romanboy man page on DragonFly
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romanboy(6) XScreenSaver manual romanboy(6)
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] [-wire‐
frame] [-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] [-orienta‐
tion-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 defor‐
mation (homotopy) between these two famous immersions of the real pro‐
jective plane was constructed by Fran�ois Ap�ry.
The real projective plane is a non-orientable surface. To make this
apparent, the two-sided color mode can be used. Alternatively, orien‐
tation 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 sin‐
gled 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 cir‐
cle. Points on the line at infinity are also used to model directions
in projective geometry. The origin can be visualized in different man‐
ners. When using distance colors, the origin is the point that is dis‐
played 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 orien‐
tation 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 orien‐
tation 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 remain‐
ing bands. This choice, however, also implies that the band at infin‐
ity 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, per‐
spective projection to the screen will be used.
There are three display modes for the projective plane: mesh (wire‐
frame), solid, or transparent. Furthermore, the appearance of the pro‐
jective 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 projec‐
tive 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 satu‐
rated colors that depend on the distance of the points on the projec‐
tive 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 satu‐
rated 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 projec‐
tive 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 deforma‐
tion 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 topo‐
logical 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 Fran�ois Ap�ry'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 pro‐
jective plane meet, i.e., where the orientation of the projec‐
tive 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 infin‐
ity is displayed in magenta. If the projective plane is dis‐
played 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 projec‐
tive 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 dis‐
played.
-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 pro‐
jection 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 ortho‐
graphic projection.
The following three options determine the rotation speed of the projec‐
tive plane around the three possible axes. The rotation speed is mea‐
sured 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 dis‐
tance 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 maxi‐
mum 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 exam‐
ine 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, mod‐
ify, distribute, and sell this software and its documentation for any
purpose is hereby granted without fee, provided that the above copy‐
right notice appear in all copies and that both that copyright notice
and this permission notice appear in supporting documentation. No rep‐
resentations 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) romanboy(6)
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