| projectiveplane(6x) | XScreenSaver manual | projectiveplane(6x) |
projectiveplane - Draws a 4d embedding of the real projective plane.
projectiveplane [--display host:display.screen] [--install] [--visual visual] [--window] [--root] [--window-id number] [--delay usecs] [--fps] [--mode display-mode] [--wireframe] [--surface] [--transparent] [--appearance appearance] [--solid] [--distance-bands] [--direction-bands] [--colors color-scheme] [--onesided-colors] [--twosided-colors] [--distance-colors] [--direction-colors] [--change-colors] [--depth-colors] [--view-mode view-mode] [--walk] [--turn] [--walk-turn] [--orientation-marks] [--projection-3d mode] [--perspective-3d] [--orthographic-3d] [--projection-4d mode] [--perspective-4d] [--orthographic-4d] [--speed-wx float] [--speed-wy float] [--speed-wz float] [--speed-xy float] [--speed-xz float] [--speed-yz float] [--walk-direction float] [--walk-speed float]
The projectiveplane program shows a 4d embedding of the real projective plane. You can walk on the projective plane, see it turn in 4d, or walk on it while it turns in 4d. The fact that the surface is an embedding of the real projective plane in 4d can be seen in the depth colors mode (using static colors): set all rotation speeds to 0 and the projection mode to 4d orthographic projection. In its default orientation, the embedding of the real projective plane will then project to the Roman surface, which has three lines of self-intersection. However, at the three lines of self-intersection the parts of the surface that intersect have different colors, i.e., different 4d depths.
The real projective plane is a non-orientable surface. To make this apparent, the two-sided color mode can be used. Alternatively, orientation markers (curling arrows) can be drawn as a texture map on the surface of the projective plane. While walking on the projective plane, you will notice that the orientation of the curling arrows changes (which it must because the projective plane is non-orientable).
The real projective plane is a model for the projective geometry in 2d space. One point can be singled out as the origin. A line can be singled out as the line at infinity, i.e., a line that lies at an infinite distance to the origin. The line at infinity, like all lines in the projective plane, is topologically a circle. Points on the line at infinity are also used to model directions in projective geometry. The origin can be visualized in different manners. When using distance colors (and using static colors), the origin is the point that is displayed as fully saturated red, which is easier to see as the center of the reddish area on the projective plane. Alternatively, when using distance bands, the origin is the center of the only band that projects to a disk. When using direction bands, the origin is the point where all direction bands collapse to a point. Finally, when orientation markers are being displayed, the origin the the point where all orientation markers are compressed to a point. The line at infinity can also be visualized in different ways. When using distance colors (and using static colors), the line at infinity is the line that is displayed as fully saturated magenta. When two-sided (and static) colors are used, the line at infinity lies at the points where the red and green "sides" of the projective plane meet (of course, the real projective plane only has one side, so this is a design choice of the visualization). Alternatively, when orientation markers are being displayed, the line at infinity is the place where the orientation markers change their orientation.
Note that when the projective plane is displayed with bands, the orientation markers are placed in the middle of the bands. For distance bands, the bands are chosen in such a way that the band at the origin is only half as wide as the remaining bands, which results in a disk being displayed at the origin that has the same diameter as the remaining bands. This choice, however, also implies that the band at infinity is half as wide as the other bands. Since the projective plane is attached to itself (in a complicated fashion) at the line at infinity, effectively the band at infinity is again as wide as the remaining bands. However, since the orientation markers are displayed in the middle of the bands, this means that only one half of the orientation markers will be displayed twice at the line at infinity if distance bands are used. If direction bands are used or if the projective plane is displayed as a solid surface, the orientation markers are displayed fully at the respective sides of the line at infinity.
The program projects the 4d projective plane to 3d using either a perspective or an orthographic projection. Which of the two alternatives looks more appealing is up to you. However, two famous surfaces are obtained if orthographic 4d projection is used: The Roman surface and the cross cap. If the projective plane is rotated in 4d, the result of the projection for certain rotations is a Roman surface and for certain rotations it is a cross cap. The easiest way to see this is to set all rotation speeds to 0 and the rotation speed around the yz plane to a value different from 0. However, for any 4d rotation speeds, the projections will generally cycle between the Roman surface and the cross cap. The difference is where the origin and the line at infinity will lie with respect to the self-intersections in the projections to 3d.
The projected projective plane can then be projected to the screen either perspectively or orthographically. When using the walking modes, perspective projection to the screen will be used.
There are three display modes for the projective plane: mesh (wireframe), solid, or transparent. Furthermore, the appearance of the projective plane can be as a solid object or as a set of see-through bands. The bands can be distance bands, i.e., bands that lie at increasing distances from the origin, or direction bands, i.e., bands that lie at increasing angles with respect to the origin.
When the projective plane is displayed with direction bands, you will be able to see that each direction band (modulo the "pinching" at the origin) is a Moebius strip, which also shows that the projective plane is non-orientable.
Finally, the colors with with the projective plane is drawn can be set to one-sided, two-sided, distance, direction, or depth. In one-sided mode, the projective plane is drawn with the same color on both "sides." In two-sided mode (using static colors), the projective plane is drawn with red on one "side" and green on the "other side." As described above, the projective plane only has one side, so the color jumps from red to green along the line at infinity. This mode enables you to see that the projective plane is non-orientable. If changing colors are used in two-sided mode, changing complementary colors are used on the respective "sides." In distance mode, the projective plane is displayed with fully saturated colors that depend on the distance of the points on the projective plane to the origin. If static colors are used, the origin is displayed in red, while the line at infinity is displayed in magenta. If the projective plane is displayed as distance bands, each band will be displayed with a different color. In direction mode, the projective plane is displayed with fully saturated colors that depend on the angle of the points on the projective plane with respect to the origin. Angles in opposite directions to the origin (e.g., 15 and 205 degrees) are displayed in the same color since they are projectively equivalent. If the projective plane is displayed as direction bands, each band will be displayed with a different color. Finally, in depth mode the projective plane is displayed with colors chosen depending on the 4d "depth" (i.e., the w coordinate) of the points on the projective plane at its default orientation in 4d. As discussed above, this mode enables you to see that the projective plane does not intersect itself in 4d.
The rotation speed for each of the six planes around which the projective plane rotates can be chosen. For the walk-and-turn mode, only the rotation speeds around the true 4d planes are used (the xy, xz, and yz planes).
Furthermore, in the walking modes the walking direction in the 2d base square of the projective plane and the walking speed can be chosen. The walking direction is measured as an angle in degrees in the 2d square that forms the coordinate system of the surface of the projective plane. A value of 0 or 180 means that the walk is along a circle at a randomly chosen distance from the origin (parallel to a distance band). A value of 90 or 270 means that the walk is directly from the origin to the line at infinity and back (analogous to a direction band). Any other value results in a curved path from the origin to the line at infinity and back.
This program is somewhat inspired by Thomas Banchoff's book "Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions", Scientific American Library, 1990.
projectiveplane accepts the following options:
The following four options are mutually exclusive. They determine how the projective plane is displayed.
The following three options are mutually exclusive. They determine the appearance of the projective plane.
The following four options are mutually exclusive. They determine how to color the projective plane.
The following options determine whether the colors with which the projective plane is displayed are static or are changing dynamically.
The following four options are mutually exclusive. They determine how to view the projective plane.
The following options determine whether orientation marks are shown on the projective plane.
The following three options are mutually exclusive. They determine how the projective plane is projected from 3d to 2d (i.e., to the screen).
The following three options are mutually exclusive. They determine how the projective plane is projected from 4d to 3d.
The following six options determine the rotation speed of the projective plane around the six possible hyperplanes. The rotation speed is measured in degrees per frame. The speeds should be set to relatively small values, e.g., less than 4 in magnitude. In walk mode, all speeds are ignored. In walk-and-turn mode, the 3d rotation speeds are ignored (i.e., the wx, wy, and wz speeds). In walk-and-turn mode, smaller speeds must be used than in the turn mode to achieve a nice visualization. Therefore, in walk-and-turn mode the speeds you have selected are divided by 5 internally.
The following two options determine the walking speed and direction.
If you run this program in standalone mode in its turn mode, you can rotate the projective plane by dragging the mouse while pressing the left mouse button. This rotates the projective plane in 3D, i.e., around the wx, wy, and wz planes. If you press the shift key while dragging the mouse with the left button pressed the projective plane is rotated in 4D, i.e., around the xy, xz, and yz planes. To examine the projective plane at your leisure, it is best to set all speeds to 0. Otherwise, the projective plane will rotate while the left mouse button is not pressed. This kind of interaction is not available in the two walk modes.
Copyright © 2013-2020 by Carsten Steger. Permission to use, copy, modify, distribute, and sell this software and its documentation for any purpose is hereby granted without fee, provided that the above copyright notice appear in all copies and that both that copyright notice and this permission notice appear in supporting documentation. No representations are made about the suitability of this software for any purpose. It is provided "as is" without express or implied warranty.
Carsten Steger <carsten@mirsanmir.org>, 06-jan-2020.
| 6.06 (11-Dec-2022) | X Version 11 |