Ellipsoid

27 Nov 2024

This is the first ellipse-related shape that we can actually compute!

Parameterizing the Volume

We use modified spherical coordinates, which we have to express in Cartesian coordinates.

\[\b{r}(t, \theta, \phi) = at \sin \theta \; \cos \phi \; \u{x} + bt \sin \theta \sin \phi \; \u{y} + ct \cos \theta \; \u{z} \\[10pt] V = \{ \b{r}(t, \theta, \phi) \ : \ 0 \leq t \leq 1 \quad 0 \leq \theta \leq \pi \quad 0 \leq \phi < 2\pi \}\]

Therefore

\[\begin{align} \frac{\partial \b{r}}{\partial t} &= a \sin \theta \cos \phi \; \u{x} + b \sin \theta \sin \phi \; \u{y} + c \cos \theta \; \u{z} \\[10pt] \frac{\partial \b{r}}{\partial \theta} &= at \cos \theta \cos \phi \; \u{x} + bt \cos \theta \sin \phi \; \u{y} - ct \sin \theta \; \u{z} \\[10pt] \frac{\partial \b{r}}{\partial \phi} &= - at \sin \theta \sin \phi \; \u{x} + bt \sin \theta \cos \phi \; \u{y} \end{align}\]

This is pretty nasty because we have to do everything in Cartesian coordinates.

\[\begin{align} dV &= \left \lvert \begin{array}{ccc} a \sin \theta \cos \phi \; dt & b \sin \theta \sin \phi \; dt & c \cos \theta \; dt \\ at \cos \theta \cos \phi \; d\theta & bt \cos \theta \sin \phi \; d\theta & - ct \sin \theta \; d\theta \\ - at \sin \theta \sin \phi \; d\phi & bt \sin \theta \cos \phi \; d\phi & 0 \end{array} \right \rvert \\[10pt] &= (- at \sin \theta \sin \phi \; d\phi) \cdot \left \lvert \begin{array}{cc} b \sin \theta \sin \phi \; dt & c \cos \theta \; dt \\ bt \cos \theta \sin \phi \; d\theta & - ct \sin \theta \; d\theta \end{array} \right \rvert - (bt \sin \theta \cos \phi \; d\phi) \cdot \left \lvert \begin{array}{cc} a \sin \theta \cos \phi \; dt & c \cos \theta \; dt \\ at \cos \theta \cos \phi \; d\theta & - ct \sin \theta \; d\theta \end{array} \right \rvert + (0) \cdot \left \lvert \begin{array}{cc} b \sin \theta \sin \phi \; dt & c \cos \theta \; dt \\ bt \cos \theta \sin \phi \; d\theta & - ct \sin \theta \; d\theta \end{array} \right \rvert \\[10pt] &= (- at \sin \theta \sin \phi \; d\phi) \left [ - bc \; t \; \sin^2 \theta \sin \phi \; dt \; d\theta - bc \; t \; \cos^2 \theta \sin \phi \; dt \; d\theta \right ] + (bt \sin \theta \cos \phi \; d\phi) \left [ ac \; t \; \sin^2 \theta \cos \phi \; dt \; d\theta + ac \; t \; \cos^2 \theta \cos \phi \; dt \; d\theta \right ] \\[10pt] &= abc \; t^2 \; \sin \theta \left [ (\sin^2 \theta + \cos^2 \theta) \sin^2 \phi + (\sin^2 \theta + \cos^2 \theta) \cos^2 \phi \right ] \; dt \; d\theta \; d\phi \\[10pt] &= abc \; t^2 \; \sin \theta \left [ \sin^2 \phi + \cos^2 \phi \right ] \; dt \; d\theta \; d\phi \\[10pt] &= abc \; t^2 \; \sin \theta \; dt \; d\theta \; d\phi \end{align}\]

Mass

\[\begin{align} M &= \int dm \\[10pt] &= \rho \int dV \\[10pt] &= \rho \int_{0}^{1} \int_{0}^{\pi} \int_{0}^{2\pi} abc \; t^2 \; \sin \theta \; dt \; d\theta \; d\phi \\[10pt] &= \rho \cdot abc \left ( \int_{0}^{1} t^2 \; dt \right ) \left ( \int_{0}^{\pi} \sin \theta \; d\theta \right ) \left ( \int_{0}^{2 \pi} d\phi \right ) \\[10pt] &= \rho \cdot abc \left ( \frac{1}{3} \right ) \left ( 2 \right ) \left ( 2 \pi \right ) \\[10pt] &= \rho \cdot \tfrac{4}{3} \pi abc \end{align}\]

Moment of Inertia About Any Diameter

\[\begin{align} I &= \int r_{axis}^2 \; dm \\[10pt] &= \rho \int r_{axis}^2 \; dV \\[10pt] &= \rho \int_{0}^{1} \int_{0}^{\pi} \int_{0}^{2 \pi} (a^2 t^2 \sin^2 \theta \cos^2 \phi + b^2 t^2 \sin^2 \theta \sin^2 \phi) \; abc \; t^2 \sin \theta \; dt \; d\theta \; d\phi \\[10pt] &= \rho \cdot abc \left ( \int_{0}^{1} t^4 \; dt \right ) \left ( \int_{0}^{\pi} \sin^3 \theta \; d\theta \right ) \left ( \int_{0}^{2\pi} (a^2 \cos^2 \phi + b^2 \sin^2 \phi) d \phi \right ) \\[10pt] &= \rho \cdot abc \left ( \frac{1}{5} \right ) \left ( \frac{4}{3} \right ) \left ( \pi a^2 + \pi b^2 \right ) \\[10pt] &= \rho \cdot \tfrac{4}{15} \pi abc (a^2 + b^2) \\[10pt] &= \tfrac{1}{5} M (a^2 + b^2) \end{align}\]

Inertia Tensor

\[I = \begin{bmatrix} \frac{1}{5} M (b^2 + c^2) & 0 & 0 \\ 0 & \frac{1}{5} M (c^2 + a^2) & 0 \\ 0 & 0 & \frac{1}{5} M (a^2 + b^2) \end{bmatrix} = \tfrac{1}{5} M \begin{bmatrix} b^2 + c^2 & 0 & 0 \\ 0 & c^2 + a^2 & 0 \\ 0 & 0 & a^2 + b^2 \end{bmatrix}\]