【讲解清晰生动,深入浅出,通俗易懂】什么是测地线(geodesi
![](https://img.haomeiwen.com/i1233356/bcfed0cfc2a3125f.gif)
测地线就是在一个三维物体的表面上找出两个点的最短距离。测地线的具体应用挺广的,比如说飞机船只的航道设计。首先我们知道在二维平面上两点之间线段最短,但若是换到三维这就没办法实现了,因为你无法穿透这个物体以寻求最短距离。所以,我们就得想办法在曲面上面寻求最短距离。因为曲面略微抽象而且路径很多让人感觉无从下手,所以看似很难找。
![](https://img.haomeiwen.com/i1233356/ed5842584d2750cc.png)
其实不然,想象一张纸(假设它的厚度是忽略不计的),你既可以平铺让它处于绝对二维状态,又可以将其折叠成不同形状使其处于三维状态。如果这样想,事情就变简单了。假设你的那张不计厚度的纸处于平面状态,纸上有两个位置不同的点,你可以很容易找到两点之间最短距离。然后,你再将纸折叠成不同形状,尽管此时面不同了,但是两点的最短距离依然还是原先那条线:因为面不管被如何折面积都是不变的。
所以要找到测地线的关键就是把曲面转化成平面的这一步。微积分里面的术语叫parametrization(参数化),先不做过多讲解。当把曲面参数化成二维面之后,我们可以通过微积分求导,最后把二维重新转回三维。
数学语言表达
![](https://img.haomeiwen.com/i1233356/386c1bf78a98e033.png)
![](https://img.haomeiwen.com/i1233356/269e01dcafdc57ee.png)
The geodesic equation
In aRiemannian manifoldMwithmetric tensorg, the lengthLof a continuously differentiable curve γ : [a,b] →Mis defined by
![](https://img.haomeiwen.com/i1233356/dae917f06a3b277e.png)
Another equivalent way of defining geodesics on a Riemannian manifold, is to define them as the minima of the followingactionorenergy functional
![](https://img.haomeiwen.com/i1233356/85c946c9c8905031.png)
TheEuler–Lagrange equationsof motion for the functionalEare then given in local coordinates by
![](https://img.haomeiwen.com/i1233356/b8e89f0e90a06b83.png)
where
![](https://img.haomeiwen.com/i1233356/3b9d67abe52013e9.png)
the Christoffel symbols of the metric
are theChristoffel symbolsof the metric. This is thegeodesic equation.
几何直观表达
![](https://img.haomeiwen.com/i1233356/991b221843e062c7.gif)
![](https://img.haomeiwen.com/i1233356/8badf513720d5094.gif)
![](https://img.haomeiwen.com/i1233356/ecd9fb4f13616a81.gif)
![](https://img.haomeiwen.com/i1233356/1aaaedc3d89c0872.jpg)
![](https://img.haomeiwen.com/i1233356/7f1a7ba1ab8ee415.gif)
![](https://img.haomeiwen.com/i1233356/339c1f9e3fa65ac7.png)
释义
1.ADJ relating to or involving the geometry of curved surfaces 曲面几何学的 (See also geodetic, geodesical)
2.N the shortest line between two points on a curved or plane surface 短程线 (Also called geodesic line)
The existence of the infinite closed geodesics of a compact no-simply connected Riemannian manifold.
紧致的非单连通黎曼流形上无穷多的闭测地线存在性问题?
Geodesics on smooth surface have many good geometric properties and there are equivalent partial differential equations and analytical methods solving it.
测地线在光滑曲面上有很好的几何性质,也有相应的测地线偏微分方程表达以及一些解析的方法来求解。
参考资料
Geodesic Deviation:https://ion.uwinnipeg.ca/~vincent/4500.6-001/Cosmology/GeodesicDeviation.htm
https://www.zhihu.com/question/22274518/answer/42849207
https://www.markushanke.net/tag/geodesic-equation/