Provided by: xscreensaver-gl_6.02+dfsg1-2ubuntu1_amd64 
NAME
projectiveplane - Draws a 4d embedding of the real projective plane.
SYNOPSIS
projectiveplane [-display host:display.screen] [-install] [-visual visual] [-window]
[-root] [-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]
DESCRIPTION
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.
OPTIONS
projectiveplane accepts the following options:
-window Draw on a newly-created window. This is the default.
-root Draw on the root window.
-install
Install a private colormap for the window.
-visual visual
Specify which visual to use. Legal values are the name of a visual class, or the
id number (decimal or hex) of a specific visual.
-delay microseconds
How much of a delay should be introduced between steps of the animation. Default
10000, or 1/100th second.
-fps Display the current frame rate, CPU load, and polygon count.
The following four options are mutually exclusive. They determine how the projective
plane is displayed.
-mode random
Display the projective plane in a random display mode (default).
-mode wireframe (Shortcut: -wireframe)
Display the projective plane as a wireframe mesh.
-mode surface (Shortcut: -surface)
Display the projective plane as a solid surface.
-mode transparent (Shortcut: -transparent)
Display the projective plane as a transparent surface.
The following three options are mutually exclusive. They determine the appearance of the
projective plane.
-appearance random
Display the projective plane with a random appearance (default).
-appearance solid (Shortcut: -solid)
Display the projective plane as a solid object.
-appearance distance-bands (Shortcut: -distance-bands)
Display the projective plane as see-through bands that lie at increasing distances
from the origin.
-appearance direction-bands (Shortcut: -direction-bands)
Display the projective plane as see-through bands that lie at increasing angles
with respect to the origin.
The following four options are mutually exclusive. They determine how to color the
projective plane.
-colors random
Display the projective plane with a random color scheme (default).
-colors onesided (Shortcut: -onesided-colors)
Display the projective plane with a single color.
-colors twosided (Shortcut: -twosided-colors)
Display the projective plane with two colors: one color one "side" and the
complementary color on the "other side." For static colors, the colors are red
and green. Note that the line at infinity lies at the points where the red and
green "sides" of the projective plane meet, i.e., where the orientation of the
projective plane reverses.
-colors distance (Shortcut: -distance-colors)
Display the projective plane with fully saturated colors that depend on the
distance of the points on the projective plane to the origin. For static colors,
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.
-colors direction (Shortcut: -direction-colors)
Display the projective plane 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.
-colors depth (Shortcut: -depth)
Display the projective plane 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.
The following options determine whether the colors with which the projective plane is
displayed are static or are changing dynamically.
-change-colors
Change the colors with which the projective plane is displayed dynamically.
-no-change-colors
Use static colors to display the projective plane (default).
The following four options are mutually exclusive. They determine how to view the
projective plane.
-view-mode random
View the projective plane in a random view mode (default).
-view-mode turn (Shortcut: -turn)
View the projective plane while it turns in 4d.
-view-mode walk (Shortcut: -walk)
View the projective plane as if walking on its surface.
-view-mode walk-turn (Shortcut: -walk-turn)
View the projective plane as if walking on its surface. Additionally, the
projective plane turns around the true 4d planes (the xy, xz, and yz planes).
The following options determine whether orientation marks are shown on the projective
plane.
-orientation-marks
Display orientation marks on the projective plane.
-no-orientation-marks
Don't display orientation marks on the projective plane (default).
The following three options are mutually exclusive. They determine how the projective
plane is projected from 3d to 2d (i.e., to the screen).
-projection-3d random
Project the projective plane from 3d to 2d using a random projection mode
(default).
-projection-3d perspective (Shortcut: -perspective-3d)
Project the projective plane from 3d to 2d using a perspective projection.
-projection-3d orthographic (Shortcut: -orthographic-3d)
Project the projective plane from 3d to 2d using an orthographic projection.
The following three options are mutually exclusive. They determine how the projective
plane is projected from 4d to 3d.
-projection-4d random
Project the projective plane from 4d to 3d using a random projection mode
(default).
-projection-4d perspective (Shortcut: -perspective-4d)
Project the projective plane from 4d to 3d using a perspective projection.
-projection-4d orthographic (Shortcut: -orthographic-4d)
Project the projective plane from 4d to 3d using an orthographic projection.
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.
-speed-wx float
Rotation speed around the wx plane (default: 1.1).
-speed-wy float
Rotation speed around the wy plane (default: 1.3).
-speed-wz float
Rotation speed around the wz plane (default: 1.5).
-speed-xy float
Rotation speed around the xy plane (default: 1.7).
-speed-xz float
Rotation speed around the xz plane (default: 1.9).
-speed-yz float
Rotation speed around the yz plane (default: 2.1).
The following two options determine the walking speed and direction.
-walk-direction float
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 (default:
83.0). 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.
-walk-speed float
The walking speed is measured in percent of some sensible maximum speed (default:
20.0).
INTERACTION
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.
ENVIRONMENT
DISPLAY to get the default host and display number.
XENVIRONMENT
to get the name of a resource file that overrides the global resources stored in
the RESOURCE_MANAGER property.
SEE ALSO
X(1), xscreensaver(1)
COPYRIGHT
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.
AUTHOR
Carsten Steger <carsten@mirsanmir.org>, 06-jan-2020.