Tagged Questions
5
votes
1answer
164 views
Geometric proof of the geodesics of a sphere?
I have seen the standard variational proof that great circles are the geodesics on the $2$-sphere. Do you know a purely geometric proof of this fact, not involving calculus of variations or ...
1
vote
1answer
58 views
Maximium of area/perimeter^2 of a function
Let $a,b$ be two reals. Does there exists a constant $C$ such that for all functions $f:[a,b]\to\mathbb{R}$, continuous on $[a,b]$ and differentiable on $(a,b)$ with $f(a)=f(b)=0$,
\begin{equation}
...
0
votes
1answer
69 views
Lines through the origin and Euler-Lagrange
What form should the minimal-length curve to $\int \sqrt{dr^2 +r^2d\theta^2\over{1-r^2}}$ take? I think I can use the Euler-Lagrange equations. So write the integral as $\int\sqrt{({dr\over ...
2
votes
1answer
70 views
Infimum length of curves
Let the unit disc $\{(x,y): r^2=x^2+y^2<1\}\subset\mathbb R^2$ be equipped with the Riemannian metric $dx^2 +dy^2\over 1-(x^2+y^2)$. Why does it follow that the shortest/infimum length of curves ...
5
votes
1answer
353 views
Shortest path on hyperboloid
On the sphere $S^2$, the shortest path between two points is the great circle path. How about $H^2$, the hyperboloid $x^2+y^2-z^2=-1, z\ge 1$, with the Euclidean distance? Is there a formula for the ...
12
votes
7answers
2k views
Why Circle encloses largest Area?
In this wikipedia, article http://en.wikipedia.org/wiki/Circle#Area_enclosed its stated that the circle is the closed curve which has the maximum area for a given arc length. First, of all i would ...
