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Math thing that currently interests me...

Suppose you had 2D parametric function

x(t), y(t)

with the property that differential vector is always a unit vector

| x'(t), y'(t) | = 1

You then convert that to polar coordinates

r'(t), φ'(t)

Since r'(t) is always 1, it can be dismissed, and φ'(t) alone describes the traced shape.

Problem: Given φ'(t), how to find the original 2D shape?

Analogy: Find the position of a car moving at constant speed, given the steering wheel angle over time.