Example of Non-Orthogonal Coordinates

[F. Russell]

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.

[Ross A. Finlayson]

Post by F. Russell
Orthogonal coordinate systems are the most common
because they are easy to use. Wikipedia has a list
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?

[F. Russell]

Post by Ross A. Finlayson
Polar coordinates?

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

[konyberg]

Post by F. Russell

Post by Ross A. Finlayson
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

[F. Russell]

Post by konyberg
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.

[konyberg]

Post by F. Russell

Post by konyberg
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

[Ross A. Finlayson]

Post by konyberg

Post by F. Russell

Post by Ross A. Finlayson
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

[b***@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?

Post by Ross A. Finlayson

Post by konyberg

Post by F. Russell

Post by Ross A. Finlayson
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

...

[F. Russell]

Post by b***@gmail.com
Are these coordinates orthogonal.
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.

[b***@gmail.com]

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.

Post by b***@gmail.com
Are these coordinates orthogonal.
https://en.wikipedia.org/wiki/Hyperbolic_coordinates
Can fruit cake use them for his
space filling manic disorder?

Post by Ross A. Finlayson

Post by konyberg

Post by F. Russell

Post by Ross A. Finlayson
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

...

[Ross A. Finlayson]

Post by b***@gmail.com
https://en.wikipedia.org/wiki/Rindler_coordinates
Is the universe space filled neither countable
nor uncountable, fruit cake to the rescue.

Post by b***@gmail.com
Are these coordinates orthogonal.
https://en.wikipedia.org/wiki/Hyperbolic_coordinates
Can fruit cake use them for his
space filling manic disorder?

Post by Ross A. Finlayson

Post by konyberg

Post by F. Russell

Post by Ross A. Finlayson
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).

[Ross A. Finlayson]

Post by Ross A. Finlayson

Post by b***@gmail.com
https://en.wikipedia.org/wiki/Rindler_coordinates
Is the universe space filled neither countable
nor uncountable, fruit cake to the rescue.

Post by b***@gmail.com
Are these coordinates orthogonal.
https://en.wikipedia.org/wiki/Hyperbolic_coordinates
Can fruit cake use them for his
space filling manic disorder?

Post by Ross A. Finlayson

Post by konyberg

Post by F. Russell

Post by Ross A. Finlayson
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. "

[Ross A. Finlayson]

Post by b***@gmail.com
Are these coordinates orthogonal.
https://en.wikipedia.org/wiki/Hyperbolic_coordinates
Can fruit cake use them for his
space filling manic disorder?

Post by Ross A. Finlayson

Post by konyberg

Post by F. Russell

Post by Ross A. Finlayson
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

[Ross A. Finlayson]

Post by F. Russell

Post by Ross A. Finlayson
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.

[FredJeffries]

Post by F. Russell
Orthogonal coordinate systems are the most common
because they are easy to use. Wikipedia has a list
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

[Steven Carlip]

Post by F. Russell
Orthogonal coordinate systems are the most common
because they are easy to use. Wikipedia has a list
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