Picture your eye at the camera's position. Look up at the flight hemisphere. You will see the visible outline (the "edge", if you will) of the hemisphere. If you draw a bunch of lines from the camera position (a point) out to all the points on that outline, the lines will trace a cone. The intersection of that cone with the hemisphere is a circle (don't take my word for it, prove it for yourself). That circle is a minor circle whose center lies on the line joining the camera's center with the sphere's center. We see the half of that circle that lies above the sphere's equator. Finally, imagine that circle slicing the hemisphere into two chunks, as if with a knife: the part that's closer to the cam, and the far part. The "near" chunk is smaller in volume than the "far" chunk. With me so far?
Now... The issue at hand is that there is a deadband in that circle's region. Alberto referred to this as the "tangent between the sphere and the camera cone" in an earlier post here. An image is composed of pixels, which are whole chunks (I'm not aware of subpixeling detection in the image processing library yet). That means we can only detect the airplane's position in video at whole numbers: pixel 920, then pixel 930, for example. There is no in-between. When the airplane passes through that outline we defined above, its 3D location "skips" between the near and the far side of the outline plane. I found that the skip is quite large -- in one case (21m lines) it's 5.6m (over 18ft!). That's unacceptable for 3D tracking.
To alleviate the issue, I am experimenting with the following idea.
Most stunt figures are performed on the far side of that tangent slice, and at 45° elevation or higher (let's ignore level/inverted flight and wingovers). If we place the camera such that it sees at least the 45° elevation or lower, AND sees the pilot's feet within the frame, then one camera will suffice. We need to see the 45° elevation because the overhead eight is executed in that region near the tangent. All other figures are performed well away from it on the far side.
The constraint then becomes the focal length: we need a wide-angle lens that will fit the picture described above in the frame.
Some numbers for instance:
Sphere radius: 63 ft (60ft lines plus a few feet of arm)
Camera location at 89 ft from sphere center, and 3 ft below equator
This results in elevation angle at tangent equal to about 43°
The minimum view angle is about 48° vertically in the camera's field of view.
For reference, a 10mm lens on an APS-C sensor with 16:9 video crop has a 68° vertical angle of view, so it *should* fit.
I'm renting such a lens this week to try out the idea.
I'm planning to go to the NYCLST 2020 Stunt Championship this weekend. If I can make an arrangement with the CD, I'd love to try it out. As a bonus, I'd get recordings of stunt performances by Expert pilots whose pattern figures are close to perfection