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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.