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You can probably get better angular resolution with lenticular sheets. You see one image if you're dead on, and the other images if you're off to the left or right.

Some napkin math says a couple inches of precision should be achievable at reasonable distances?

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Low-tech solution in my driveway: Put the car into the desired position, then hang up a couple of old tennis balls on strings to just touch it.
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Do hanging tennis balls (and their expensive gadget relatives, laser pointers) help with backing in straight?

In my mental model, the tennis ball tells you when you've backed in / pulled in the appropriate distance. I'm not seeing how it would help me back in straight / centered in my garage, or at least not until I've already backed in so far that I'd need to pull out again to try again.

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Usually "good enough". Garages aren't big enough to be completely crazy in angles. People pick a consistent location for the tennis ball. Their face, a sticker, a mirror, etc. If they are off, its not going to line up correctly.
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My fire hall has spray-paint lines to align the rear wheels with in the mirror, including a short cross-line at the stopping point when backing in. Works great!
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Could you maybe arrange some hints for yourself in the surrounding landscaping?

I have done similar in the past with what I call "taking mental snapshots", but it's always easier if there are things you can change to simplify the process. i.e. make sure to align the side mirrors with the flower pots when I'm backed up enough to begin straightening out (direction of the misalignment tells which way to adjust).

The extreme end of this would, of course, be putting lines down for the rear tires. I'm just curious if there's a better way using what you already have so it doesn't look dorky.

Ultimately the only trick to backing up perfectly is the distance to each side being equal as you reverse in as straight of a line as you can, so I'm not sure it takes more than two surrounding reference points being compared to another two points on the car. Distance to each side is comparing the lower corners of the rear window to the lower corners of the garage opening. Flower pots or something else further away gives you better "angular resolution" when you're straightening up (no idea what else to call it).

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