Question

How do you find the parametric equations of a circle?

Answer 1

We'll start with the parametric equations for a circle:

`y = rsin t`
`x = rcos t`

where `t` is the parameter and `r` is the radius.

If you know that the implicit equation for a circle in Cartesian coordinates is `x^2 + y^2 = r^2` then with a little substitution you can prove that the parametric equations above are exactly the same thing.

We will take the equation for `x`, and solve for `t` in terms of `x`:

`x/r = cos t`
`t = arccos (x/r)`

Now substitute into the equation for `y`. This eliminates the parameter `t` and gives us an equation with only `x` and `y`.

`y = rsin arccos(x/r)`

`sin arccos(x/r)` is equal to `sqrt(r^2 - x^2)/r`. This is apparent if one sketches a right triangle, letting `theta = arccos(x/r)`:

Thus, `sin theta = sqrt(r^2 - x^2)/r`. So now we have

`y = r*sqrt(r^2 - x^2)/r`

This simplifies to

`y = sqrt(r^2 - x^2)`

If we square this entire deal and solve for `r`, we get:

`r^2 = x^2 + y^2`

which is precisely the equation for a circle in Cartesian coordinates.