The geodesy tag has no wiki summary.
1
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2answers
14 views
Vector Field Generating Variation Along Curve
I'm learning a proof of the fact that length extremising curves are geodesics of the Levi-Civita connection, and have found something I don't understand. The argument states the following.
Suppose ...
6
votes
1answer
49 views
Hamiltonian for Geodesic Flow
I'm trying to prove that geodesic flow on the cotangent bundle $T^* M$ is generated by the Hamiltonian vector field $X_H$ where
$$H = \frac{1}{2}g^{ij}p_i p_j$$
but I am stuck. Could somebody show ...
4
votes
0answers
39 views
Levi-Civita Connection for 2-dimensional Riemannian manifold
I'm trying to show the following. Suppose $(M, g)$ is a $2$-dimensional Riemannian manifold with connection $\nabla$. Suppose also that $\nabla$ is metric compatible, and that length extremizing ...
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0answers
17 views
Geodesic on a Hilbert manifold
Given a Hilbert manifold $\mathcal H$ (always using the natural Hilbert inner product) and a geodesic $\Gamma(t)$ in this manifold, can one show that the projection of this geodesic onto a submanifold ...
1
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0answers
23 views
Boltz method for solving normal equations
Recently I came across an interesting comment in a geodetic paper which follows as:
"Initially, the
normal equations were solved using the Gaussian method
of successive elimination. This method, ...
5
votes
1answer
96 views
Geodesic of a Surface in $\mathbb{R}^3$
I'm not familiar with geodesics. How can I show that a curve $c$ given by
$c(t)=(t,f(t)\cos{\alpha},f(t)\sin{\alpha})$ for $\alpha$ constant is a geodesic on $M$ where
$M=\left\{(x,y,z) \in \Bbb{R}^3 ...
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0answers
106 views
Lagrange multipliers for finding geodesics on a sphere
The problem statement, all variables and given/known data
Find the geodesics on a sphere $g(x,y,z)=x^{2}+y^{2}+z^{2}−1=0$
arc-length element $ds=\sqrt{dx^{2}+dy^{2}+dz^{2}}$
Relevant equations
...
2
votes
1answer
47 views
Geodesic First Variation
I'm trying to prove that if the first variation of length vanishes then the curve $\gamma$ must be an affinely parameterised geodesic. In the following $T=\dot{\gamma}$.
So I've attacked the ...
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0answers
125 views
Maximum latitude of a great circle
1 - I am trying to figure out the longitude at which a geodetic great circle reaches its apex. (I have a point and the azimuth at that point identifying the circle)
I have found a good resource that ...
4
votes
2answers
124 views
Is there an “opposite” of a geodesic?
If I understand correctly, a geodesic between two points $a$ and $b$ is the "most direct" path from $a$ to $b$. Geodesics on a plane are straight lines, geodesics on a sphere are great circle arcs. ...
2
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2answers
222 views
Properties of geodesics on ruled surfaces
I am trying to solve the following problem:
Show that a unit-speed curve $\gamma$ with nowhere vanishing curvature is a geodesic on the ruled surface $\sigma(u,v)=\gamma(u)+v\delta(u)$, where ...
1
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0answers
75 views
Completelly cover area with minimum number of maxed circles NP-completeness (or harder) proof
everyone.
I'm looking for paper with proof of NP-completeness following, or similar problem.
Given:
Area $S \subset \mathbb{N}^2$, let it be convex or rectangular, I believe it doesn't matter
...
1
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1answer
66 views
Distances are different by ~100-200m
I'm measuring distance of 2 points on Google Map and then in my program converting them into ECEF using this formula. Then using Pythagorean theorem to calculate distance between those 2 points. ...
4
votes
1answer
114 views
Geodesic on Spheres
We can join the north pole and the south pole of an sphere by an unlimited number of geodesics.
1: Is this property still valid if we take any manifold that is diffeomorphic to the sphere, i.e. are ...
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0answers
36 views
Is there an action functional whose critical points are the geodesics of a arbitrary connection on TM?
Geodesics of the Levi-Civita connection may be defined as the critical points of the action functional $S[\gamma]=\int \lvert\dot{\gamma}\rvert\,dt$ (or square it, if you like). The Euler-Lagrange ...
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0answers
48 views
Geodesic On Compact Manifolds
Let $M$ be a compact Riemmanian manifold. Let $G$ denote the set of all geodesics of $M$. If $\gamma\in G$ let $l(\gamma)$ denote its length. Let $$S=\sup\{l(\gamma): \gamma\in G\}$$
Suppose ...
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0answers
63 views
What property does this equation calculate?
It's pretty difficult to Google for the meaning of a formula.
This is the equation, it has to do with ellipses and GIS coordinates.
$$\nu =\frac{ a} {\sqrt{(1 - (e^2 \cdot \sin(\varphi))^2)}}$$
$a$ ...
3
votes
1answer
185 views
Convert LLA (long, lat, alt) to flat earth model
I would like to divide the globe into 1000 $\times$ 1000 meter geodesic squares, and then map any long / lat to the applicable square.
The altitude of each block would be the altitude of the earth at ...
3
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1answer
75 views
how to solve differential equation $y^4 = k^2 (y^2 + y'^2\csc^2\alpha)$?
What's the solution of the differential equation $y^4 = k^2 (y^2 + y'^2\csc^2\alpha)$, where $y$ is a function of $x$ and $\alpha$ is a constant?
Polynomial solutions don't seem to work, because the ...
4
votes
4answers
332 views
What is the geodesic between a point and a line (geodesic between two points) on an oblate spheroid?
I found a similiar question that also asks for the distance from a point to a line but works on a sphere.
Now I'm trying to figure out the length of the geodesic line ...
6
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3answers
292 views
Why are we interested in closed geodesics?
There's a lot of work about the existence, number and other properties of closed geodesics on a Riemannian manifold (belonging to some specific class of manifolds).
In the case of geodesics ...
2
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0answers
450 views
Solving geodesic problems with Euler-Lagrange equation
This is the question:
Problem B.1 Two cities - Tel-Aviv, Israel and SanDiego, CA - have the same latitude 32
◦ N, but, different
longitudes: Tel-Aviv is 34
◦
E and San-Diego is 117
◦ W.
What is the ...
1
vote
1answer
93 views
Point on a surface with no geodesics passing through
Take an orientable surface $S_g^s$ of genus $g$ with no boundaries and $s$ points removed and fix a complete hyperbolic metric of finite area (assuming that the Euler characteristic allows an ...
2
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1answer
164 views
Effect of curvature of spacetime on intrinsic geometric properties (under general relativity)
If a space has curvature, then the curvature can be seen intrinsically by finding sums of angles in triangles made of geodesics. Under general relativity, space-time is curved on local scales. On ...
2
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1answer
808 views
How to find a (longitude, latitude) point on a circle when given only the center (longitude, latitude) point and radius measured in Feet
How do I find a (longitude, latitude) point (any will do) on a circle where the only info I have is a (longitude, latitude) center point, and a radius measured in Feet (ft.)?
