How do you find the parametric equations of a circle?
1 Answer
We'll start with the parametric equations for a circle:
y = rsin t
x = rcos t
where
If you know that the implicit equation for a circle in Cartesian coordinates is
We will take the equation for
x/r = cos t
t = arccos (x/r)
Now substitute into the equation for
y = rsin arccos(x/r)
Thus,
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^2 = x^2 + y^2
which is precisely the equation for a circle in Cartesian coordinates.