How to define a finite objects with parametric equations

Question

I never had seen parametric equations, but while trying to learn line integrals through Wikipedia, quickly found these equations are remarkable. Some can represent things for which more normal equations or functions are needed, if at all possible.

However, the page about them in Wikipedia is not very long, and I didn't learn as much about them as I would like. Being so, I didn't find described any limitation for them. They can make many more finite shapes than normal equations.

But can define stuff with clear ends, such as a curve with two separate end-points, or just a line segment? I have no idea how to find these hypothetical parametric equations, as I said, I didn't find limitations, nor examples to this. Is this possible, or does it require "parametric inequations"?

Answer 1

$$\begin{align} x &= 0 \\ y &= \sin t \end{align}$$

Answer 2

The parametric equations $$x = t^2 \qquad y = 0$$ give a line with one endpoint. The parametric equations $$x = \sin(t) \qquad y = 0$$ give a line with two end points.

Answer 3

You do this by defining an interval for the parameter.

Take a simple line: $x = y = z = t.$ The (infinite) line implies that the parameter $t$ can take on any value.

The line segment from $(-1, -1, -1)$ to $(2, 2, 2)$ means restricting $t$ to the interval $[-1, 2]$.