Provided by: xscreensaver-gl_5.34-2ubuntu1_amd64 
NAME
romanboy - Draws a 3d immersion of the real projective plane that smoothly deforms between the Roman
surface and the Boy surface.
SYNOPSIS
romanboy [-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] [-view-mode view-mode] [-walk] [-turn] [-no-deform] [-deformation-speed float]
[-initial-deformation float] [-roman] [-boy] [-surface-order number] [-orientation-marks] [-projection
mode] [-perspective] [-orthographic] [-speed-x float] [-speed-y float] [-speed-z float] [-walk-direction
float] [-walk-speed float]
DESCRIPTION
The romanboy program shows a 3d immersion of the real projective plane that smoothly deforms between the
Roman surface and the Boy surface. You can walk on the projective plane or turn in 3d. The smooth
deformation (homotopy) between these two famous immersions of the real projective plane was constructed
by François Apéry.
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 immersed projective plane can 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, or
direction. 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.
The rotation speed for each of the three coordinate axes around which the projective plane rotates can be
chosen.
Furthermore, in the walking mode 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.
By default, the immersion of the real projective plane smoothly deforms between the Roman and Boy
surfaces. It is possible to choose the speed of the deformation. Furthermore, it is possible to switch
the deformation off. It is also possible to determine the initial deformation of the immersion. This is
mostly useful if the deformation is switched off, in which case it will determine the appearance of the
surface.
As a final option, it is possible to display generalized versions of the immersion discussed above by
specifying the order of the surface. The default surface order of 3 results in the immersion of the real
projective described above. The surface order can be chosen between 2 and 9. Odd surface orders result
in generalized immersions of the real projective plane, while even numbers result in a immersion of a
topological sphere (which is orientable). The most interesting even case is a surface order of 2, which
results in an immersion of the halfway model of Morin's sphere eversion (if the deformation is switched
off).
This program is inspired by François Apéry's book "Models of the Real Projective Plane", Vieweg, 1987.
OPTIONS
romanboy 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 four 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.
The following three 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 3d.
-view-mode walk (Shortcut: -walk)
View the projective plane as if walking on its surface.
The following options determine whether the surface is being deformed.
-deform Deform the surface smoothly between the Roman and Boy surfaces (default).
-no-deform
Don't deform the surface.
The following option determines the deformation speed.
-deformation-speed float
The deformation speed is measured in percent of some sensible maximum speed (default: 10.0).
The following options determine the initial deformation of the surface. As described above, this is
mostly useful if -no-deform is specified.
-initial-deformation float
The initial deformation is specified as a number between 0 and 1000. A value of 0 corresponds to
the Roman surface, while a value of 1000 corresponds to the Boy surface. The default value is
1000.
-roman This is a shortcut for -initial-deformation 0.
-boy This is a shortcut for -initial-deformation 1000.
The following option determines the order of the surface to be displayed.
-surface-order number
The surface order can be set to values between 2 and 9 (default: 3). As described above, odd
surface orders result in generalized immersions of the real projective plane, while even numbers
result in a immersion of a topological sphere.
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 random
Project the projective plane from 3d to 2d using a random projection mode (default).
-projection perspective (Shortcut: -perspective)
Project the projective plane from 3d to 2d using a perspective projection.
-projection orthographic (Shortcut: -orthographic)
Project the projective plane from 3d to 2d using an orthographic projection.
The following three options determine the rotation speed of the projective plane around the three
possible axes. 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.
-speed-x float
Rotation speed around the x axis (default: 1.1).
-speed-y float
Rotation speed around the y axis (default: 1.3).
-speed-z float
Rotation speed around the z axis (default: 1.5).
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. 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 walk mode.
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 © 2013-2014 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.