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回転面の性質 📂幾何学

回転面の性質

概要

曲線 α(t)=(r(t),z(t))\boldsymbol{\alpha}(t) = \left( r(t), z(t) \right)zz-軸に対して回転させて得られた回転面x\mathbf{x}としよう。

x(t,θ)=(r(t)cosθ,r(t)sinθ,z(t)) \mathbf{x}(t, \theta) = \left( r(t)\cos \theta, r(t)\sin \theta, z(t) \right)

回転面の様々な性質について説明する。

性質

読みやすさのため、r=r(t)r = r(t)z=z(t)z = z(t)を以下のようにする。

  • 偏微分

x1=xt=(r˙cosθ,r˙sinθ,z˙)x2=xθ=(rsinθ,rcosθ,0)x11=xtt=(r¨cosθ,r¨sinθ,z¨)x12=x21=xtθ=(r˙sinθ,r˙cosθ,0)x22=xθθ=(rcosθ,rsinθ,0) \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*}

n=1rr˙2+z˙2(rz˙cosθ,rz˙sinθ,rr˙) \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)

[gij]=[r˙2+z˙200r2] \begin{bmatrix} g_{ij} \end{bmatrix} = \begin{bmatrix} \dot{r}^{2} + \dot{z}^{2} & 0 \\ 0 & r^{2} \end{bmatrix}

[gkl]=[1r˙2+z˙2001r2] \begin{bmatrix} g^{kl} \end{bmatrix} = \begin{bmatrix} \dfrac{1}{\dot{r}^{2} + \dot{z}^{2}} & 0 \\ 0 & \dfrac{1}{r^{2}} \end{bmatrix}

[Lij]=1r˙2+z˙2[r˙z¨z˙r¨00rz˙] \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}

[Lij]=1r˙2+z˙2[r˙z¨z˙r¨r˙2+z˙200z˙r] \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)=(r˙z¨z˙r¨)(r˙2+z˙2)2z˙r K = \det(L) = \dfrac{(\dot{r}\ddot{z} - \dot{z}\ddot{r})}{(\dot{r}^{2} + \dot{z}^{2})^{2}} \dfrac{\dot{z}}{r}

証明

単位法線ベクトル

x1×x2=(r˙cosθ,r˙sinθ,z˙)×(rsinθ,rcosθ,0)=(rz˙cosθ,rz˙sinθ,rr˙) \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*} n=x1×x2x1×x2=1rr˙2+z˙2(rz˙cosθ,rz˙sinθ,rr˙) \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*}

第1基本形式

g11=gtt=xt,xt=(r˙cosθ,r˙sinθ,z˙)(r˙cosθ,r˙sinθ,z˙)=r˙2+z˙2 \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*}

g12=g21=gtθ=xt,xθ=(r˙cosθ,r˙sinθ,z˙)(rsinθ,rcosθ,0)=0 \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*}

g22=gθθ=xθ,xθ=(rsinθ,rcosθ,0)(rsinθ,rcosθ,0)=r2 \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*}

第2基本形式

L11=Ltt=xtt,n=1rr˙2+z˙2(r¨cosθ,r¨sinθ,z¨)(rz˙cosθ,rz˙sinθ,rr˙)=1rr˙2+z˙2(rr¨z˙cos2θrr¨z˙sin2θ+rr˙z¨)=1r˙2+z˙2(rr˙z¨r¨z˙) \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*}

L12=L21=Ltθ=xtθ,n=1rr˙2+z˙2(r˙sinθ,r˙cosθ,0)(rz˙cosθ,rz˙sinθ,rr˙)=1rr˙2+z˙2(rr˙z˙cosθsinθrr˙z˙cosθsinθ)=0 \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*}

L22=Lθθ=xθθ,n=1rr˙2+z˙2(rcosθ,rsinθ,0)(rz˙cosθ,rz˙sinθ,rr˙)=1rr˙2+z˙2(r2z˙cos2θ+r2z˙sin2θ)=1r˙2+z˙2(rz˙) \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*}

ワインガルテンマップ

[Lij]=[L11L12L21L22]=[gli][Lik]=1r˙2+z˙2[1r˙2+z˙2001r2][r˙z¨z˙r¨00rz˙]=1r˙2+z˙2[r˙z¨z˙r¨r˙2+z˙200z˙r] \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*}


  1. Richard S. Millman and George D. Parker, Elements of Differential Geometry (1977), p101 ↩︎

  2. Richard S. Millman and George D. Parker, Elements of Differential Geometry (1977), p108 ↩︎