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Example of Non-Orthogonal Coordinates

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  Example of Non-Orthogonal Coordinates "F. Russell" <fr@random.info> - 2018-06-28 15:37 +0000
    Re: Example of Non-Orthogonal Coordinates "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2018-06-28 09:45 -0700
      Re: Example of Non-Orthogonal Coordinates "F. Russell" <fr@random.info> - 2018-06-28 16:57 +0000
        Re: Example of Non-Orthogonal Coordinates konyberg <konyberg@online.no> - 2018-06-28 10:27 -0700
          Re: Example of Non-Orthogonal Coordinates "F. Russell" <fr@random.info> - 2018-06-28 17:41 +0000
            Re: Example of Non-Orthogonal Coordinates konyberg <konyberg@online.no> - 2018-06-28 11:14 -0700
          Re: Example of Non-Orthogonal Coordinates "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2018-06-28 10:42 -0700
            Re: Example of Non-Orthogonal Coordinates bursejan@gmail.com - 2018-06-28 10:49 -0700
              Re: Example of Non-Orthogonal Coordinates "F. Russell" <fr@random.info> - 2018-06-28 18:10 +0000
              Re: Example of Non-Orthogonal Coordinates bursejan@gmail.com - 2018-06-28 11:12 -0700
                Re: Example of Non-Orthogonal Coordinates "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2018-06-28 12:09 -0700
                  Re: Example of Non-Orthogonal Coordinates "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2018-06-28 14:05 -0700
              Re: Example of Non-Orthogonal Coordinates "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2018-06-28 11:14 -0700
        Re: Example of Non-Orthogonal Coordinates "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2018-06-28 10:28 -0700
    Re: Example of Non-Orthogonal Coordinates FredJeffries <fredjeffries@gmail.com> - 2018-06-29 06:20 -0700
    Re: Example of Non-Orthogonal Coordinates Steven Carlip <carlip@physics.ucdavis.edu> - 2018-06-29 14:06 -0700

#416650 — Example of Non-Orthogonal Coordinates

Orthogonal coordinate systems are the most common
because they are easy to use.  Wikipedia has a list
of orthogonal coordinate systems:

https://en.wikipedia.org/wiki/Orthogonal_coordinates

I need some examples of non-orthogonal coordinate systems.
The reason is that ALL textbooks present non-orthogonal
systems in a totally general way and I want to practice on some
actual examples rather than dry generalized systems.

Can anyone provide an example or examples of non-orthogonal
coordinate systems?  By an example I mean the actual functions
that relate the Cartesian variables (x, y, z) to the
non-orthogonal coordinates (q1, q2, q3): x=f1(q1, q2, q3),
y=f2(q1, q2, q3), z=f3(q1, q2, q3) or the inverse relations.

I have searched and searched but can find no examples of
non-orthogonal coordinates.  Skew coordinates are non-orthogonal
but I am looking for more complicated examples.

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#416652

On Thursday, June 28, 2018 at 8:38:22 AM UTC-7, F. Russell wrote:
> Orthogonal coordinate systems are the most common
> because they are easy to use.  Wikipedia has a list
> of orthogonal coordinate systems:
> 
> https://en.wikipedia.org/wiki/Orthogonal_coordinates
> 
> I need some examples of non-orthogonal coordinate systems.
> The reason is that ALL textbooks present non-orthogonal
> systems in a totally general way and I want to practice on some
> actual examples rather than dry generalized systems.
> 
> Can anyone provide an example or examples of non-orthogonal
> coordinate systems?  By an example I mean the actual functions
> that relate the Cartesian variables (x, y, z) to the
> non-orthogonal coordinates (q1, q2, q3): x=f1(q1, q2, q3),
> y=f2(q1, q2, q3), z=f3(q1, q2, q3) or the inverse relations.
> 
> I have searched and searched but can find no examples of
> non-orthogonal coordinates.  Skew coordinates are non-orthogonal
> but I am looking for more complicated examples.

Polar coordinates?

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#416657

On Thu, 28 Jun 2018 09:45:28 -0700, Ross A. Finlayson wrote:

> 
> Polar coordinates?
>

Polar coordinates are orthogonal.  They are on the list
of Wikipedia coordinate system that I cited.

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#416660

torsdag 28. juni 2018 18.57.56 UTC+2 skrev F. Russell følgende:
> On Thu, 28 Jun 2018 09:45:28 -0700, Ross A. Finlayson wrote:
> 
> > 
> > Polar coordinates?
> >
> 
> Polar coordinates are orthogonal.  They are on the list
> of Wikipedia coordinate system that I cited.

Sorry about the title of this article :)

http://terpconnect.umd.edu/~mattland/plasmanotes/20120311-04_General_coordinate_systems.pdf

KON

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#416665

On Thu, 28 Jun 2018 10:27:35 -0700, konyberg wrote:

> 
> http://terpconnect.umd.edu/~mattland/plasmanotes/20120311-04_General_coordinate_systems.pdf
> 

I found this file during my search.  It does NOT contain an actual example
of a non-orthogonal system.  It just mentions psi, theta, and zeta as
general non-orthogonal coordinates.  There is no function that relates these
coordinates to a Cartesian system and nothing specific can be calculated.

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#416674

torsdag 28. juni 2018 19.42.20 UTC+2 skrev F. Russell følgende:
> On Thu, 28 Jun 2018 10:27:35 -0700, konyberg wrote:
> 
> > 
> > http://terpconnect.umd.edu/~mattland/plasmanotes/20120311-04_General_coordinate_systems.pdf
> > 
> 
> I found this file during my search.  It does NOT contain an actual example
> of a non-orthogonal system.  It just mentions psi, theta, and zeta as
> general non-orthogonal coordinates.  There is no function that relates these
> coordinates to a Cartesian system and nothing specific can be calculated.

There is a reference to Arfken and Weber, “Mathematical	Methods	for	Physicists”

https://isidore.co/calibre/get/pdf/4469/CalibreLibrary

Great book!

KON

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#416666

On Thursday, June 28, 2018 at 10:27:42 AM UTC-7, konyberg wrote:
> torsdag 28. juni 2018 18.57.56 UTC+2 skrev F. Russell følgende:
> > On Thu, 28 Jun 2018 09:45:28 -0700, Ross A. Finlayson wrote:
> > 
> > > 
> > > Polar coordinates?
> > >
> > 
> > Polar coordinates are orthogonal.  They are on the list
> > of Wikipedia coordinate system that I cited.
> 
> Sorry about the title of this article :)
> 
> http://terpconnect.umd.edu/~mattland/plasmanotes/20120311-04_General_coordinate_systems.pdf
> 
> KON

I see that those measurements aren't in terms of 
the directions of the bases, but, don't they just 
transform back to orthogonal and orthonormal 
co-ordinates?

If it's instead that the measurements are "outside 
the space", then that's still in the containing space.
That they have "principal components" in the space 
of the original vector space, has they transform back, 
otherwise they don't ("broken vector space" about here 
"continuous" or "smooth" manifolds).


Thanks, excuse me

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#416668

Are these coordinates orthogonal. 
I am too lazy to figure it out:
https://en.wikipedia.org/wiki/Hyperbolic_coordinates

Can fruit cake use them for his
space filling manic disorder?

Am Donnerstag, 28. Juni 2018 19:42:34 UTC+2 schrieb Ross A. Finlayson:
> On Thursday, June 28, 2018 at 10:27:42 AM UTC-7, konyberg wrote:
> > torsdag 28. juni 2018 18.57.56 UTC+2 skrev F. Russell følgende:
> > > On Thu, 28 Jun 2018 09:45:28 -0700, Ross A. Finlayson wrote:
> > > 
> > > > 
> > > > Polar coordinates?
> > > >
> > > 
> > > Polar coordinates are orthogonal.  They are on the list
> > > of Wikipedia coordinate system that I cited.
> > 
> > Sorry about the title of this article :)
> > 
> > http://terpconnect.umd.edu/~mattland/plasmanotes/20120311-04_General_coordinate_systems.pdf
> > 
> > KON
> 
> I see that those measurements aren't in terms of 
> the directions of the bases, but, don't they just 
> transform back to orthogonal and orthonormal 
> co-ordinates?
> 
> If it's instead that the measurements are "outside 
> the space", then that's still in the containing space.
> That they have "principal components" in the space 
> of the original vector space, has they transform back, 
> otherwise they don't ("broken vector space" about here 
> "continuous" or "smooth" manifolds).
> 
> 
> Thanks, excuse me

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#416672

On Thu, 28 Jun 2018 10:49:27 -0700, bursejan wrote:

> Are these coordinates orthogonal. 
> I am too lazy to figure it out:
> https://en.wikipedia.org/wiki/Hyperbolic_coordinates
> 

Hyperbolic coordinates, as described by the relation:

x = cos(u)*cos(v)*cos(phi)
y = cos(u)*cos(v)*sin(phi)
z = sinh(u)*sin(v)

are orthogonal.

The calculation of the metric tensor shows this.

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#416673

Oh, there is a whole family of coordinates:
https://en.wikipedia.org/wiki/Rindler_coordinates

Is the universe space filled neither countable
nor uncountable, fruit cake to the rescue.

Am Donnerstag, 28. Juni 2018 19:49:34 UTC+2 schrieb burs...@gmail.com:
> Are these coordinates orthogonal. 
> I am too lazy to figure it out:
> https://en.wikipedia.org/wiki/Hyperbolic_coordinates
> 
> Can fruit cake use them for his
> space filling manic disorder?
> 
> Am Donnerstag, 28. Juni 2018 19:42:34 UTC+2 schrieb Ross A. Finlayson:
> > On Thursday, June 28, 2018 at 10:27:42 AM UTC-7, konyberg wrote:
> > > torsdag 28. juni 2018 18.57.56 UTC+2 skrev F. Russell følgende:
> > > > On Thu, 28 Jun 2018 09:45:28 -0700, Ross A. Finlayson wrote:
> > > > 
> > > > > 
> > > > > Polar coordinates?
> > > > >
> > > > 
> > > > Polar coordinates are orthogonal.  They are on the list
> > > > of Wikipedia coordinate system that I cited.
> > > 
> > > Sorry about the title of this article :)
> > > 
> > > http://terpconnect.umd.edu/~mattland/plasmanotes/20120311-04_General_coordinate_systems.pdf
> > > 
> > > KON
> > 
> > I see that those measurements aren't in terms of 
> > the directions of the bases, but, don't they just 
> > transform back to orthogonal and orthonormal 
> > co-ordinates?
> > 
> > If it's instead that the measurements are "outside 
> > the space", then that's still in the containing space.
> > That they have "principal components" in the space 
> > of the original vector space, has they transform back, 
> > otherwise they don't ("broken vector space" about here 
> > "continuous" or "smooth" manifolds).
> > 
> > 
> > Thanks, excuse me

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#416683

On Thursday, June 28, 2018 at 11:12:14 AM UTC-7, burs...@gmail.com wrote:
> Oh, there is a whole family of coordinates:
> https://en.wikipedia.org/wiki/Rindler_coordinates
> 
> Is the universe space filled neither countable
> nor uncountable, fruit cake to the rescue.
> 
> Am Donnerstag, 28. Juni 2018 19:49:34 UTC+2 schrieb burs...@gmail.com:
> > Are these coordinates orthogonal. 
> > I am too lazy to figure it out:
> > https://en.wikipedia.org/wiki/Hyperbolic_coordinates
> > 
> > Can fruit cake use them for his
> > space filling manic disorder?
> > 
> > Am Donnerstag, 28. Juni 2018 19:42:34 UTC+2 schrieb Ross A. Finlayson:
> > > On Thursday, June 28, 2018 at 10:27:42 AM UTC-7, konyberg wrote:
> > > > torsdag 28. juni 2018 18.57.56 UTC+2 skrev F. Russell følgende:
> > > > > On Thu, 28 Jun 2018 09:45:28 -0700, Ross A. Finlayson wrote:
> > > > > 
> > > > > > 
> > > > > > Polar coordinates?
> > > > > >
> > > > > 
> > > > > Polar coordinates are orthogonal.  They are on the list
> > > > > of Wikipedia coordinate system that I cited.
> > > > 
> > > > Sorry about the title of this article :)
> > > > 
> > > > http://terpconnect.umd.edu/~mattland/plasmanotes/20120311-04_General_coordinate_systems.pdf
> > > > 
> > > > KON
> > > 
> > > I see that those measurements aren't in terms of 
> > > the directions of the bases, but, don't they just 
> > > transform back to orthogonal and orthonormal 
> > > co-ordinates?
> > > 
> > > If it's instead that the measurements are "outside 
> > > the space", then that's still in the containing space.
> > > That they have "principal components" in the space 
> > > of the original vector space, has they transform back, 
> > > otherwise they don't ("broken vector space" about here 
> > > "continuous" or "smooth" manifolds).
> > > 
> > > 
> > > Thanks, excuse me

Well, the uncountable's pretty much 
irrelevant to real analysis thus 
continuum mechanics thus physics, so ....

Physics has much to do with the continuous 
and discrete as about discrete observation 
in the continuous space of events as about 
parallel transport, you know....

Here it's a mode of abstraction of notation 
about that the path integral has sum-of-histories, 
"SR is local", "space-time is flat but it's 
curving".

"The integral curves of the timelike unit vector field 
 e → 0 {\displaystyle \scriptstyle {\vec {e}}_{0}} \scriptstyle\vec{e}_0 
give a timelike congruence, consisting of the world lines 
of a family of observers called the Rindler observers. 
In the Rindler chart, these world lines appear as the 
vertical coordinate lines 
 x = x 0 , y = y 0 , z = z 0 {\displaystyle \scriptstyle x\;=\;x_{0},\;y\;=\;y_{0},\;z\;=\;z_{0}} \scriptstyle x \;=\; x_0,\; y \;=\; y_0,\; z \;=\; z_0. 

Using the coordinate transformation above, we find that 
these correspond to hyperbolic arcs in the original 
Cartesian chart."
 -- https://en.wikipedia.org/wiki/Rindler_coordinates#The_Rindler_observers

"... the world lines of our Rindler observers are the analogs 
of a family of concentric circles in the Euclidean plane, so 
we are simply dealing with the Lorentzian analog of a fact 
familiar to speed skaters: in a family of concentric circles, 
inner circles must bend faster (per unit arc length) than the 
outer ones."

Oh, ok Sagnac.

Or, "brainiac".

So, these "non-orthogonal co-ordinates" are yet 
relevant about that they are making neat lines 
(for usual traction in the linear).

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#416706

On Thursday, June 28, 2018 at 12:09:36 PM UTC-7, Ross A. Finlayson wrote:
> On Thursday, June 28, 2018 at 11:12:14 AM UTC-7, burs...@gmail.com wrote:
> > Oh, there is a whole family of coordinates:
> > https://en.wikipedia.org/wiki/Rindler_coordinates
> > 
> > Is the universe space filled neither countable
> > nor uncountable, fruit cake to the rescue.
> > 
> > Am Donnerstag, 28. Juni 2018 19:49:34 UTC+2 schrieb burs...@gmail.com:
> > > Are these coordinates orthogonal. 
> > > I am too lazy to figure it out:
> > > https://en.wikipedia.org/wiki/Hyperbolic_coordinates
> > > 
> > > Can fruit cake use them for his
> > > space filling manic disorder?
> > > 
> > > Am Donnerstag, 28. Juni 2018 19:42:34 UTC+2 schrieb Ross A. Finlayson:
> > > > On Thursday, June 28, 2018 at 10:27:42 AM UTC-7, konyberg wrote:
> > > > > torsdag 28. juni 2018 18.57.56 UTC+2 skrev F. Russell følgende:
> > > > > > On Thu, 28 Jun 2018 09:45:28 -0700, Ross A. Finlayson wrote:
> > > > > > 
> > > > > > > 
> > > > > > > Polar coordinates?
> > > > > > >
> > > > > > 
> > > > > > Polar coordinates are orthogonal.  They are on the list
> > > > > > of Wikipedia coordinate system that I cited.
> > > > > 
> > > > > Sorry about the title of this article :)
> > > > > 
> > > > > http://terpconnect.umd.edu/~mattland/plasmanotes/20120311-04_General_coordinate_systems.pdf
> > > > > 
> > > > > KON
> > > > 
> > > > I see that those measurements aren't in terms of 
> > > > the directions of the bases, but, don't they just 
> > > > transform back to orthogonal and orthonormal 
> > > > co-ordinates?
> > > > 
> > > > If it's instead that the measurements are "outside 
> > > > the space", then that's still in the containing space.
> > > > That they have "principal components" in the space 
> > > > of the original vector space, has they transform back, 
> > > > otherwise they don't ("broken vector space" about here 
> > > > "continuous" or "smooth" manifolds).
> > > > 
> > > > 
> > > > Thanks, excuse me
> 
> Well, the uncountable's pretty much 
> irrelevant to real analysis thus 
> continuum mechanics thus physics, so ....
> 
> Physics has much to do with the continuous 
> and discrete as about discrete observation 
> in the continuous space of events as about 
> parallel transport, you know....
> 
> Here it's a mode of abstraction of notation 
> about that the path integral has sum-of-histories, 
> "SR is local", "space-time is flat but it's 
> curving".
> 
> "The integral curves of the timelike unit vector field 
>  e → 0 {\displaystyle \scriptstyle {\vec {e}}_{0}} \scriptstyle\vec{e}_0 
> give a timelike congruence, consisting of the world lines 
> of a family of observers called the Rindler observers. 
> In the Rindler chart, these world lines appear as the 
> vertical coordinate lines 
>  x = x 0 , y = y 0 , z = z 0 {\displaystyle \scriptstyle x\;=\;x_{0},\;y\;=\;y_{0},\;z\;=\;z_{0}} \scriptstyle x \;=\; x_0,\; y \;=\; y_0,\; z \;=\; z_0. 
> 
> Using the coordinate transformation above, we find that 
> these correspond to hyperbolic arcs in the original 
> Cartesian chart."
>  -- https://en.wikipedia.org/wiki/Rindler_coordinates#The_Rindler_observers
> 
> "... the world lines of our Rindler observers are the analogs 
> of a family of concentric circles in the Euclidean plane, so 
> we are simply dealing with the Lorentzian analog of a fact 
> familiar to speed skaters: in a family of concentric circles, 
> inner circles must bend faster (per unit arc length) than the 
> outer ones."
> 
> Oh, ok Sagnac.
> 
> Or, "brainiac".
> 
> So, these "non-orthogonal co-ordinates" are yet 
> relevant about that they are making neat lines 
> (for usual traction in the linear).

"The generalization essential [sic] involves constructing 
an appropriate orthonormal tetrad and then transporting it 
along the given trajectory using the Fermi–Walker transport rule. "

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#416675

On Thursday, June 28, 2018 at 10:49:34 AM UTC-7, burs...@gmail.com wrote:
> Are these coordinates orthogonal. 
> I am too lazy to figure it out:
> https://en.wikipedia.org/wiki/Hyperbolic_coordinates
> 
> Can fruit cake use them for his
> space filling manic disorder?
> 
> Am Donnerstag, 28. Juni 2018 19:42:34 UTC+2 schrieb Ross A. Finlayson:
> > On Thursday, June 28, 2018 at 10:27:42 AM UTC-7, konyberg wrote:
> > > torsdag 28. juni 2018 18.57.56 UTC+2 skrev F. Russell følgende:
> > > > On Thu, 28 Jun 2018 09:45:28 -0700, Ross A. Finlayson wrote:
> > > > 
> > > > > 
> > > > > Polar coordinates?
> > > > >
> > > > 
> > > > Polar coordinates are orthogonal.  They are on the list
> > > > of Wikipedia coordinate system that I cited.
> > > 
> > > Sorry about the title of this article :)
> > > 
> > > http://terpconnect.umd.edu/~mattland/plasmanotes/20120311-04_General_coordinate_systems.pdf
> > > 
> > > KON
> > 
> > I see that those measurements aren't in terms of 
> > the directions of the bases, but, don't they just 
> > transform back to orthogonal and orthonormal 
> > co-ordinates?
> > 
> > If it's instead that the measurements are "outside 
> > the space", then that's still in the containing space.
> > That they have "principal components" in the space 
> > of the original vector space, has they transform back, 
> > otherwise they don't ("broken vector space" about here 
> > "continuous" or "smooth" manifolds).
> > 
> > 
> > Thanks, excuse me

It just seems that "orthogonal co-ordinates" 
are not matching up with "orthogonal bases" 
where they are many non-orthogonal bases of 
a vector space (all vector spaces have an 
orthogonal basis) but here those would all still not 
be "extra-dimensional values projected back 
down to the manifold" or "extra-dimensional".

Or, "nonorthogonal coordinates are 
not about a vector space."

https://en.wikipedia.org/wiki/Vector_space#Basis_and_dimension

This is where manifolds are Euclidean, but then 
they are having that only "locally", but there's 
yet that each "locale" is its own "origin" or 
"zero vector".

So, these "non-orthogonal coordinates" are 
not arising in the Euclidean and spaces with 
an orthogonal vector basis.

You need not excuse me if I err, 
but I readily correct myself 
and would rather know.

Thanks good luck

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#416662

On Thursday, June 28, 2018 at 9:57:56 AM UTC-7, F. Russell wrote:
> On Thu, 28 Jun 2018 09:45:28 -0700, Ross A. Finlayson wrote:
> 
> > 
> > Polar coordinates?
> >
> 
> Polar coordinates are orthogonal.  They are on the list
> of Wikipedia coordinate system that I cited.

Some kind of fractal, so basically (as it were) 
each box in the integer lattice has its own 
self-contained but mutually incompatible 
"addressing mode" of its values?  (An 
encoding scheme?)

I don't know how "co-ordinate" and "not 
reversible" go together without "broken 
geometry" or "broken continuity", i.e., 
"broken vector space".

Thanks for pointing that out.

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#416792

On Thursday, June 28, 2018 at 8:38:22 AM UTC-7, F. Russell wrote:
> Orthogonal coordinate systems are the most common
> because they are easy to use.  Wikipedia has a list
> of orthogonal coordinate systems:
> 
> https://en.wikipedia.org/wiki/Orthogonal_coordinates
> 
> I need some examples of non-orthogonal coordinate systems.
> The reason is that ALL textbooks present non-orthogonal
> systems in a totally general way and I want to practice on some
> actual examples rather than dry generalized systems.
> 
> Can anyone provide an example or examples of non-orthogonal
> coordinate systems?  By an example I mean the actual functions
> that relate the Cartesian variables (x, y, z) to the
> non-orthogonal coordinates (q1, q2, q3): x=f1(q1, q2, q3),
> y=f2(q1, q2, q3), z=f3(q1, q2, q3) or the inverse relations.
> 
> I have searched and searched but can find no examples of
> non-orthogonal coordinates.  Skew coordinates are non-orthogonal
> but I am looking for more complicated examples.

Let's get rid of (possibly confusing) subscripts and do
(x, y, z) |-> (u, v, w)

How about
u = xy + x^2
v = y*(e^x) + 42
w = xyz - z((x^2)y)^3

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#416823

On 6/28/18 8:37 AM, F. Russell wrote:
> Orthogonal coordinate systems are the most common
> because they are easy to use.  Wikipedia has a list
> of orthogonal coordinate systems:
> 
> https://en.wikipedia.org/wiki/Orthogonal_coordinates
> 
> I need some examples of non-orthogonal coordinate systems.
> The reason is that ALL textbooks present non-orthogonal
> systems in a totally general way and I want to practice on some
> actual examples rather than dry generalized systems.
> 
> Can anyone provide an example or examples of non-orthogonal
> coordinate systems?  By an example I mean the actual functions
> that relate the Cartesian variables (x, y, z) to the
> non-orthogonal coordinates (q1, q2, q3): x=f1(q1, q2, q3),
> y=f2(q1, q2, q3), z=f3(q1, q2, q3) or the inverse relations.

Almost all commonly used spatial coordinate systems are
orthogonal, simply because they're usually easier to work
with.  One exception is the case of coordinates in a crystal
lattice, as another poster mentioned, but these are just
skewed.

If you're willing to add time as a coordinate, the most common
nonorthogonal coordinate systems are those that describe a
rotating body.  Start with spherical or cylindrical coordinates
and make a shift \phi -> \phi - \omega t, where omega is the
angular velocity.  A standard exercise is to then work out
the Coriolis effect in the new coordinates.

There are other variations of this idea -- for instance, the
rest frame coordinates for an object moving along an arbitrary
trajectory.  But for spatial coordinates alone, I can't think
of any commonly used examples more complicated than skew.

Steve Carlip

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