Lecture 012

Spherical Visual

Spherical

$\begin{cases} r^2 = x^2 + y^2\\ \tan \theta = \frac{y}{x}\\ \theta = \arctan(\frac{y}{x}) + \pi \text{ if (x,y) in quadrant 2 or 3}\\ \theta = \arctan(\frac{y}{x}) \text{ if (x,y) in quadrant 1 or 4, output negative theta on quadrant 4}\\ x = r\cos \theta\\ y = r\sin \theta\\ \rho = \sqrt{x^2 + y^2 + z^2}\\ \cos \varphi = \frac{z}{\sqrt{x^2 + y^2 + z^2}}\\ x = r \cos \theta = \rho \sin \varphi \cos \theta\\ y = \rho \sin \varphi \sin \theta\\ z = \rho \cos \varphi\\ r = \rho \sin \varphi\\ \end{cases}$

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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