Is there a way to calculate this transformation?

Namaste & G'day!

Imagine a helicopter view of a Polo ground. It's length & breadth are known.




Now you are seated where the blue dot is. Your view is such:



How do mathematicians calculate the distance travelled by a ball from the second perspective?

From the top view, this would be trivial.

But now your view is transformed.

I think of the second prespective in cylindrical coordinates (r,θ). θ is “easy” to determine, r is more difficult. In a perfect world, one could measure the diameter of the ball to determine its distance. There are other experimental techniques, but I am unsure exactly what you are looking for.

Suppose the eye-point location is at the center of the polar coordinates (##r_{eye}=0##) and the angle, ##\theta##, of the polar coordinates of the ball are known. The distance to the ball location, ##r##, remains to be determined. Assuming a flat earth, ##r## can be calculated using trigonometry. You would need to know the distance to the tree-line. That tree-line has sides and its distance would require some calculations that depend on the direction.