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Winfried Weber Group moderator
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Travels on the earth
The next feature is part of (sphere) geometry. This category does not exist (yet), so I post it to “Algebra”.

Sometimes I discuss with some friends « problems about travels on the earth ».
The earth is seen as a complete regular sphere with
radius R = 6366,385 km, so
U = 2 * R * pi = 40.000 km.

The earth is graduated with the “normal used coordinator system”, s.m.
latitude 0° = equator
longitude 0° = Greenwich-line

We got the solution for all this problems (perhaps I will give some funny examples later) except of one:

Our travel starts at (0°,0°) at the equator and goes the following way:
(0°,0°) -> (n°, n°) (n from 0 to 90) -> (90°,90°) = north pole

How far is this travel?

Does a mathematical exact solution (algebraic or analytic) exist, or you can find only an approximation? Which approximation?

(for example: to go directly from (0°,0°) to north pole we need 0.25 * U = 10.000 km)

Best regards,
Winfried Weber
This post was changed on 24 Jul 2006 at 07:12 pm by Dr. Dirk Kehrwald .
- 24 Jul 2006, 10:05 am
Winfried Weber wrote: > The next feature is part of (sphere) geometry. This category does not > exist (yet), so I post it to “Algebra”. > > Sometimes I discuss with some friends « problems about travels on the > earth ». > The earth is seen as a complete regular sphere with > radius R = 6366,385 km, so > U = 2 * R * pi = 40.000 km. > > The earth is graduated with the “normal used coordinator system”, > s.m. > latitude 0° = equator > longitude 0° = Greenwich-line > > We got the solution for all this problems (perhaps I will give some > funny examples later) except of one: > > Our travel starts at (0°,0°) at the equator and goes the following > way: > (0°,0°) -> (n°, n°) (n from 0 to 90) -> (90°,90°) = north pole > > How far is this travel? > > Does a mathematical exact solution (algebraic or analytic) exist, or > you can find only an approximation? Which approximation? > > (for example: to go directly from (0°,0°) to north pole we need 0.25 > * U = 10.000 km) > > Best regards, > Winfried Weber
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Winfried Weber Group moderator
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Re^2: Travels on the earth
Hi Thorsten,
the "great circle distance" and the "loxodrome "(constant azimuth) are a n o t h e r paths.

By discussing the loxodroms somebody thought the path looks like
(0°,0°) -> (n°, n°) -> (90°, 90°) (n runs up from 0 to 90)

That is of course n o t the loxodrome, but what path is it?
Interesting question: Is it calculable?

Best regards,
Winfried
- 24 Jul 2006, 12:04 pm
Winfried Weber wrote: > Hi Thorsten, > the "great circle distance" and the "loxodrome "(constant azimuth) > are a n o t h e r paths. > > By discussing the loxodroms somebody thought the path looks like > (0°,0°) -> (n°, n°) -> (90°, 90°) (n runs up from 0 to 90) > > That is of course n o t the loxodrome, but what path is it? > Interesting question: Is it calculable? > > > Best regards, > Winfried
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Winfried Weber Group moderator
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Re: Travels on earth
Dr. Thorsten Maier wrote:
Hi Winfried,

basically it is possible to calculate the length of every (piecewise smooth) path on the sphere by using line-integrals... provided one can find a suitable parametrization, of course.
In your case a parametrization can be given by using spherical coordinates and by letting longitude and latitude simultaniously tend from 0 to 90 degrees. If haven't done anything wrong, the line integral (in this case) leads to an elliptic integral which cannot be calculated explicitely. Nevertheless, with MATLAB (function quad) I was able to numerically calculate the length of the path to be appr. 12.169 km (radius of earth 6.371km).
Hi Thorsten,

Thanks foryour help.
I don't have the possibility to use MATLAB.
Perhaps you could try it again with U = 40.000 km, that means R=6366,2 km.

My friend calculated (with EXCEL) 12.159,6837 km.

More interesting is the following:
He added an argument (but without proof) and I don't know, if he is right:

"On the sphere with Radius r the lenght of the path is calculated:
r * lenght of f(x) = arcsin(x) between 0 and 1.

This means to calculate r * Integral (from 0 to 1) over sqrt(1+1/(1-x^2)) dx "

Is this the right solution?

Is it possible to find a „Stammfunktion” for sqrt(1+1/(1-x^2)) ?

Best regards,
Winfried Weber
- 26 Jul 2006, 10:26 am
Winfried Weber wrote: > Dr. Thorsten Maier wrote: > > Hi Winfried, > > > > basically it is possible to calculate the length of every (piecewise > > smooth) path on the sphere by using line-integrals... provided one > > can find a suitable parametrization, of course. > > In your case a parametrization can be given by using spherical > > coordinates and by letting longitude and latitude simultaniously tend > > from 0 to 90 degrees. If haven't done anything wrong, the line > > integral (in this case) leads to an elliptic integral which cannot be > > calculated explicitely. Nevertheless, with MATLAB (function quad) I > > was able to numerically calculate the length of the path to be appr. > > 12.169 km (radius of earth 6.371km). > > Hi Thorsten, > > Thanks foryour help. > I don't have the possibility to use MATLAB. > Perhaps you could try it again with U = 40.000 km, that means > R=6366,2 km. > > My friend calculated (with EXCEL) 12.159,6837 km. > > More interesting is the following: > He added an argument (but without proof) and I don't know, if he is > right: > > "On the sphere with Radius r the lenght of the path is calculated: > r * lenght of f(x) = arcsin(x) between 0 and 1. > > This means to calculate r * Integral (from 0 to 1) over > sqrt(1+1/(1-x^2)) dx " > > Is this the right solution? > > Is it possible to find a „Stammfunktion” for sqrt(1+1/(1-x^2)) ? > > Best regards, > Winfried Weber
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Mathematics / Mathematik / Matemática

Travels on the earth

Re^2: Travels on the earth

Re: Travels on earth