Equation of Circle/Parametric

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Theorem

The equation of a circle embedded in the Cartesian plane with radius $R$ and center $\tuple {a, b}$ can be expressed as a parametric equation:

$\begin {cases} x = a + R \cos t \\ y = b + R \sin t \end {cases}$


Proof

Let the point $\tuple {x, y}$ satisfy the equations:

$x = a + R \cos t$
$y = b + R \sin t$

By the Distance Formula, the distance between $\tuple {x, y}$ and $\tuple {a, b}$ is:

$\sqrt {\paren {\paren {a + R \cos t} - a}^2 + \paren {\paren {b + R \sin t} - b}^2}$

This simplifies to:

$\sqrt {R^2 \cos^2 t + R^2 \sin^2 t} = R \sqrt {\cos^2 t + \sin^2 t}$

Then by Sum of Squares of Sine and Cosine, this distance equals $R$.

Therefore the distance between points satisfying the equation and the center is constant and equal to the radius.

$\blacksquare$


Sources