Write vector and parametric equations of curves and convert between vector, parametric and cartesian forms

What this dot point is asking

SCSA wants you to move fluently between the vector form of a curve, its parametric component equations, and its cartesian equation, and to recognise the standard curves these describe.

From vector to parametric form

A curve in the plane can be written as a position vector that depends on a parameter:

As $t$ ranges over an interval, the tip of $\mathbf{r}(t)$ sweeps out the curve. The two component functions $x = x(t)$ and $y = y(t)$ are the parametric equations of the same curve.

Eliminating the parameter

The resulting relation between $x$ and $y$, free of $t$, is the cartesian equation. Always note any restriction the parameter places on the range of $x$ or $y$, since the parametric curve may be only part of the full cartesian graph.

Recognising standard curves

Some parametrisations recur. A linear $\mathbf{r}(t) = \mathbf{a} + t\mathbf{b}$ traces a straight line. The pair $x = a\cos t$, $y = a\sin t$ traces a circle of radius $a$, since squaring and adding gives $x^2 + y^2 = a^2$. The pair $x = a\cos t$, $y = b\sin t$ traces an ellipse.