In my study of coordinate geometry, I am using the textbook provided by S.L. Loney. In the textbook, he derives various formulas for rectangular coordinates as well as coordinates inclined at a general angle $\omega$. Which makes me ask: is the study of non-rectangular coordinates ever put to use? Is there any profit out of the practice of it? As of now I am merely studying it to further understand the general case of coordinate axes, however, it would be neat if there was actually some case or reason for which non-rectangular are better suited and more comfortable than rectangular coordinates. I could think of one in the area of mathematics alone: to derive formulas more easily. To derive the formula for the area of a hexagon, for example, the oblique axes could be set to be two intersecting sides of the hexagon. So the proof flows much more easier. I would love to hear of more places I can put the knowledges into use. Thank you in advance.
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4Do you know polar coordinates (https://en.wikipedia.org/wiki/Polar_coordinate_system)? – zkutch Aug 04 '23 at 03:12
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1Barycentric coordinates are oriented along the sides of a triangle. Used in every triangulation of surfaces in computer graphics and triangular meshes as basis for PDE solvers. – Lutz Lehmann Aug 04 '23 at 04:11
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[A] Check out "Oblique Co-Ordinates" & "Curvilinear Co-Ordinates" & "Lorentzian Plane" [B] When the Problem at hand (Physics/Electrical/Mechanical) itself has vectors (Potential/Intensity/velocity/force) along 2 lines , then the Co-Ordinates along those 2 lines will be suitable. – Prem Aug 04 '23 at 04:42
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1See Example of a Problem Made Easier with Skew Coordinates and my comment to Can the coordinate axes be scaled differently in pure math? – Dave L. Renfro Aug 04 '23 at 06:27
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1Non-rectangular coordinates are frequently useful when items are arranged in a non-rectangular grid, for example in drawing a hexagonal tiling in computer graphics. – awkward Aug 04 '23 at 13:57