Provided by: xscreensaver-gl_6.02+dfsg1-2ubuntu2_amd64 
NAME
etruscanvenus - Draws a 3d immersion of a Klein bottle that smoothly deforms between the
Etruscan Venus surface, the Roman surface, the Boy surface surface, and the Ida surface.
SYNOPSIS
etruscanvenus [-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] [-no-
change-colors] [-view-mode view-mode] [-walk] [-turn] [-no-deform] [-deformation-speed
float] [-initial-deformation float] [-etruscan-venus] [-roman] [-boy] [-ida]
[-orientation-marks] [-projection mode] [-perspective] [-orthographic] [-speed-x float]
[-speed-y float] [-speed-z float] [-walk-direction float] [-walk-speed float]
DESCRIPTION
The etruscanvenus program shows a 3d immersion of a Klein bottle that smoothly deforms
between the Etruscan Venus surface, the Roman surface, the Boy surface, and the Ida
surface. You can walk on the Klein bottle or turn it in 3d. Topologically, all surfaces
are Klein bottles, even the Roman and Boy surfaces, which are doubly covered and therefore
appear to be an immersed real projective plane. The smooth deformation between these
surfaces was constructed by George K. Francis.
The Klein bottle 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 Klein bottle. While walking on the Klein bottle, you
will notice that the orientation of the curling arrows changes (which it must because the
Klein bottle is non-orientable). Since all the surfaces except the Ida surface have
points where the surface normal is not well defined for some points, walking is only
performed on the Ida surface.
As mentioned above, the Roman and Boy surfaces are doubly covered and therefore appear to
be an immersed real projective plane. Since some of the parameter names are based on this
interpretation of the surface, the geometry of the real projective plane will be briefly
disussed. 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. Direction and distance bands refer to this
interpretation of the surface. If direction bands are used, the bands extend from the
origin of the projective plane in different directions to the line at infinity and back to
the origin. If distance bands are used, the bands lie at constant distances to the
origin. The same interpretation is used for distance and direction colors. Although
there is no conceptually equivalent geometric interpretation for the two Klein bottle
surfaces (the Etruscan Venus and Ida surfaces), the smooth deformation between the
surfaces results in a natural extension of these concepts to the Klein bottle surfaces.
The immersed surfaces can be projected to the screen either perspectively or
orthographically. When using the walking mode, perspective projection to the screen will
be used.
There are three display modes for the Klein bottle: mesh (wireframe), solid, or
transparent. Furthermore, the appearance of the surface can be as a solid object or as a
set of see-through bands. The bands can be distance bands or direction bands, as
explained above.
The colors with with the surface is drawn can be set to one-sided, two-sided, distance, or
direction. In one-sided mode, the surface is drawn with the same color on both sides of
the underlying triangles. In two-sided mode, the surface is drawn with one color on one
side of the underlying triangles and the complementary color on the other side. Since the
surface actually only has one side, the color jumps from red to green along a line on the
surface. This mode enables you to see that the surface is non-orientable. In distance
mode, the surface is displayed with fully saturated colors that depend on the distance of
the points on the projective plane to the origin, as described above. If the surface is
displayed as distance bands, each band will be displayed with a different color. In
direction mode, the surface is displayed with fully saturated colors that depend on the
angle of the points on the projective plane with respect to the origin (see above for an
explanation). If the surface is displayed as direction bands, each band will be displayed
with a different color. The colors used to color the surface can either be static or can
be changed dynamically.
The rotation speed for each of the three coordinate axes around which the Klein bottle
rotates can be chosen.
Furthermore, in the walking mode the walking direction in the 2d base square of the
surface 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. 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 along a direction band. Any other value results in a curved path along the
surface. As noted above, walking is performed only on the Ida surface.
By default, the immersion of the Klein bottle smoothly deforms between the Etruscan Venus
surface, the Roman surface, the Boy surface, and the Ida surface. 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. A value of 0 corresponds to the Etruscan
Venus surface, a value of 1000 to the Roman surface, a value of 2000 to the Boy surface,
and a value of 3000 to the Ida surface.
This program is inspired by George K. Francis's book "A Topological Picturebook",
Springer, 1987, by George K. Francis's paper "The Etruscan Venus" in P. Concus, R. Finn,
and D. A. Hoffman: "Geometric Analysis and Computer Graphics", Springer, 1991, and by a
video entitled "The Etruscan Venus" by Donna J. Cox, George K. Francis, and Raymond L.
Idaszak, presented at SIGGRAPH 1989.
OPTIONS
etruscanvenus 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 Klein bottle is
displayed.
-mode random
Display the Klein bottle in a random display mode (default).
-mode wireframe (Shortcut: -wireframe)
Display the Klein bottle as a wireframe mesh.
-mode surface (Shortcut: -surface)
Display the Klein bottle as a solid surface.
-mode transparent (Shortcut: -transparent)
Display the Klein bottle as a transparent surface.
The following four options are mutually exclusive. They determine the appearance of the
Klein bottle.
-appearance random
Display the Klein bottle with a random appearance (default).
-appearance solid (Shortcut: -solid)
Display the Klein bottle as a solid object.
-appearance distance-bands (Shortcut: -distance-bands)
Display the Klein bottle as see-through bands that lie at increasing distances
from the origin (see above for an explanation).
-appearance direction-bands (Shortcut: -direction-bands)
Display the Klein bottle as see-through bands that lie at increasing angles with
respect to the origin (see above for an explanation).
The following five options are mutually exclusive. They determine how to color the Klein
bottle.
-colors random
Display the Klein bottle with a random color scheme (default).
-colors onesided (Shortcut: -onesided-colors)
Display the Klein bottle with a single color.
-colors twosided (Shortcut: -twosided-colors)
Display the Klein bottle with two colors: one color on one "side" and the
complementary color on the "other side."
-colors distance (Shortcut: -distance-colors)
Display the Klein bottle with fully saturated colors that depend on the distance
of the points on the projective plane to the origin (see above for an
explanation). If the Klein bottle is displayed as distance bands, each band will
be displayed with a different color.
-colors direction (Shortcut: -direction-colors)
Display the Klein bottle with fully saturated colors that depend on the angle of
the points on the projective plane with respect to the origin (see above for an
explanation). If the Klein bottle is displayed as direction bands, each band will
be displayed with a different color.
The following options determine whether the colors with which the Klein bottle are
displayed are static or are changing dynamically.
-change-colors
Change the colors with which the Klein bottle is displayed dynamically (default).
-no-change-colors
Use static colors to display the Klein bottle.
The following three options are mutually exclusive. They determine how to view the Klein
bottle.
-view-mode random
View the Klein bottle in a random view mode (default). The walking mode will be
randomly selected in approximately 10% of the cases.
-view-mode turn (Shortcut: -turn)
View the Klein bottle while it turns in 3d.
-view-mode walk (Shortcut: -walk)
View the Klein bottle as if walking on its surface.
The following options determine whether the surface is being deformed.
-deform Deform the surface smoothly between the Etruscan Venus surface, the Roman surface,
the Boy surface surface, and the Ida surface (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 4000. A value of 0
corresponds to the Etruscan Venus surface, a value of 1000 to the Roman surface, a
value of 2000 to the Boy surface, and a value of 3000 to the Ida surface. The
default value is 0.
-etruscan-venus
This is a shortcut for -initial-deformation 0.
-roman This is a shortcut for -initial-deformation 1000.
-boy This is a shortcut for -initial-deformation 2000.
-ida This is a shortcut for -initial-deformation 3000.
The following options determine whether orientation marks are shown on the Klein bottle.
-orientation-marks
Display orientation marks on the Klein bottle.
-no-orientation-marks
Don't display orientation marks on the Klein bottle (default).
The following three options are mutually exclusive. They determine how the Klain bottle
is projected from 3d to 2d (i.e., to the screen).
-projection random
Project the Klein bottle from 3d to 2d using a random projection mode (default).
-projection perspective (Shortcut: -perspective)
Project the Klein bottle from 3d to 2d using a perspective projection.
-projection orthographic (Shortcut: -orthographic)
Project the Klein bottle from 3d to 2d using an orthographic projection.
The following three options determine the rotation speed of the Klein bottle 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 Klein bottle (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 along a direction band. Any other value results
in a curved path along the surface. As noted above, walking is performed only on
the Ida surface.
-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 Klein
bottle by dragging the mouse while pressing the left mouse button. This rotates the Klein
bottle in 3d. To examine the Klein bottle at your leisure, it is best to set all speeds
to 0. Otherwise, the Klein bottle 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 © 2019-2020 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>, 05-jan-2020.