parametric curves

A parametric curve is a curve which is defined by a two dimensional equation $P$ of one parameter $t$. The two coordinates of the vector $P(t)$ are the $x$ and $y$ coordinates of the point of the curve corresponding to a particular value of the parameter.

One curve can be defined by several different parametric equations like $P_1$ and $P_2$. This means that for each $t_1$ in the range of the first equation, another value $t_2$ in the range of the second equation can be found such that $P_1(t_1)=P_2(t_2)$. The relationship between $t_1$ and $t_2$ can be either simple or very complex depending on the equations. The $f$ function that transforms $t_1$ into $t_2$, $t_2=f(t_1)$, is monotonic. If $t_2$ increases when $t_1$ increases, the two equations define the same orientation for the curve, otherwise they define opposite orientations.