Change of direction of rotation with change of perspective

Question

Take an apple. Rotate it around its vertical axis to the right (relative to you). If you look at it from above (your head above the apple) and take a point there while the apple is rotating, you will notice that the point´s trajectory is a circle going counterclockwise. If you look at it from below (your head below the apple) and take a point there, you will notice this time that the point`s trajectory is a circle going clockwise.

My question is why exactly does this happen? Why does the change of perspective changes how we see the direction of rotation of the point, knowing that we´re still seeing that the apple is rotating to the right? Is it some property of the sphere? Or what is it?

A detailed answer would be very much appreciated. Because most of what i find on the net is just citing that the change of perspective changes the direction, but why?

Thank you.

Answer 1

direction relative to you is dependent on your system of coordinates, that is, where are you located and where are you heading. The simplest case would be linear motion. If I shoot a bullet heading in the direction of the bullet's motion, the bullet will move in the positive x axis (we assume that the positive direction is the direction of the observer's heading). But if I shoot the bullet looking backwards, the bullet will move in the negative direction, even if we are in the same physical situation. It is about how motion differs across different systems of coordinates.

Exactly the same happens with your rotating example. The system of coordinates has changed in direction (from positive down to positive up)

Answer 2

When you change the point of perspective from 'looking above' to 'looking below', this is a parity transformation:

$$x\to -x;\ y\to-y;\ z\to-z$$

or for a function in spherical coordinates:

$$f(r, \theta, \phi)\to f(r, \pi-\theta, \pi+\phi)$$

Now suppose a sphere is oriented such that it is rotating about the $z$ axis, and the sphere is being looked at directly down it. Then the $\phi$ part of the coordinates (the angle that is changing w.r.t the rotation) changes as

$$\sin(\phi)\to-\sin(\phi),$$

so the direction that the sphere appears to be precessing appears to be reversed.

Answer 3

It is not a property of a sphere- it arises from your perception of left and right.

If you hold an object if front of you at eye level and rotate it fractionally clockwise, you will see that the upper part of the object moves to your right while the lower moves to your left. Likewise, the right hand side of the object is moving down while the left is moving up. It is that combination of motions that you call clockwise. To someone who is standing facing you, viewing the object from the other side, they see all of the directions of motion reversed. The top of the object is moving to their left, the bottom to their right, and so on, so to them it is rotating counter-clockwise.

So the effect is simply a consequence of the fact that what you consider to be the left and right hand sides of an object depends upon the direction from which you are looking at it. If you hold a pen horizontally lengthways-on you might have its nib to the right- if I am facing you I will say that you are holding the pen with its nib to the left. The switch in direction of rotation is no more than that.

Answer 4

All motion is relevant to the observers frame of reference. By changing your frame of reference from the top "Northern" axis to the bottom "Southern" axis you have reversed your viewpoint.