Angled parking bays along a kerb
Angles, Parallel Lines and Translation
A car park serves hundreds of drivers and not one of them had to learn a second manoeuvre. The same turn of the wheel works in every bay. For that to hold, the bays cannot have been drawn one by one: each has to be the previous one, slid along the kerb. That single demand rules out almost everything the line-marking machine could have painted.
Three ways to see it
The driver runs parallel to the kerb, turns the wheel through one fixed angle and rolls in. At the next bay the very same movement happens, just a few metres along. The manoeuvre path in this scene was drawn once and copied to the neighbouring bays by sliding it — nothing else was done to it. If the bays sat at different angles, every bay would demand its own manoeuvre and the car park would be unusable.
Switch layers — the scene stays put
What's really going on
The angle of a parking bay is not a matter of taste; it is what a constraint leaves behind. A car park can teach one manoeuvre and no more. That forces every bay to be the previous one translated — and translation is the single transformation that carries a shape without turning or resizing it, which is to say the single transformation that leaves DIRECTION untouched. The moment direction is preserved the bay lines have no option but to be parallel; the kerb, crossing all of them, becomes a transversal; and the pair of angles a transversal makes cannot drift along the row. So 'corresponding angles are equal' is not a rule to memorise here — it is another way of saying that the same manoeuvre works in every bay. The size of the angle is not arbitrary either: a shallower angle makes the turn easier but eats more kerb per car. The car park is a bargain struck between those two pressures — but whichever angle wins, it has to repeat.
The equation
y = tan θ · (x − k·d)
θ is the bay angle, d the translation step along the kerb, k the bay's index. tan θ is the slope of the line, that is Δy / Δx. Only x has k·d taken from it — a translation slides a line, it cannot tilt one. Because the slope does not depend on k, every bay line is parallel to the rest and the kerb (y = 0) meets all of them at the same angle θ.
Same concept, elsewhere
Memorising one example gets you nowhere. You need to spot the concept wherever it turns up.
Escalator — every step is the one before it shifted by a fixed vector, which is why they all ride at a single slope
Zebra crossing — equally spaced parallel bands cut by the road edge, so the same pair of angles repeats all the way over
Aircraft seat row — one row is designed, then translated down the cabin at a fixed pitch, so every passenger sits down the same way
Plough furrows — the tractor repeats one manoeuvre, so the furrows come out parallel and evenly spaced, with the field edge cutting across them all
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