Provided by: xscreensaver-gl_6.08+dfsg1-1ubuntu3_amd64 
NAME
sphereeversion - Displays a sphere eversion.
SYNOPSIS
sphereeversion [--display host:display.screen] [--install] [--visual visual] [--window]
[--root] [--window-id number] [--delay usecs] [--fps] [--eversion-method method]
[--analytic] [--corrugations] [--mode display-mode] [--surface] [--transparent]
[--appearance appearance] [--solid] [--parallel-bands] [--meridian-bands] [--graticule
mode] [--colors color-scheme] [--twosided-colors] [--parallel-colors] [--meridian-colors]
[--earth-colors] [--deformation-speed float] [--projection mode] [--perspective]
[--orthographic] [--surface-order order] [--lunes-1] [--lunes-2] [--lunes-4] [--lunes-8]
[--hemispheres-1] [--hemispheres-2] [--speed-x float] [--speed-y float] [--speed-z float]
DESCRIPTION
The sphereeversion program shows a sphere eversion, i.e., a smooth deformation (homotopy)
that turns a sphere inside out. During the eversion, the deformed sphere is allowed to
intersect itself transversally. However, no creases or pinch points are allowed to occur.
The sphere can be deformed with two eversion methods: analytic or corrugations. The
analytic sphere eversion method is described in the following paper: Adam Bednorz, Witold
Bednorz: "Analytic sphere eversion using ruled surfaces", Differential Geometry and its
Applications 64:59-79, 2019. The corrugations sphere eversion method is described in the
video "Outside In" by the Geometry Center (Bill Thurston, Silvio Levy, Delle Maxwell,
Tamara Munzner, Nathaniel Thurston, David Ben-Zvi, Matt Headrick, et al.), 1994, and the
accompanying booklet: Silvio Levy: "Making Waves - A Guide to the Ideas Behind Outside
In", A K Peters, Wellesley, MA, 1995. See also the section "Brief Description of the
Corrugations Sphere Eversion Method" below.
The deformed sphere can be projected to the screen either perspectively or
orthographically.
There are three display modes for the sphere: solid, transparent, or random. If random
mode is selected, the mode is changed each time an eversion has been completed.
The appearance of the sphere can be as a solid object, as a set of see-through bands, or
random. The bands can be parallel bands or meridian bands, i.e., bands that run along the
parallels (lines of latitude) or bands that run along the meridians (lines of longitude)
of the sphere. If random mode is selected, the appearance is changed each time an
eversion has been completed.
For the analytic sphere eversion, it is also possible to display a graticule (i.e., a
coordinate grid consisting of parallel and meridian lines) on top of the surface. The
graticule mode can be set to on, off, or random. If random mode is selected, the
graticule mode is changed each time an eversion has been completed.
The colors with with the sphere is drawn can be set to two-sided, parallel, meridian,
earth, or random. In two-sided mode, the sphere is drawn with red on one side and green
on the other side (analytic eversion) or with gold on one side and purple on the other
side (corrugations eversion). In parallel mode, the sphere is displayed with colors that
run from blue to white to orange on one side of the surface and from magenta to black to
green on the other side. The colors are aligned with the parallels of the sphere in this
mode. In meridian mode, the the sphere is displayed with colors that run from blue to
white to orange to black and back to blue on one side of the surface and from magenta to
white to green to black and back to magenta on the other side. The colors are aligned
with the meridians of the sphere in this mode. In earth mode, the sphere is drawn with a
texture of earth by day on one side and with a texture of earth by night on the other
side. Initially, the earth by day is on the outside and the earth by night on the inside.
After the first eversion, the earth by night will be on the outside. All points of the
earth on the inside and outside are at the same positions on the sphere. Since an
eversion transforms the sphere into its inverse, the earth by night will appear with all
continents mirror reversed. If random mode is selected, the color scheme is changed each
time an eversion has been completed.
By default, the sphere is rotated to a new viewing position each time an eversion has been
completed. In addition, it is possible to rotate the sphere while it is deforming. The
rotation speed for each of the three coordinate axes around which the sphere rotates can
be chosen arbitrarily. For best effects, however, it is suggested to rotate only around
the z axis while the sphere is deforming.
For the analytic sphere eversion, it is possible to define a surface order of the sphere
eversion as random or as a value between 2 and 5. This determines the the complexity of
the deformation. For higher surface orders, some z-fighting might occur around the
central stage of the eversion, which might lead to some irregular flickering of the
displayed surface if it is displayed as a solid object. For odd surface orders, z-
fighting will occur very close to the central stage of the eversion since the deformed
sphere is a doubly covered Boy surface (for surface order 3) or a doubly covered
generalized Boy surface (for surface order 5) in this case. If you find this distracting,
you should set the surface order to 2. If a random surface order is selected, the surface
order is changed each time an eversion has been completed.
BRIEF DESCRIPTION OF THE CORRUGATIONS SPHERE EVERSION METHOD
The corrugations sphere eversion method is described in detail in the video and booklet
mentioned above. Briefly, the method works as follows: Imagine the sphere cut into eight
spherical lunes (spherical biangles). Now imagine each lune to be a belt. The ends of
the belt (which correspond to the north and south poles of the sphere) are pushed past
each other. This creates a loop in the belt. If the belt were straightened out, it would
contain a 360 degree rotation. This rotation can be removed by rotating each end of the
belt by 180 degrees. Finally, the belt is pushed to the opposite side of the sphere,
which causes the side of the belt that initially was inside the sphere to appear on the
outside.
The method described so far only works for a single lune (belt) and not for the entire
sphere. To make it work for the entire sphere, corrugations (i.e., waves) must be added
to the sphere. This happens in the first phase of the eversion. Then, the method
described above is applied to the eight lunes. Finally, the corrugations are removed to
obtain the everted sphere.
To see the eversion for a single lune, the option --lunes-1 can be used. Using this
option, the eversion, as described above, is easier to understand. It is also possible to
display two lunes using --lunes-2 and four lunes using --lunes-4. Using fewer than eight
lunes reduces the visual complexity of the eversion and may help to understand the method.
Furthermore, it is possible to display only one hemisphere using the option
--hemispheres-1. This allows to see what is happening in the center of the sphere during
the eversion. Note that the north and south half of the sphere move in a symmetric
fashion during the eversion. Hence, the eversion is actually composed of 16 semi-lunes
(spherical triangles from the equator to the poles) that all deform in the same manner.
By specifying --lunes-1 --hemispheres-1, the deformation of one semi-lune can be observed.
Note that the options described above are only intended for educational purposes. They
are not used if none of them are explicitly specified.
OPTIONS
sphereeversion accepts the following options:
--window
Draw on a newly-created window. This is the default.
--root Draw on the root window.
--window-id number
Draw on the specified 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 three options are mutually exclusive. They determine which sphere eversion
method is used.
--eversion-method random
Use a random sphere eversion method (default).
--eversion-method analytic (Shortcut: --analytic)
Use the analytic sphere eversion method.
--eversion-method corrugations (Shortcut: --corrugations)
Use the corrugations sphere eversion method.
The following three options are mutually exclusive. They determine how the deformed
sphere is displayed.
--mode random
Display the sphere in a random display mode (default).
--mode surface (Shortcut: --surface)
Display the sphere as a solid surface.
--mode transparent (Shortcut: --transparent)
Display the sphere as a transparent surface.
The following four options are mutually exclusive. They determine the appearance of the
deformed sphere.
--appearance random
Display the sphere with a random appearance (default).
--appearance solid (Shortcut: --solid)
Display the sphere as a solid object.
--appearance parallel-bands (Shortcut: --parallel-bands)
Display the sphere as see-through bands that lie along the parallels of the
sphere.
--appearance meridian-bands (Shortcut: --meridian-bands)
Display the sphere as see-through bands that lie along the meridians of the
sphere.
The following three options are mutually exclusive. They determine whether a graticule is
displayed on top of the sphere. These options only have an effect if the analytic sphere
eversion method is selected.
--graticule random
Randomly choose whether to display a graticule (default).
--graticule on
Display a graticule.
--graticule off
Do not display a graticule.
The following five options are mutually exclusive. They determine how to color the
deformed sphere.
--colors random
Display the sphere with a random color scheme (default).
--colors twosided (Shortcut: --twosided-colors)
Display the sphere with two colors: red on one side and green on the other side
(analytic eversion) or gold on one side and purple on the other side (corrugations
eversion).
--colors parallel (Shortcut: --parallel-colors)
Display the sphere with colors that run from from blue to white to orange on one
side of the surface and from magenta to black to green on the other side. The
colors are aligned with the parallels of the sphere. If the sphere is displayed
as parallel bands, each band will be displayed with a different color.
--colors meridian (Shortcut: --meridian-colors)
Display the sphere with colors that run from from blue to white to orange to black
and back to blue on one side of the surface and from magenta to white to green to
black and back to magenta on the other side. The colors are aligned with the
meridians of the sphere. If the sphere is displayed as meridian bands, each band
will be displayed with a different color.
--colors earth (Shortcut: --earth-colors)
Display the sphere with a texture of earth by day on one side and with a texture
of earth by night on the other side. Initially, the earth by day is on the
outside and the earth by night on the inside. After the first eversion, the earth
by night will be on the outside. All points of the earth on the inside and
outside are at the same positions on the sphere. Since an eversion transforms the
sphere into its inverse, the earth by night will appear with all continents mirror
reversed.
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 three options are mutually exclusive. They determine how the deformed
sphere is projected from 3d to 2d (i.e., to the screen).
--projection random
Project the sphere from 3d to 2d using a random projection mode (default).
--projection perspective (Shortcut: --perspective)
Project the sphere from 3d to 2d using a perspective projection.
--projection orthographic (Shortcut: --orthographic)
Project the sphere from 3d to 2d using an orthographic projection.
The following option determines the order of the surface to be displayed. This option
only has an effect if the analytic sphere eversion method is selected.
--surface-order order
The surface order can be set to random or to a value between 2 and 5 (default:
random). This determines the the complexity of the deformation.
The following four options are mutually exclusive. They determine how many lunes of the
sphere are displayed. These options only have an effect if the corrugations sphere
eversion method is selected.
--lunes-1
Display one of the eight lunes that form the sphere.
--lunes-2
Display two of the eight lunes that form the sphere.
--lunes-4
Display four of the eight lunes that form the sphere.
--lunes-8
Display all eight lunes that form the sphere (default).
The following two options are mutually exclusive. They determine how many hemispheres of
the sphere are displayed. These options only have an effect if the corrugations sphere
eversion method is selected.
--hemispheres-1
Display only one hemisphere of the sphere.
--hemispheres-2
Display both hemispheres of the sphere (default).
The following three options determine the rotation speed of the deformed sphere 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.
--speed-x float
Rotation speed around the x axis (default: 0.0).
--speed-y float
Rotation speed around the y axis (default: 0.0).
--speed-z float
Rotation speed around the z axis (default: 0.0).
INTERACTION
If you run this program in standalone mode, you can rotate the deformed sphere by dragging
the mouse while pressing the left mouse button. This rotates the sphere in 3d. To
examine the deformed sphere at your leisure, it is best to set all speeds to 0.
Otherwise, the deformed sphere will rotate while the left mouse button is not pressed.
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.
XSCREENSAVER_WINDOW
The window ID to use with --root.
SEE ALSO
X(1), xscreensaver(1),
https://profs.etsmtl.ca/mmcguffin/eversion/,
http://www.geom.uiuc.edu/docs/outreach/oi/software.html
COPYRIGHT
Copyright © 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.
Parts of the code in this program are based on the program "sphereEversion 0.4" by Michael
J. McGuffin, which, in turn, is based on the program "Evert" developed by Nathaniel
Thurston at the Geometry Center. The modified code is used with permission.
AUTHOR
Carsten Steger <carsten@mirsanmir.org>, 01-jun-2020.