by 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
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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