4D cordiates

Higher-dimensional geometry (previously "Polyshapes").

4D cordiates

Postby Frisk-256 » Mon May 13, 2024 7:07 pm

So I know in 3D there is Cartesian, Cylindrical, and spherical coordinates. So I am going to try to list and name 4D variants of these

Quickly listing of the 3D ones
Cartesian = (x , y , z)
-Clasic
Cylindrical = (cos(θ1)r1 , sin(θ1)r1 , z)
-Polar coordinates extruded
Spherical = (cos(θ1)sin(θ2)r1 , sin(θ1)sin(θ2)r1 , cos(θ2)r1)
-Polar coordinates revolved into a sphere

Now the 4D ones
Cartesian = (x , y , z , w)
-Clasic
Cubeinder = (cos(θ1)r1 , sin(θ1)r1 , z , w)
-Polar extruded 2 times
DuoCylindrical = (cos(θ1)r1 , sin(θ1)r1 , cos(θ2)r2 , sin(θ2)r2)
-Cartesian product of the polar coordinate system and the polar coordinate system
Spherinder = (cos(θ1)sin(θ2)r1 , sin(θ1)sin(θ2)r1 , cos(θ2)r1 , w)
-Spherical coordinates extruded
Hyperspherical = (cos(θ1)sin(θ2)sin(θ3)r1 , sin(θ1)sin(θ2)sin(θ3)r1 , cos(θ2)sin(θ3)r1 , cos(θ3)r1)
-Spherical coordinates revolved into a hypersphere

Is there any I missed? Is there any interesting ones beyond 4D?
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Re: 4D cordiates

Postby wxyhly » Tue May 14, 2024 3:39 pm

There's another hyperspheric coordinates: Hopf Coordinates.

Hopf Coordinates = (cos(θ1)cos(θ3)r , sin(θ1)cos(θ3)r , cos(θ2)sin(θ3)r , sin(θ2)sin(θ3)r)

it seems like a combination of duocylindrical coordinates and another 2D polar coordinates on r1 and r2. Hopf coordinates is easier to use for rotating objects like rigid bodies. imagine 4Ders on a 4D planet, they might prefer Hopf coordinates instead of previous hyperspherical one, because double rotation occurs on 4D planet, and there's no polar axis but two perpendicular circular equators instead. Here the varying direction θ3 is latitude with range 0-90 degree, the varying directions θ1 and θ2 are both two different longitudes with range 0-360 degree.
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Re: 4D cordiates

Postby mr_e_man » Thu May 16, 2024 7:44 am

Yes, you missed one:
(cos(θ1)cos(θ2)r1 , sin(θ1)cos(θ2)r1 , cos(θ3)sin(θ2)r1 , sin(θ3)sin(θ2)r1)

In general, if you have an n-dimensional coordinate system, you can make an (n+1)-dimensional coordinate system, by taking any one Cartesian coordinate, let's call it u, and splitting it into two coordinates, cos(θ)u and sin(θ)u. This produces different hyperspherical coordinate systems.

In particular, the above is gotten from 3D spherical coordinates (cos(θ1)cos(θ2)r, sin(θ1)cos(θ2)r, sin(θ2)r) by replacing the third coordinate u=sin(θ2)r with (cos(θ3)u, sin(θ3)u).
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
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Re: 4D cordiates

Postby Frisk-256 » Fri May 17, 2024 2:44 pm

So is this just another hypersphere?
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