Different Parametrizations, Same Curve

A parametrization is a set of instructions, not just a shape. x(t) and y(t) tell a point where to be at every moment t, so the same picture in the xy-plane can come from infinitely many different instruction sets. A circle of radius 1 can be traced by x=cos(t), y=sin(t), or by x=cos(3t), y=sin(3t) which goes three times faster, or by x=cos(-t), y=sin(-t) which runs backwards. Eliminate t from any of these and you get the same equation, x²+y²=1, because the geometry does not care how fast or which way you got there.

The key move is substitution inside t itself. Replacing t with 2t in x(t)=cos(t), y(t)=sin(t) does not change which points appear, since cos(2t) and sin(2t) still only output values that satisfy x²+y²=1. What changes is the schedule: the point that used to arrive at angle π at time t=π now arrives there at time t=π/2, so the whole trip is compressed to half the time. This is exactly like driving the same highway route at double the speed: same road, same destinations, different clock.

This connects directly to the idea of eliminating the parameter from 4.1 and 4.2: no matter which version of t you use, solving for t and substituting back gives one rectangular equation for the curve. It also sets up vectors, where position, velocity, and speed depend on how a path is parametrized even though the path itself stays fixed.