某曲线在单叶旋转双曲面上的测地曲率

青青子衿

求出该曲线\(\,r\left(t\right)=\Big\{\,\cos^3t,\,\sin^3t,\,\cos\left(2t\right)\,\Big\}\,\)在单叶旋转双曲面上的测地曲率。

1. Show[ParametricPlot3D[{Cos[t]^3, Sin[t]^3, Cos[2 t]}, {t, 0, 2 \[Pi]}],
2. ContourPlot3D[4 (x^2 + y^2) - 3 z^2 == 1, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}]]

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？

hbghlyj

Asymptote HTML format
import graph3;

size(8cm);
currentprojection=perspective(5,4,3);

draw(graph(new triple (real t) {return (cos(t)^3, sin(t)^3, cos(2t));}, 0, 2pi), red + 1bp);

surface s = surface(new triple (pair t) {real r = t.x;return (r*cos(t.y), r*sin(t.y), sqrt((4*r^2-1)/3));}, (1/2,0), (1,2pi),30);

draw(s, lightblue,render(merge=true));
draw(zscale3(-1)*s, lightblue,render(merge=true));

xaxis3("$x$",-2,2);yaxis3("$y$",-2,2);zaxis3("$z$",-2,2);

hbghlyj

Geodesic Curvature Problems
The following problems require you to review the definition of geodesic curvature of a curve $γ$ on a surface.
The most straightforward formula for $κ_g$ in this context is \begin{equation}\kappa_{g}=\vec{\kappa} \cdot(\vec{n} \times \vec{T})=\frac{\gamma^{\prime \prime}(t) \cdot\left(\vec{n} \times \gamma^{\prime}(t)\right)}{\left\|\gamma^{\prime}(t)\right\|^{3}}\label1\end{equation}$\vv n$ being the surface normal.

hbghlyj

将曲线上的点旋转$s$得双曲面的一个参数化
$$\sigma(s,t)=\left\{\cos (s) \cos ^3(t)-\sin (s) \sin ^3(t),\sin (s) \cos ^3(t)+\cos (s) \sin ^3(t),\cos(2t)\right\}$$
曲线就能表示为$\gamma(t)=\sigma(0,t)$

1. \[Sigma]={Cos[s] Cos[t]^3-Sin[s] Sin[t]^3,Cos[t]^3 Sin[s]+Cos[s] Sin[t]^3,Cos[2t]};
2. n=Normalize[Cross[D[\[Sigma],s],D[\[Sigma],t]]]/.s->0//ComplexExpand//FullSimplify

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$$\vv n=\frac{\left\{-4 \sqrt{2} \cos ^3(t),-4 \sqrt{2} \sin ^3(t),3 \sqrt{2} \cos (2 t)\right\}}{\sqrt{21 \cos (4 t)+29}}$$代入\eqref{1}

1. n={-4 Sqrt[2] Cos[t]^3,-4 Sqrt[2] Sin[t]^3,3 Sqrt[2] Cos[2 t]}/Sqrt[21Cos[4t]+29];
2. \[Gamma]={Cos[t]^3,Sin[t]^3,Cos[2 t]};
3. D[\[Gamma],{t,2}].Cross[n,D[\[Gamma],t]]/ComplexExpand[Norm[D[\[Gamma],t]]]^3//TrigReduce//FullSimplify

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$$\kappa_g=-\frac{12 \cos (2 t)}{5 \sqrt{1-\cos (4 t)} \sqrt{21 \cos (4 t)+29}}$$不知道对不对