Elliptics

Higher-dimensional geometry (previously "Polyshapes").

Elliptics

Postby Aale de Winkel » Sat Jan 10, 2004 10:32 am

This thread introduction summarizes the discusion in the 4-d coordinate thread, and brings up the possibilities of elliptical shapes. The hyperellipse serves as starting point, which cal be extruded into the hyperellipsecylinder or coned into the hyperellipsecone. Also the possibilities of "[elliptic] [oblate/prolate] lathing" are introduces, giving on the square (2d) either the cylinder (3d) or a duocone (3d).

The hyperellipse (or n-ellipse) is defined by the following equation:

(x/R)[sub]k[/sub] (x/R)[sup]k[/sup] = 1 (k = 1..n) or more regularly: [sub]k=1[/sub]∑[sup]n[/sup] (x[sub]k[/sub]/R[sub]k[/sub])[sup]2[/sup] = 1

The polar coördinates: { IR[sup]n-1[/sup] => IR[sup]n[/sup] }
x[sub]i[/sub] = R[sub]i[/sub] [sub]k=1[/sub]Π[sup]i-1[/sup]sin(θ[sub]k[/sub]) cos(θ[sub]i-1[/sub]) i = 1..n-1
x[sub]n[/sub] = R[sub]n[/sub] [sub]k=1[/sub]Π[sup]n-1[/sup]sin(θ[sub]k[/sub])

is an IR[sup]n-1[/sup] => IR[sup]n[/sup] mapping, though the values of the involved θ's could be limited to [0,2π) [0,π][sup]n-2[/sup] in case one wants the mapping to be an isomorphism (so from an (n-1)d "filled hyperbeam" onto the nd hyperellipse).

The coördinates could be taken thus that R[sub]i[/sub] >= R[sub]i+1[/sub] i = 1..n-1, the hypersphere can be seen as a degenerate hyperellipse having all R's equal.

The 2-ellipse is regularly called the ellipse, the 3-ellipse or ellipsoid has two degerenate recognized variations, lathing the ellipse around the major axis is callled prolate spheroid, lathing around the minor axis the oblate spheroid (to meet the conditions on R's posed above an interchange of axes x[sub]2[/sub] <-> x[sub]3[/sub] should be applied (ie a transposition))

The previous section introduces 2 kinds of lathing "[elliptic] prolate lathing" and "[eliptic] oblate lathing". The added 'elliptic' here introduces the possibillity to rotate along an elliptic arc in stead of the commonly circular arc, parametrizable by the arc's hight which might be smaller equal or larger then the lathed length (in case larger a transposition might be in order).
Oblate lathing thus takes the longest length, while prolate lathing take the smallest length of the figure. In case one does this to a square the oblate lathing thus gives the duocone, while the prolate lathing gives the cylinder

references:
http://home.wanadoo.nl/aaledewinkel/Enc ... hapes.html
http://mathworld.wolfram.com/Ellipse.html
http://mathworld.wolfram.com/Ellipsoid.html
http://mathworld.wolfram.com/OblateSpheroid.html
http://mathworld.wolfram.com/ProlateSpheroid.html
.
Aale de Winkel
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twisted polygon rings

Postby Aale de Winkel » Wed Jan 14, 2004 8:31 pm

The following forms a workthrough of what happens when one rings a polygon and rotates the polygon simultaneously.

The points of a p-polygon are given by
p-gon: [ r cos(2πk/p) , r sin(2πk/p) ] ; k= 0..p-1
moving this around the point [ R , 0 ] gives
[ R + r cos(2πk/p) , r sin(2πk/p) ]
Ringing this through the third dimension around the y-axis gives
R(p-gon): [ { R + r cos(2πk/p) } cos(φ) , r sin(2πk/p) , { R + r cos(2πk/p) } sin(φ) ]
Now gradually rotating the p-polygon such that for φ=2π the polygon coincides with the one for φ=0 is given by:
R[sub]2πl/p[/sub](p-gon): [ { R + r cos((2πk+φl)/p) } cos(φ) , r sin((2πk+φl)/p) , { R + r cos((2πk+φl)/p) } sin(φ) ]
when l=0 there is no tordation of the ring, l can be any integer which rotates the polygon 2πl/p through one full circle around the y-axis.

for p=2 and l=1 this gives the normal möbius belt, for other values of p one has a tordated polygon ring which are simular to the möbius belt, but in stead of starting with a line one starts with a polygon.

Note: when l and p are relatively prime, like the möbius belt, any R[sub]2πl/p[/sub](p-gon) consists of a single edged ribbon.

(this might be called 3d polygontwisters, but polygondude seems to have those not existing in prime dimensions)
Aale de Winkel
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