4d - > 3d projection models

Higher-dimensional geometry (previously "Polyshapes").

4d - > 3d projection models

Postby thigle » Sat Mar 11, 2006 1:09 am

though there's quite a bit of info on different 4d objects, there's not that much info on projection models.

so some representations are made through central projection (of which the orthogonal is just a special case). then there's this sterographic projection made by Sullivan?, with sexy bubble curved hypercube.

what other kinds of projection do you people use ? any order in all the different projections ? i don't really understand clearly any of these when applied to 4d -> 3d case. :cry:

for exemple these cases of n-cube projected into n-1 space:
as for 3d->2d case, the cube(made of wireframe) casts in orthogonal light 2 limit-shadows: one is a square other is a hexagon with diagonals (face-on,vertex-on)
is it so that for 3d->4d case vertex-first, hypercube projects as rhombic dodeca, and cell-first as double cube (2 cubes at same position with other 6 cubes between these, apparently nowhere) ?

then i wonder:
in 3-2case, looking at the shadow of the 3-coordinate base it goes from 6-rayed hexagonal star to a cross with one dimension hidden in the depth of the picture plane.
in 4-3 case, looking at the shadow of the 4-coordinate base, it goes from 4 axies containing the diagonals of the cube, similar to diamond arrangement of carbon atoms, but doubled, with maraldi (or bubble-link) angle between root vectors, to 3 orthogonal axies of cube-faces centres connections.
where is the 4th coordinate axis hidden ?

can anyone explain projection from 4 to 3d ? as detailed as possible ?
thigle
Tetronian
 
Posts: 390
Joined: Fri Jul 29, 2005 5:00 pm

Postby wendy » Sat Mar 11, 2006 8:57 am

All of the azithmal projections work in 4D. These are point-centric views of the sphere, the projection replaces circular with spheric symmetry.

For the azymithmal projections, let F be the focus, or mapping point.

STEREOGRAPHIC
--------------------

The stereographic projection places the observer at a pole, and the map tangent to the opposite pole. The ray from F through P strikes the map at P'. Stereographic maps preserve angle, so at any locality, the angles are correct.

GNOMIC
----------

The gnomic projection places F at the centre of the glome, and maps P in the same manner, ie F - P - P' are in line. In the gnomic projection, any great circle becomes a straight line.

ORTHGRAPHIC
-----------------

The orthographic projection places F at deep infinity, so that P-P' are always parallel.


LINE-PROJECTIONS
-----------------------

Line projections, such as mercartor's or idlewile, do not do well in four dimensions. There are some exceptions, though.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Postby thigle » Sat Mar 11, 2006 9:54 am

thank you.

still, from the GNOMIC, what if F is offset from centre of the glome, letsay r/2, parallely to mapping plane ? is it still considered gnomic until F enters glome's hypersurface, until it "touches it" ? then it turns to stereographic-deformed (because F is not polar to the point of contact of the glome & the map ?

for the LINE projections, why is it that they don't do well in fourspace ?
thigle
Tetronian
 
Posts: 390
Joined: Fri Jul 29, 2005 5:00 pm

Postby wendy » Sun Mar 12, 2006 9:06 am

Hi

You can indeed offset F anywhere in the space. But these do not generally lead to useful projections, in the manner that the the centre, the opposite point, and the point at infinity do.

LINE PROJECTIONS
----------------------

A line-projection preserves a line, eg the equator. One can have transverse projections, where some other great circle fills this role. But since the bulk of projections are equator-based, we shall describe this.

The most common projection is Mercartor's. The feature here is that an lexidrome is mapped as straight. This means, that if one draws a straight line on the map, it crosses eg 30 W at the angle and position as drawn on the map. Its general utility comes from having a device that can measure angles from north at any given location: ie a compass.

The reason that Mercator's projection does not work in 4d, is that there is no obvious reason that there ought be a N and S magnetic pole.

Instead, we turn to physics, and note that in 4D, there are essentially two orthogonal rotations. The nature of physics says that there will be a transfer of energy from one state to the other, and it is therefore safe to assume that these two states of rotation will become the same.

This means, that on a 4d earth, every part of the planet rotates around the centre, and relative to space, a given location that has star X as zentith, will always have it.

The longitude is as on earth, a dividing space, where at a given longitude, it is 9am. The actual shape of this is a half-glome, but it stretches into a complete sphere (3d).

Lattitude is then orthogonal to longitude, each lattitude point becomes a circle. The lattitude where X is zenith, is but one point on the lattitude-sphere (eg like 30 N).

The glome is then presented as a prism of lattitude and longitude. This is a glomolatrix (circle-surface) * glomohedrix (sphere-surface) prism.

The sphere surface is that, where every point represents a circle that has the same points of the sky as a zenith-point. It does not "rotate" in much the same way that the N-S arc does not rotate.

At some ring on this sphere lies the "elliptic", or surface where the sun is overhead. The sun moves on a different circle, and so it moves around.

The effect on the lattitude sphere is that it appears to rotate once a year. That is, you have not only time-zones, but date-zones. At any given point, it is summer, at another, autumn, at another, spring, at another, winter.

The circle marked out by the sun makes for a centre (ie Tropic ring), and a point opposite (ie artic ring). The tropic ring has equatorial climate, the artic ring has polar climate. Unlike the earth, the sun is confined to the tropics, and instead of tropics of Capricorn and Cancer, one has also the tropics of Leo, Aquarius, Aries, etc.

You can then plot the whole lot onto a prism, with climata, annula, and longitude.

Or you could do the thing radially, where centre = midnight, outside = 24 oclock, and the rising-spheres shown as outward-rays. You don't need to make the centre solid.

Or you can use some kind of sphere-projection, made into a prism.

But, once you go past the point-projections, one has to mind the way things are spinning, and the ability to derive directions.

W
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Postby thigle » Sun Mar 12, 2006 4:07 pm

Hi.
very cool indeed.
you wrote:
You can indeed offset F anywhere in the space. But these do not generally lead to useful projections, in the manner that the the centre, the opposite point, and the point at infinity do.

yep. i was asking because of looking for some continuity between the projections, similar to like between circles, ellipses, parabolas and hyperbolas, i was thinking if there aint some model that unifies these projection somehow, or similar to the model you gave for allspace and different curvatures with F, sphere and plane. some overall structure. anyway.

But, once you go past the point-projections, one has to mind the way things are spinning, and the ability to derive directions.

past the point-projections mean that the ray-bundle is not form a point but for exemple all normals to a plane, or through a line, or something similar ?
i still cannot navigate your wheels very vell though, :cry: my ability to derive directions is amateurish at best. :?
thigle
Tetronian
 
Posts: 390
Joined: Fri Jul 29, 2005 5:00 pm

Postby wendy » Tue Mar 14, 2006 9:43 am

The general utility of a map is more what one wants it do do for you. But then again, a lot of what happens map-wise is due to the compass and the time-peice. Still, if you don't glark terrestial magnetism in 4d, it's hard to feed it into maps.

For example, there is a variation of the meactor projection, called the mecca projection, where for each locality, one has correctly the longitude lines, and the rays through some point (eg mecca), as straight lines. This has the advantage of being able to determine the angle to mecca for praying.

The more common projections, then are based on some unfolding of the surface into three dimensions. You could, go for polytope-net projection, which works in any dimension: project the sphere onto a polytope, and unfold said polytope flat.

So until we can get what a real 4d world ought look like, it's hardly pointful discussing what mercartor's projection might look like.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Postby Marek14 » Wed Mar 15, 2006 7:45 am

I was thinking about 3D "maps" of 4D environment, and their potential use in strategy games :) Take the old classic Civilization, for example. Wouldn't it be fun to have 3D map, using 4th dimension to explain things like "bridges" or "peaks"?
Marek14
Pentonian
 
Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

Postby wendy » Wed Mar 15, 2006 8:28 am

It is interesting to think of what sort of lattice, and what sort of moves to allow in a civ-style game. All versions of civ use a {4,4}. Battles would be rather interesting, because there are more ways that one can attack.

Civ1 and Civ2 had river as a terrain type, but in civ3, it became a border type (ie something between land-squares). In 4d, there is little point making rivers into a border-type, because they are typically latrous. Instead, they could be like roads.

You should not really be needing bridge-building, either. Really. I suppose, you could loose a movement-third for changing from a road to a river, representing getting on or off the barge.

Peaks should not present a problem by themselves, but hedrous forms of mountions might form when one tectonic plate mounts another. Since also in 3d, a typical way of forming surface features is by crater strikes (eg the polab basin), one might have the four-dimensional world with mountians representing one tectonic plate riding up on an other (ie a hedrous mass), and another formed by teelous (point-like) basins representing ancient meteor-strikes.

One might get peaks by way of volcanos, such as the glass-house mountians, not far from the City.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Postby thigle » Wed Mar 15, 2006 2:43 pm

is there an entrance into the Underworld from the City ?
thigle
Tetronian
 
Posts: 390
Joined: Fri Jul 29, 2005 5:00 pm

Postby Marek14 » Thu Mar 16, 2006 6:50 am

Actually, it always kinda puzzled me that no tile-based game, AFAIK, uses hyperbolic tiling :)

However, I think that you are wrong about Bridge Building. Apart from 1D rivers and 3D oceans, there could be also 2D water bodies, which would have to be traversed by bridge.

One aspect of 3D civilization would be that cities could grow that much bigger: in 2D, cities can get resources from 20 squares which can be approximated as all squares with Euclidean distance < 2 sqrt(2), and includes distances (1,0) (4 squares) (1,1) (4 squares) (2,0) (4 squares) and (2,1) (8 squares). In 3D, you would have:

(1,0,0) - 6 cubes
(1,1,0) - 12 cubes
(1,1,1) - 8 cubes
(2,0,0) - 6 cubes
(2,1,0) - 24 cubes
(2,1,1) - 24 cubes

80 cubes for one city, giving it 4 times the growth potential :)

Actually, I wonder if this is typical result, i.e. if higher-dimensional planet would indeed allow for easier growth of civilization.
Marek14
Pentonian
 
Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

Postby PWrong » Sat Apr 01, 2006 9:00 am

However, I think that you are wrong about Bridge Building. Apart from 1D rivers and 3D oceans, there could be also 2D water bodies, which would have to be traversed by bridge.

I agree that 2D water bodies are possible, but how would they form? A river is caused by water running from the top of a mountain to the bottom. It tries to take the shortest path, so the river would be 1D. Once it gets to the bottom, it forms a stationary 3D lake. To get a 2D lake, you'd need a 2D valley for it to sit in. I think erosion would eventually turn it into a roughly spherical valley.
User avatar
PWrong
Pentonian
 
Posts: 1599
Joined: Fri Jan 30, 2004 8:21 am
Location: Perth, Australia

Postby wendy » Wed Apr 05, 2006 9:32 am

It is quite easy for hedrous (2d form) lakes to form in four dimensions.

Consider, for example, the nature of plate-tectonics, and such. This creates a 3d plate climbing on an other, leaving a hedrous depression, which fills with water..

Still, bridge building is not the sort of thing that one uses to cover rift-lakes, even in ordinary civ. I can't think of a way that a non-passable hamma (meandering and forking of rivers) might form in 4d.

W
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia


Return to Other Geometry

Who is online

Users browsing this forum: No registered users and 9 guests

cron