DeepCurves in space

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §12.2 Plane Motion: Projectiles and Cycloids, #36 · Multivariable Calculus · Curves in space

Explain why … and … for the point … on the witch…

Explain why $x = 2 a cot θ$ and $y = 2 a sin^{2} θ$ for the point $P$ on the witch of Agnesi. Eliminate $θ$ to find the $x y$ equation.
Note: Maria Agnesi wrote the first three-semester calculus text ('Hôpital didn't do integral calculus). The word "witch" is a total mistranslation, nothing to do with her or the curve.

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §12.2 Plane Motion: Projectiles and Cycloids, #36

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