Spatial Transformations

Higher-dimensional geometry (previously "Polyshapes").

Spatial Transformations

Postby pat » Wed Oct 04, 2006 2:01 pm

As part of a more general 4D (nD?) Viewer, I am working on a system that lets you specify the transformations to get from 4D to 2D. I am striving for maximum flexibility.

As part of this, I want to put together a list of useful/desirable transformations. Here's what I have so far.

  • E<sup>n</sup> -> E<sup>n+1</sup>
    • inclusion
  • E<sup>n</sup> -> E<sup>n</sup>
    • affine transformation (rotation, translation, skewing, general affine translation)
    • hyperbolic projection
    • inverse hyperbolic projection
  • E<sup>n</sup> -> E<sup>n-1</sup>
    • orthographic projection
    • perspective projection
  • E<sup>n</sup> -> S<sup>n-1</sup>
    • normalize
  • S<sup>n</sup> -> E<sup>n+1</sup>
    • embed
  • E<sup>n</sup> -> S<sup>n</sup>
    • inverse stereographic projection
  • S<sup>n</sup> -> E<sup>n</sup>
    • stereographic projeciton
  • S<sup>n</sup> -> S<sup>n</sup>
    • rotation

So, to get a four-dimensional object to the screen, you need to get from E<sup>4</sup> to E<sup>2</sup>. One possible chain would be: E<sup>4</sup> -affine-> E<sup>4</sup> -orthographic-> E<sup>3</sup> -normalize-> S<sup>2</sup> -rotation-> S<sup>2</sup> -stereographic-> E<sup>2</sup>.

Any suggestions for other transformations?

Certainly, I will have some canned sequences built-in so that you don't have to start from the ground up.
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Postby wendy » Thu Oct 05, 2006 7:26 am

There are always the ever-useful cartographic representations.

For example, there is a projections as

En -> En
  • Projective geometry (ie infinity -> line, lines as straight. This applies for Sn -> Sn too!
  • Stereographic projection (ie inversion)

The downwards projections usually involve projecting a space onto a point. This is eg, delatrification (line -> point), or dehedrification (plane to point). You can still get useful pictures of dehedrified d4, for example, even by way of simple coordinate-loss.

The process has been developed by me as lace cities for polytopes.

In practice, one has things like conformal, projective and isoperimetric projections.

  • Conformal projections preserve angle. Examples are (outside of direct insertion), stereographic (S), inversion (E), and poincare (H), or the beltrami-poincare halfplane (H).
  • Projective projections preserve straightness. Examples are the gnomic (S), projective (E), and beltrami-klein (H) projections
  • Isoperimetric preserves circumferences of concentric circles. This effectively simulates the way things will appear in the target geometry, if standing on a circle. A thing a mile away in this projection will appear to be as if it were, eg an euclidean mile away.
  • Mercartor, which, for lines perpendicular to a line, a straight line that crosses these at an angle represents a loxodrome. The H mercartor is of finite width!
  • Cylindric, for which there is a common line, and the area is preserved.


For higher dimensions, there operates different modes of rotation, and the corresponding projection needs to take this into account. One of the most interesting projections in 4d is the toric projection. This can be implemented radially, passing through a sphere.

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Postby pat » Thu Oct 05, 2006 2:47 pm

wendy wrote:En -> En
  • Projective geometry (ie infinity -> line, lines as straight. This applies for Sn -> Sn too!
  • Stereographic projection (ie inversion)


Hmmm... maybe I'm not following what you mean by "Projective Geometry". You said, En -> En, so, I'm assuming you don't mean En -> Pn. And for the moment, I want to stick to transformations where points stay points.

And, you're right... I can do inversion and such. I had thought about cylindric... I'm not sure why I left it off the list. I hadn't thought about the usual cartographic projections.... interesting. And, thanks for listing so many Conformal projections, too. I will have to look at them closer to see if there is a common language in which to express them. From C -> C (effectively E2 -> E2), there are lots of conformal projections that can be expressed as rational polynomials over C. But, I'll have to look at which ones I can make En -> En. Thanks...

For higher dimensions, there operates different modes of rotation, and the corresponding projection needs to take this into account. One of the most interesting projections in 4d is the toric projection. This can be implemented radially, passing through a sphere.

Every rotation in En can be represented by an nxn orthogonal matrix, no? And every nxn orthogonal matrix is a rotation in En. Any rotation in n-dimensions is equivalent to a sequence of rotations which leave all but two axises fixed, no?

And, my Google search for 'toric projection' didn't lead to anything glarkable. Could you describe the toric projection? or point me to some place that does describe it?

Thanks....
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Postby wendy » Fri Oct 06, 2006 7:43 am

The rotation that leaves all but two axies fixed is what i call a great arrow. That is, it's a great circle with an arrow on it. I suppose you could construct all rotations out of great arrows. In four dimensions, one constructs all rotations, including great arrows, out of clifford rotations.

With the "projective geometry", there is a loss of detail in some geometries. Those that maintain an infinity have no point-loss, because the antipodal point is not in real space.

When you project a plane by projective geometry, imagine that you have an observer U, above the ground, and that lines are transferred from the horizontal to the vertical, by passing planes through the line on the vertical, and through the point U.

A point P is transfromed from the horizontal to the vertical, by continuing a line from P to U to its image P'.

What this gives is the actual projective, as used by artists.

Hyperbolic geometry is completely preserved by the projective as well.

And, my Google search for 'toric projection' didn't lead to anything glarkable.


Projections, such as the mercartor and cylinderic, rely on the nature of the Earth's equator and its role in defining longitude and lattitude. In four dimensions, you don't have an equator as such.

Instead, what you have is a double rotation (ie in wx, yz). The nature of physics says that energy is transferred from rotational modes, so you get a rotation where wx, yz are at the same frequency: a clifford rotation. These are representable by quarterions. (The great-arrow rotations are effected by quarterions by using xZx', where x' is some kind of complement of x.

Since in a clifford-rotation, every point goes around the centre, one gets lattitude by the rising of say, the zenith star. Any point that has star X as zenith is on the same line of longitude.

One, can, for a pair of different points, calculate the differences of lattitude, because star X will peak in the sky at say, 60 degrees, then the distance between the lattitudes is 30 degrees (ie 90-60). If one makes a map of distances between lattitudes, one gets a sphere surface (glomochorix), where the angles are doubled. A pair of orthogonal great circles appears as antipodal on the sphere.

Longitude corresponds to the rotation, and is measured out in the time of the day: all points at say 150E is at 10pm at night, for example.

One can then take the product of longitude (circle-surface) and lattitude (sphere-surface) to get a circle-sphere torus. Unwapping this leads to interesting projections.

1. You can preserve the sphere-surface, and unwrap the circle into the radially to the sphere. The rotation is then outwards, and reappearing into the centre.

2. You can unwrap the sphere, and the circle, to give a (sphere-projection) * (circle-projection) prism. Since the circle-projection is a line (but still dividable at different points, cf different breaks in the mercartor projection).

You then can project the sphere in different ways. This is in nature, a stack of rectangles (from the thin line in x to the thin line in y). Rotations follow the diagonals of these lines, but can be skewed to go left-to-right.

The next orientation assumes that the earth is tilted: that is, the zodiac is a line that is not parallel to the earth-rotation, and there is a single pair of great arrows, for which each is also right-parallel to these (ie the line of zodiac is equidistant from these lines).

This presents a second and third axis, which effectively turns your lattitude-sphere into (like the earth), where N pole is icy cold, and S pole is tropics. [as if you doubled the distance from the north pole]. The earth-like longitude becomes the year-seasons. That is, for all points that is on the lattitude sphere at 140 E might be having spring, while 40 W is in autumn.

This gives 3 coordinates, (climate, time of year, time of day), which makes a box-shaped thing, and this is the toric projection.

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Postby pat » Fri Oct 06, 2006 3:20 pm

wendy wrote:In four dimensions, one constructs all rotations, including great arrows, out of clifford rotations.


Okay, but a clifford rotation is just the successive application of two planar rotations which just happen to be in orthogonal planes. But, if I'm following you, you might like to specify a plane, then specify the angular velocity in that plane and the angular velocity in the orthogonal plane rather than having to do that as two separate steps. That's perfectly fine.

A point P is transfromed from the horizontal to the vertical, by continuing a line from P to U to its image P'.


Okay, I'm all over that, I think.

This gives 3 coordinates, (climate, time of year, time of day), which makes a box-shaped thing, and this is the toric projection.


I will have to read this again a couple of times, but I'm pretty sure I kept up with that.

Thanks a bunch...
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Postby Hugh » Fri Oct 06, 2006 6:35 pm

Hi Pat. I came across this page a while ago: http://users.adelphia.net/~44mrf/hierarchy(1).html

I like how figures 8, 9 and 10 explain things. I hope there is something there that you can use. :)
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Postby pat » Fri Oct 06, 2006 9:52 pm

Thanks, Hugh...
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Postby Hugh » Fri Oct 06, 2006 10:33 pm

I know you probably already knew all that stuff.

I liked the "double bubble" two-sphere configuration mentioned on that page in figure 10. It's an interesting way to see 4d represented in 3d. :)
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