The physics
A path can come as two rules, x(t) and y(t). There is one rule for each coordinate. At any moment t, the particle is at (x(t), y(t)).
To find the shape of the path, remove t. Solve one rule for t and put it into the other. Or use an identity such as cos² t + sin² t = 1.
x = at, y = bt → line y = (b/a) x
x = t, y = t² → parabola y = x²
x = a cos t, y = b sin t → ellipse (x/a)² + (y/b)² = 1