Properties of Rotational Surfaces
Overview
Consider a surface obtained by rotating the curve $\boldsymbol{\alpha}(t) = \left( r(t), z(t) \right)$ about the $z-$ axis, denoted $\mathbf{x}$.
$$ \mathbf{x}(t, \theta) = \left( r(t)\cos \theta, r(t)\sin \theta, z(t) \right) $$
This document discusses various properties of the rotational surface.
Properties
For better readability, let’s denote $r = r(t)$, $z = z(t)$ as follows:
- Partial Derivations
$$ \begin{align*} \mathbf{x}_{1} &= \mathbf{x}_{t} = \left( \dot{r}\cos\theta, \dot{r}\sin\theta, \dot{z} \right) \\ \mathbf{x}_{2} &= \mathbf{x}_{\theta} = \left( -r\sin\theta, r\cos\theta, 0 \right) \\ \mathbf{x}_{11} &= \mathbf{x}_{tt} = \left( \ddot{r}\cos\theta, \ddot{r}\sin\theta, \ddot{z} \right) \\ \mathbf{x}_{12} = \mathbf{x}_{21} &= \mathbf{x}_{t\theta} = \left( -\dot{r}\sin\theta, \dot{r}\cos\theta, 0 \right) \\ \mathbf{x}_{22} &= \mathbf{x}_{\theta\theta} = \left( -r\cos\theta, -r\sin\theta, 0 \right) \\ \end{align*} $$
$$ \mathbf{n} = \dfrac{1}{r\sqrt{\dot{r}^{2} + \dot{z}^{2}}}\left( -r\dot{z}\cos\theta, -r\dot{z}\sin\theta, r\dot{r} \right) $$
$$ \begin{bmatrix} g_{ij} \end{bmatrix} = \begin{bmatrix} \dot{r}^{2} + \dot{z}^{2} & 0 \\ 0 & r^{2} \end{bmatrix} $$
$$ \begin{bmatrix} g^{kl} \end{bmatrix} = \begin{bmatrix} \dfrac{1}{\dot{r}^{2} + \dot{z}^{2}} & 0 \\ 0 & \dfrac{1}{r^{2}} \end{bmatrix} $$
$$ \begin{bmatrix} L_{ij} \end{bmatrix} = \dfrac{1}{\sqrt{\dot{r}^{2} + \dot{z}^{2}}} \begin{bmatrix} \dot{r}\ddot{z} - \dot{z}\ddot{r} & 0 \\ 0 & r\dot{z} \end{bmatrix} $$
$$ \begin{bmatrix} {L^{i}}_{j} \end{bmatrix} = \dfrac{1}{\sqrt{\dot{r}^{2} + \dot{z}^{2}}} \begin{bmatrix} \dfrac{\dot{r}\ddot{z} - \dot{z}\ddot{r}}{\dot{r}^{2} + \dot{z}^{2}} & 0 \\ 0 & \dfrac{\dot{z}}{r} \end{bmatrix} $$
$$ K = \det(L) = \dfrac{(\dot{r}\ddot{z} - \dot{z}\ddot{r})}{(\dot{r}^{2} + \dot{z}^{2})^{2}} \dfrac{\dot{z}}{r} $$
Proofs
Unit Normal Vector
$$ \begin{align*} \mathbf{x}_{1} \times \mathbf{x}_{2} &= \left( \dot{r}\cos\theta, \dot{r}\sin\theta, \dot{z} \right) \times \left( -r\sin\theta, r\cos\theta, 0 \right) \\ &= \left( -r\dot{z}\cos\theta, r\dot{z}\sin\theta, r\dot{r} \right) \end{align*} $$ $$ \begin{align*} \mathbf{n} &= \dfrac{\mathbf{x}_{1} \times \mathbf{x}_{2}}{\left| \mathbf{x}_{1} \times \mathbf{x}_{2} \right|} \\ &= \dfrac{1}{r\sqrt{\dot{r}^{2} + \dot{z}^{2}}}\left( -r\dot{z}\cos\theta, -r\dot{z}\sin\theta, r\dot{r} \right) \end{align*} $$
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First Fundamental Form
$$ \begin{align*} g_{11} = g_{tt} &= \left\langle \mathbf{x}_{t}, \mathbf{x}_{t} \right\rangle = \left( \dot{r}\cos\theta, \dot{r}\sin\theta, \dot{z} \right) \cdot \left( \dot{r}\cos\theta, \dot{r}\sin\theta, \dot{z} \right) \\ &= \dot{r}^{2} + \dot{z}^{2} \end{align*} $$
$$ \begin{align*} g_{12} = g_{21} = g_{t\theta} &= \left\langle \mathbf{x}_{t}, \mathbf{x}_{\theta} \right\rangle = \left( \dot{r}\cos\theta, \dot{r}\sin\theta, \dot{z} \right) \cdot \left( -r\sin\theta, r\cos\theta, 0 \right) \\ &= 0 \end{align*} $$
$$ \begin{align*} g_{22} = g_{\theta\theta} &= \left\langle \mathbf{x}_{\theta}, \mathbf{x}_{\theta} \right\rangle =\left( -r\sin\theta, r\cos\theta, 0 \right) \cdot \left( -r\sin\theta, r\cos\theta, 0 \right) \\ &= r^{2} \end{align*} $$
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Second Fundamental Form
$$ \begin{align*} L_{11} =L_{tt} = \left\langle \mathbf{x}_{tt}, \mathbf{n} \right\rangle &= \dfrac{1}{r\sqrt{\dot{r}^{2} + \dot{z}^{2}}} \left( \ddot{r}\cos\theta, \ddot{r}\sin\theta, \ddot{z} \right) \cdot \left( -r\dot{z}\cos\theta, -r\dot{z}\sin\theta, r\dot{r} \right) \\ &= \dfrac{1}{r\sqrt{\dot{r}^{2} + \dot{z}^{2}}} \left( -r\ddot{r}\dot{z}\cos^{2}\theta - r\ddot{r}\dot{z}\sin^{2}\theta + r\dot{r}\ddot{z} \right) \\ &= \dfrac{1}{\sqrt{\dot{r}^{2} + \dot{z}^{2}}} \left( r\dot{r}\ddot{z} -\ddot{r}\dot{z} \right) \end{align*} $$
$$ \begin{align*} L_{12} = L_{21} =L_{t\theta} = \left\langle \mathbf{x}_{t\theta}, \mathbf{n} \right\rangle &= \dfrac{1}{r\sqrt{\dot{r}^{2} + \dot{z}^{2}}} \left( -\dot{r}\sin\theta, \dot{r}\cos\theta, 0 \right) \cdot \left( -r\dot{z}\cos\theta, -r\dot{z}\sin\theta, r\dot{r} \right) \\ &= \dfrac{1}{r\sqrt{\dot{r}^{2} + \dot{z}^{2}}} \left( r\dot{r}\dot{z}\cos\theta\sin\theta - r\dot{r}\dot{z}\cos\theta\sin\theta \right) \\ &= 0 \end{align*} $$
$$ \begin{align*} L_{22} =L_{\theta\theta} = \left\langle \mathbf{x}_{\theta\theta}, \mathbf{n} \right\rangle &= \dfrac{1}{r\sqrt{\dot{r}^{2} + \dot{z}^{2}}} \left( -r\cos\theta, -r\sin\theta, 0 \right) \cdot \left( -r\dot{z}\cos\theta, -r\dot{z}\sin\theta, r\dot{r} \right) \\ &= \dfrac{1}{r\sqrt{\dot{r}^{2} + \dot{z}^{2}}} \left( r^{2}\dot{z}\cos^{2}\theta + r^{2}\dot{z}\sin^{2}\theta \right) \\ &= \dfrac{1}{\sqrt{\dot{r}^{2} + \dot{z}^{2}}} \left( r\dot{z} \right) \end{align*} $$
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Weingarten Map
$$ \begin{align*} \begin{bmatrix} {L^{i}}_{j} \end{bmatrix} &= \begin{bmatrix} {L^{1}}_{1} & {L^{1}}_{2} \\ {L^{2}}_{1} & {L^{2}}_{2} \end{bmatrix} = \begin{bmatrix} g^{li} \end{bmatrix} \begin{bmatrix} L_{ik} \end{bmatrix} \\ &= \dfrac{1}{\sqrt{\dot{r}^{2} + \dot{z}^{2}}} \begin{bmatrix} \dfrac{1}{\dot{r}^{2} + \dot{z}^{2}} & 0 \\ 0 & \dfrac{1}{r^{2}} \end{bmatrix} \begin{bmatrix} \dot{r}\ddot{z} - \dot{z}\ddot{r} & 0 \\ 0 & r\dot{z} \end{bmatrix} \\ &= \dfrac{1}{\sqrt{\dot{r}^{2} + \dot{z}^{2}}} \begin{bmatrix} \dfrac{\dot{r}\ddot{z} - \dot{z}\ddot{r}}{\dot{r}^{2} + \dot{z}^{2}} & 0 \\ 0 & \dfrac{\dot{z}}{r} \end{bmatrix} \end{align*} $$
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