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.