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projectiveplane(6) XScreenSaver manual projectiveplane(6)
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]
[-twosided-colors] [-distance-colors] [-direction-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: 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 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, 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, the line at infinity is the line that is displayed as fully
saturated magenta. When two-sided 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 two-sided, distance, direction, or depth. In two-sided mode, 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. 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. The origin is displayed in red, 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 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 twosided (Shortcut: -twosided-colors)
Display the projective plane with two colors: red on one "side"
and green on the "other side." 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. The origin is displayed in red, 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 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 (C) 2005-2014 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>, 03-oct-2014.
X Version 11 5.34 (24-Oct-2015) projectiveplane(6)