Parametric Form of a Circle

Parametric Form of a Circle

Setting up the angle parameter. Let $P(x,y)$ be any point on a circle centred at the origin $O$ with radius $r$. Let $\theta$ be the angle that $OP$ makes with the positive direction of the X-axis. Drop a perpendicular $PM$ from $P$ to the X-axis; then $OMP$ is a right triangle with hypotenuse $OP=r$, base $OM=x$, and height $PM=y$. From the right-triangle definitions of sine and cosine,

so

This pair of equations is the parametric form of the circle $x^2+y^2=r^2$, with $\theta$ as the parameter — as $\theta$ ranges over $[0,2\pi)$, the point $(r\cos\theta,r\sin\theta)$ traces out the whole circle exactly once.

Shifted centre. For a circle centred at $(h,k)$ instead of the origin, the same idea (now measuring the angle from a horizontal line through the centre) gives

so any point on this circle can be written as $(h+r\cos\theta,\ k+r\sin\theta)$.

Why bother with a parameter? The parametric form uses a single variable $\theta$ instead of two variables $x,y$ tied together by an equation, which is often more convenient in calculations — for instance when a point's exact coordinates aren't needed, only its angular position, or when working with tangents at a specific point identified by its angle (Section 6.3.1).

Solved Example 1 — parametric equation of $x^2+y^2-6x+4y-3=0$

Step 1 — complete the square. Group the $x$ and $y$ terms: …