Remember:
x = r \cos x because initial angle \theta is at x direction
r = \rho \sin \varphi because initial angle \varphi is NOT at r direction
Cylinder Coordinates: (r, \theta, z)
z = k: a plane
r = k \iff x^2 + y^2 = k^2: a cylinder
\theta = k: half plane
z = r^2 \cos^2 \theta \iff z = x^2: cylinder over a parabola
r = 3 \sin \theta: circular cylinder over circle of radius \frac{3}{2} centered at (0, \frac{3}{2})
r = z \iff x^2 + y^2 = z^2 (z \geq 0): half-cone
r^2 + z^2 = 9: sphere
Spherical Coordinates: (\rho, \theta, \varphi) | \rho > 0 \land 0 \leq \varphi \leq \pi
p = k: sphere at radius \sqrt{k} centered at (0, 0, 0)
\theta = k: half plane
\varphi = k: half cone
\rho = 2 \cos \varphi \implies x^2 + y^2 + z^2 = 2z: sphere centered at (0, 0, 1)
\rho \sin \varphi = 4 \implies r = 4: cylinder
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