Parametric circle
Theorem 8.1 (Parametric circle). For \(0 \leq t \leq 1\),
\[ x(t) = h + r \cos 2\pi t, \qquad y(t) = k + r \sin 2\pi t \]
traces the circle with centre \((h, k)\) and radius \(r\) exactly once, anticlockwise, starting and ending at \((h + r, k)\). Replacing \(2\pi t\) by \(\theta _0 + (\theta _1 - \theta _0)t\) traces only the arc from angle \(\theta _0\) to angle \(\theta _1\).