What is the parametric equation of the curve obtained by the intersection of the sphere $$x^2 + y^2 + z^2 = a^2$$ and the cylinder $$x^2 + y^2 = ay.$$
I do not have any idea how to solve it Please help
2026-03-29 12:04:00.1774785840
Viviani's Curve
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Start with $x,y$ from the cylinder, which is a circle of radius $\frac a2$ centered at $\left(0,\frac a2\right)$, then compute the $z$ from the sphere: $$ \begin{align} x&=\frac a2\sin(\theta)\\ y&=\frac a2-\frac a2\cos(\theta)\\[3pt] z&=a\cos(\theta/2) \end{align} $$ for $\theta\in[0,4\pi]$.