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I am working on a problem right now that I think is out of my league mathematically. I will do my best to describe the problem, please feel free to ask for clarification in the comments!

I'd like to figure out the equations to model a circular/elliptical shape on the ground, when the source (above the ground) is rotating. For example, if I was to stand and point a flashlight at the ground, the projected light would be circular. If I angle the flashlight (rotating my wrist L and R) the shape becomes elliptical. Now the confusing part- Let's say there is an ant on the ground, and i need to rotate my wrist left and right to scan for this ant with my flashlight. When I find the ant, I want to keep the flashlight fixed on it, but I am moving away from the ant holding the flashlight at constant velocity. A corrective angle perpendicular to the flashlights rotation will need to be applied for the flashlight to stay focused on the ant. I believe this may use matrices but I have also attempted a calculus solution, I just have ended up so tangled up that I get lost. If any part of that isn't clear or if this is just a nonsense question feel free to let me know, thank you in advance for any help/advice!

Alex Ravsky
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pbthehuman
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  • You might be better off simplifying your cone of light to just its central axis, then arrange to keep the ant on that axis my rotating the flashlight. That axis is a line, so is a simpler object. – Eric Towers Apr 07 '20 at 16:09
  • @eric towers updated the statement to clarify that the ants original position is not known,thank you! – pbthehuman Apr 07 '20 at 16:14
  • See https://math.stackexchange.com/questions/3103023/viewing-a-circle-from-different-angles-is-the-result-always-an-ellipse/3103301, https://math.stackexchange.com/q/2351662/265466 and https://math.stackexchange.com/q/2346896/265466, for starters. – amd Apr 07 '20 at 19:14
  • Do you want the resulting ellipse to be centered on the ant, or just have the ant somewhere within the ellipse? If the latter, then @EricTowers suggestion greatly simplifies the calculations. – amd Apr 07 '20 at 19:16
  • @amd I would like the ant somewhere in the ellipse – pbthehuman Apr 08 '20 at 15:10
  • In that case, point the light cone’s axis at the ant, as EricTowers suggests. Then the questions that I linked above cover the necessary calculations. – amd Apr 08 '20 at 17:16

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