Say we are looking at a football field from a broadcast view. To give us another tool, we also have a transform of that view to a topdown view. So our two views are like so: enter image description here

How would one simulate zoom/tilt transformations on the broadcast view and/or the topdown view?

To assist, I have the pan transform already. You find the convergence of the long quadrilaterals lines on the topdown view and then rotate the quadrilateral around the convergence. But things get more tricky when you move into zoom/tilt:

Tilt seems to require finding the point of perspective on the broadcast view on a 3D coordinate system? I will try this but are there any simpler methods?

Zoom in seems to require shifting the corner points of the quadrilateral on the topdown view towards the middle of the quadrilateral by the same proportion?

EDIT: I'm doing this to build a tool to match broadcast scenes of a football match to different perspective views of the field. That way we can transform the view into a topdown view using a saved broadcast to topdown-view homography. To do this we need a set of all possible broadcast views of the field... to create those we can just use the different perspective tranforms from a set of original manually created views (retrieved from football broadcasts). Here is a sample paper I'm using through out this.

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    $\begingroup$ Do you think you could provide some more details about what you are trying to do? In general, you would represent your field rectangle by physical coordinates, place the camera on "the side" and then run a perspective projection. Your intuition about the Zoom is correct. If you are trying to match the virtual camera view to the real camera view, this is called "camera view matching" or "perspective matching". It is a non-linear opt problem and there are different ways of doing it depending on what are you trying to do. $\endgroup$ – A_A Oct 13 '19 at 12:45
  • $\begingroup$ do you know all three spatial coordinates of the object (or the lines defining the object) and you want to project or map those 3-space coordinates to a 2-space "window" that would be your computer screen? that's not so hard. $\endgroup$ – robert bristow-johnson Oct 14 '19 at 4:34
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    $\begingroup$ Isn't there also lens distortion you need to take into account? $\endgroup$ – endolith Oct 14 '19 at 18:03
  • $\begingroup$ @robertbristow-johnson I'm not sure what the object you're referring to is. I made a tool which allows me to get coordinates of four points on the original broadcast image and then the associated 4 points on the topdown image. This allows me to create a homography H from broadcast to topdown. But the problem I'm now trying to solve is to transform either the Topdown or Broadcast view points to simulate pan/zoom/tilt of the broadcast camera. - How does any of that relate to 3-space coordinates or 2-space 'windows'? (: $\endgroup$ – Finn Maunsell Oct 15 '19 at 9:27