4D cordiates

[Frisk-256]

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?

[wxyhly]

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.

[mr_e_man]

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).

[Frisk-256]

So is this just another hypersphere?

[wendy]

The phase space is modelled on diagrams where various properties eg temperture and preesure are use as coordinates. It is the source in thermodynamic wher on consides a gas represented as a point in 6N dimensions where N is the number of particles. But like Klitzing's incidenc martices it only deals with the simplist cases.
We imagine the space represents a control for a spinning object. In 2d. it is a line that you can slide a sontrol to make the thing spin faster.The 3d case consist of setting the spin arrow at the south pole. making this a different length makth the planst spin faster.
4d is a littlte tricky. it is contolled by 2 3d spheres. each one makes a left or right clifford (Hopf) rotation. you get six degrees of rotation. the controls align against 3d axies, tif you have X,Y at x,0 or 0,y you get isoclinal rotion, if the size of x=y then it is wheel rotation. because here x-y is the orthoganal version, we are not in the business of tumbling dry the cabin crew. settig x y to different speeds gives lissajour rotation, of which petrie rotation is an example.
the coordinates of a 4d planet will follow the apparent motion of the sun and stars. the effect of equalise to isoclinea will move the control point closer to x,0 or 0,y. the addition rotation will cause a surface that seeks to reduce one of the axies to zero. under isoclina;l rotation, stars appear to follow a great circle path around the earth/ stars rise in the east, and come their cumulationm on the plane that bisects the sky. thie point of highest rise lies on a hemi3sphere. the track of the stars is as this hemisphere is turn like a page of a book, this comes from the hemispher being turned on its contact wih the horiz on when te page is turned the zenith traces a line between the east point and the west. If you project this hemisphere on a sphere whose diameter is from the observer to the zenith, it produdu the same result for all obsevers, this is like the gimbil that holds the globe.for any observer just roll the sphere so that the zenith is point out wards, and project this sphere onto a hemi3phere, this will crate a poinrs in the sky where each star follows the tracks of.
If the sun follows in the sky an orbit of offosite sign , the risings of the sun will strike the lattitude sphere in a circle, durimg the cause of a year it rises at incresing points on a cicle. So thrre sre seasons as the sun rises at different altiduds. Imagin e a 3d sphere like the earth . the tropic of capricorn represents the 'tropics'. this traces out a torus in the ske the track of the sun forms a tightle wound like a slinky. anything inside is sub tropical the antipoles is an other toris the artiv circle. the lings of lattide on the gimble become climata, and run from the ice north to the sunny south. you get season zones like the time zone tells you that the sun rise in at certain times of the siderial, the season zone tell where it is winter and where it is sping.
polar bears can reach the peguins.