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We should have always asked more persistently why the arc of the cycloid is 8r while the circumference is 2πr. As a matter of kinematics, it makes no sense. The same point draws both, so why the 21% miss? I will be told that it is because with the circumference, the circle is not moving along an x-axis, but with the cycloid, it is. It is the difference between a rolling circle and a non-rolling circle. It is this lateral movement that adds the 21%. But whoever is telling me this is missing a very important point: in the kinematic circle I am talking about, the circle is also rolling. If you are in an orbit, for instance, the circle is not moving laterally, but a point on the circle is moving. The circle is rolling in place, and it is moving exactly like the point in the cycloid. Therefore, we see it is not the lateral motion that adds the 21%, it is the rolling alone. A static circle and a circle drawn by motion are not the same. The number π works only on the given static circle, in which there is no motion, no time, and no drawing. Any real-world circle drawn in time by a real object cannot be described with π.

If we study the generation of the cycloid closely, we find more evidence of this, since the arc of the cycloid isn't some sort of integration of the circumference with the distance rolled. It can't be, because some point on the circle is always contiguous with the flat surface. We would have to slide the circle in order to add any of the x-distance traveled
What is actually happening is that with the cycloid, the x-to-y integration of distances is explicitly including time, as you see here:

droid wrote:Just another unfunny attention-whore time waster, with gems like this:

We should have always asked more persistently why the arc of the cycloid is 8r while the circumference is 2πr. As a matter of kinematics, it makes no sense. The same point draws both, so why the 21% miss? I will be told that it is because with the circumference, the circle is not moving along an x-axis, but with the cycloid, it is. It is the difference between a rolling circle and a non-rolling circle. It is this lateral movement that adds the 21%. But whoever is telling me this is missing a very important point: in the kinematic circle I am talking about, the circle is also rolling. If you are in an orbit, for instance, the circle is not moving laterally, but a point on the circle is moving. The circle is rolling in place, and it is moving exactly like the point in the cycloid. Therefore, we see it is not the lateral motion that adds the 21%, it is the rolling alone. A static circle and a circle drawn by motion are not the same. The number π works only on the given static circle, in which there is no motion, no time, and no drawing. Any real-world circle drawn in time by a real object cannot be described with π.

If we study the generation of the cycloid closely, we find more evidence of this, since the arc of the cycloid isn't some sort of integration of the circumference with the distance rolled. It can't be, because some point on the circle is always contiguous with the flat surface. We would have to slide the circle in order to add any of the x-distance traveled
What is actually happening is that with the cycloid, the x-to-y integration of distances is explicitly including time, as you see here:

Flat-earthers, No-forests-on earth, and all these other time wasters should be publicly tarred and feathered, seriously