Why may geodesic not be the shortest path on a surface?

The point is that local minimization does not imply global minimization. Local minimization says there is no nearby path that is shorter. That does not guarantee that there is no shorter path. Two comments give examples where you can find a local minimum in the sense that no nearby path is shorter, but if you are clever enough to find a very different path you will find it shorter. It is similar to the failures of greedy algorithms. In the path case, we assume that the path we want is reachable with small perturbations of the path we have. The examples show where that is not the case. In failures of a greedy algorithm, early choices constrain the global solution, and a later choice may show that the early choice was not correct.


Attach a image example for the above explanation, The geodesic between A and B longer than the curve from A to B:

enter image description here

Tags:

Differential Geometry

Geodesic

Related

How different can equivalent categories be? What are better approximations to $\pi$ as algebraic though irrational number? What are the conditions on $a$ such that the polynomial $x^4-2ax^2+x+a^2-a$ has four distinct real roots? Limit with natural log in the denominator: $\lim_{x\to1}{\frac{x^2 - 1}{\ln x}}$ Prove that $-\sqrt{2}\leq \sin\theta+\cos\theta\leq\sqrt{2}$ An upper bound of binary entropy What is the limit of $\frac{1}{\sqrt{n}}\sum_{k=1}^n\frac{1}{\sqrt{2k-1}+\sqrt{2k+1}}$? Prove that the family of curves $\frac{x^2}{a^2+\lambda}+\frac{y^2}{b^2+\lambda}=1$,where $\lambda$ is a parameter,is self orthogonal. Limit involving Gamma function How do you calculate a sum over a polynomial? What is the appropriate method to find the value of $1$ - $1\over 7$ + $1\over 13$ - ... upto infinite terms? $A \in B$ vs. $A \subset B$ for proofs

Recent Posts