The formula for 3D rotation of the perspective of an image in 2D space

Question

Do you know what the formula is for the 3D rotation of the perspective of a 3D object in a 2D space. For example, imagine that we got a picture of a 3D object. So, we have the projected picture of that 3D object in a 2D space. Next time, we rotate the 3D object by z-axis and got another projected picture in 2D space. What will be the formula of the transformation of each point in the second 2D projected picture with respect to the first one in a 2D space. Actually, two 2D projected pictures of the 3D rotated object are being compared.

Answer 1

This is indeed undefined when you are missing the depth information.

If the rotation axis is fixed, all points are moving on ellipses of various extents, as seen on these cylinders, so the new positions aren't completely undefined.

If you have two pictures of the same object, stereoscopy allows you to reconstruct the 3D model.

Answer 2

There is no answer to this question. At least not in the sense that you seem to be asking about. The easiest way to see this is to imagine two points of the 3D object such that in the 2D picture one is behind the other. That means that in the 2D picture they have the same position.

However, once we rotate the 3D object, the two points will separate in the 2D picture. That means that two things that had the same coordinates in the picture before the rotation will have different coordinates after the rotation. This, again, means that the transformation you're asking for is not a function, and thus cannot be applied in any consistent manner to the 2D picture.

Another way of looking at it is this: How something in the picture moves when you rotate the object depends on how far away from you it is. But there is no consistent way of determining distance away from the viewer in a 2D picture (although modern software is getting quite good at calculations like this from video). Therefore there is no way to determine, from the picture, how some part of the object will move when you rotate it.

In order to do what you want, you need to go back to the 3D object, rotate that, and then project down to a 2D picture again.

Answer 3

The 2D picture $P(R(O))$ of the object $O$ after the rotation $R$ is not determined by just the picture $P(O)$ and the rotation $R$. For example, consider a ccube, originally viewed in a direction aligned with a long diagonal, from far off. That object could equally well be a hexagonal prism, of arbitrary length. SO when I apply some rotation, I have no clue what it will look like.

Answer 4

Let $u$ be a vector of the object, then its projection will be $u' = Pu$. Now an additional rotation will be $R$, which leads to the projection $u''=PRu$. What you are looking for seems to be a transformation $T$ with $$ u'' = T u' \iff \\ PR u = T P u $$ Formally this would be $$ T = P R P^{-1} $$ but in practice many projections are not invertible, so an affine $T$ can not be derived that simple.

I would need to look at the specific $P$ and $R$ to get an idea how bad the situation is.