Are Oblique Coordinate Systems More Useful Than Orthogonal Systems in 2-D Space?

[rkmurtyp]

In a coordinate system two axes are inclined at an acute angle θ. Is this coordinate system different from a coordinate system in which the axes are inclined at an angle (180 - θ)? if we look at the four quardents in either of the above set of axes, both are included giving the impression that the two are same. Is that true?

 

[facenian]

I'm not sure if I understand you question but regarding rotational symmetry both systems are physically the same(disrigarding direccions on the axes)

 

[rkmurtyp]

I'm not sure if I understand you question but regarding rotational symmetry both systems are physically the same(disrigarding direccions on the axes)

Let me pose my question in a different way. Do two intersecting lines constitue a coordinate system?

1. If yes, does it have four quardents? or just one region in which coordinates of any point has positive numbers (for example (3.23,4)) only?

2. If no, then what constitutes a cordinate system?

 

[Nugatory]

Let me pose my question in a different way. Do two intersecting lines constitute a coordinate system?

Yes, they can serve as the axes of a coordinate system. There's a formal mathematical definition of "coordinate system" which is very general and allows for all sorts of weird configurations.

does it have four quardents? or just one region in which coordinates of any point has positive numbers (for example (3.23,4)) only?

Four regions.

 

[rkmurtyp]

Yes, they can serve as the axes of a coordinate system. There's a formal mathematical definition of "coordinate system" which is very general and allows for all sorts of weird configurations.



Four regions.

What advantage do we get in an arbitrary 2-D coordinate system that we don't get in a 2-D orthogonal coordinate system?

The beautiful symmetry we have in an ortogonal coordinate system is lost in a non orthogonal (for example an oblique) coordinate system. Hence my problem (question) above.

 

[HallsofIvy]

Oblique coordinates can be useful if there are two non-orthogonal lines upon which you are given special information. Another example is in solving a hyperbolic differential equation where it would simplify the equation to use the characteristic lines as axes. And they are not generally orthogonal.

Also, while one can always find coordinates on a general surface that are orthogonal at a specific point but not generally orthogonal anywhere else.

 

[rkmurtyp]

Oblique coordinates can be useful if there are two non-orthogonal lines upon which you are given special information. Another example is in solving a hyperbolic differential equation where it would simplify the equation to use the characteristic lines as axes. And they are not generally orthogonal.

Also, while one can always find coordinates on a general surface that are orthogonal at a specific point but not generally orthogonal anywhere else.

I am only interested in understanding analyses in 2-D space. So, please let me know, if I have two non orthogonal lines, what special information is required to make that coordinate system more useful in comparision to the orthogonal coordinate system?