. Not through some imaginary point chosen to try to make an argument.

A circle is most definitely not an ellipse.
Circle: All points equidistant from a single point.
Ellipse: All points equidistant from two points.

Put those two points close enough together and your ellipse may look like a circle, but it won't be. They are never, ever the same thing.

A circle is just an ellipse where the two foci are coincident. It's analogous to the statement "a square is a rectangle", because a square is just a rectangle with four equal sides.

This doesn't appear to be correct. (Not sure why you didn't look it up before committing yourself.)

Several people have given you the correct explanation of why the path is circular, but you haven't listened to anyone. "Nope" is all that's left.

No, they aren't convincing.

Using the center of the crank isn't "wrong".

(Using the imaginary point is a "circular" argument.)

graph paper. Draw it out 🙄

You don't need graph paper. The distance to the center of the crank obviously varies.



(The distance to some imaginary point doesn't vary but that point is not doing anything physically.)

One might be able to argue that using the imaginary point is equally valid to using the center of the crank but using the center of the crank isn't wrong.

I don't think the imaginary point is equally (or more) valid than the center of the crank.

You really have to make the argument that the imaginary point is the only valid point and that the center of the crank can just be dismissed.

This is exactly it. The center of the circle is the center of the circle, not the center of the BB spindle.

It's an imaginary point. Using the center of the crank is (at least) equally valid.

Then start with the general equation for a circle and introduce a translation -- the resulting equation is still a circle.

Around an imaginary point. The physical point of rotation is still the center of the crank.

No, you're looking at it wrong. Do your drawing again, but this time, make the stack height equal to the crank length. That would mean that as the crank spindle passes through Bottom Dead Center, the pedal top will be in line with the BB center, yes? So the BB center CANNOT BE the center of the path described by the pedal top.

A circle is a circle. It doesn't matter if the center of the circle is an imaginary point in space or the Rock of Gibraltar -- it's still a circle. You're out of your depth on this one.

No, it's not. The center of a circle is a defined point, defined by being equidistant from all points of the circle. You are choosing an arbitrary point and then trying to bend the geometry around it. That ain't how it works.

A tautology.

The crank center isn't "arbitrary". (Calling it "arbitrary" is bizarre.)

This is a "frame of reference" issue. The imaginary point is one frame of reference. The center of the crank is another.

A tautology.

This is a frame of reference issue. You and others think the imaginary point is the only correct frame of reference for some reason.

It's really only valid to make to make the argument that "it's a circle". It's a tautological argument.

You are most definitely wrong on this point.

This.

*sigh*
It's not a tautology, it's inherent in the definition of a circle - "A circle is a perfectly round, two-dimensional shape where all points on its edge are the same distance from a center point." So the center is ALWAYS AND BY DEFINITION the point in space equidistant from all the points on the edge.

It's mind boggling that you have to explain what a circle is. It's depressing that your effort will be in vain.