Aale de Winkel wrote:adding on
-duocircle; r,θ,φ
x = r cos(θ)
y = r sin(θ)
z = r cos(φ)
w = r sin(φ)
or simpler notated: [x,y,z,w] = r [cos(θ),sin(θ),cos(φ),sin(φ)]
alkaline wrote:this one is wrong:
x[sub]n[/sub] = r { [sub]k=1[/sub]Π[sup]n-1[/sup] sin(θ[sub]k[/sub]) } sin(θ[sub]n-1[/sub])
It creates one too many sine's (it duplicates the last one). it should be this:
x[sub]n[/sub] = r { [sub]k=1[/sub]Π[sup]n-1[/sup] sin(θ[sub]k[/sub]) }
.
alkaline wrote:i good way to prevent missing new posts from a thread is to look at the page icons at the top of the post, to the left of "Posted: 08 Jan 2004 ...". The new posts will be in brown, old ones in white. Just scroll up to the first brown one then read down from there.
alkaline wrote:As for as duocircle coordinates goes, a single r for all of the coordinates limits the coordinate system to only describing points on a sphere; it isn't a generalized coordinate system. In contrast, the coordinates of the spherical coordinate system can describe points anywhere in realmspace. The glomar coordinate system is also a general system. Both spherical and glomar coordinate systems could describe ellipses/hyperellipses, but the duocircle system with a single radius couldn't.
Aale de Winkel wrote:I don't know what you mean by "generalized coordinate system". If you mean that a duoellipse is impossible I don't know what else one should call the points:
[ R[sub]1[/sub] cos(θ[sub]1[/sub]) , R[sub]2[/sub] sin(θ[sub]1[/sub]) , R[sub]3[/sub] cos(θ[sub]2[/sub]) , R[sub]4[/sub] sin(θ[sub]2[/sub]) ]
.
so having 6 parameters to this "duo-ellipse"
alkaline wrote:Look at a single plane, for example - what does [ R[sub]1[/sub] cos(θ[sub]1[/sub]) , R[sub]2[/sub] sin(θ[sub]1[/sub]) ] describe?
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Aale de Winkel wrote:The hyperpolar.avi somehow bolts
Windows Media Player for Windows XP
version 8.00.00.4477
Laplace's operator is a real thing, and not a fait of coordinate-system. It has a solution in any coordinate system.
On the other hand, i have not figured out things like the shape of winter (ie what areas are in winter) in 4d. If the shape of '4-pm' is any thing to go by, this is simply a half glomohedrix, (ie the hemi-3-sphere), which is rotated around the world.
wendy wrote:However, the nature of 4-space is that it may not necessarily be dominated by a simple radiant field. It may be some kind of polarised thing that radiates perpendicular to some kind of dipole affair, which would give rise to stable orbits from a radiant field.
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