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+.TH XScreenSaver 1 "" "X Version 11"
+.SH NAME
+sphereeversion - Displays a sphere eversion.
+.SH SYNOPSIS
+.B sphereeversion
+[\-display \fIhost:display.screen\fP]
+[\-install]
+[\-visual \fIvisual\fP]
+[\-window]
+[\-root]
+[\-delay \fIusecs\fP]
+[\-fps]
+[\-mode \fIdisplay-mode\fP]
+[\-surface]
+[\-transparent]
+[\-appearance \fIappearance\fP]
+[\-solid]
+[\-parallel-bands]
+[\-meridian-bands]
+[\-graticule \fImode\fP]
+[\-surface-order \fIorder\fP]
+[\-colors \fIcolor-scheme\fP]
+[\-twosided-colors]
+[\-parallel-colors]
+[\-meridian-colors]
+[\-deformation-speed \fIfloat\fP]
+[\-projection \fImode\fP]
+[\-perspective]
+[\-orthographic]
+[\-speed-x \fIfloat\fP]
+[\-speed-y \fIfloat\fP]
+[\-speed-z \fIfloat\fP]
+.SH DESCRIPTION
+The \fIsphereeversion\fP 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.
+.PP
+The deformed sphere can be projected to the screen either
+perspectively or orthographically.
+.PP
+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.
+.PP
+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.
+.PP
+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.
+.PP
+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.
+.PP
+The colors with with the sphere is drawn can be set to two-sided,
+parallel, meridian, or random.  In two-sided mode, the sphere is drawn
+with red on one side and green on the other side.  In parallel mode,
+the sphere is displayed with colors that run from red to cyan on one
+side of the surface and from green to violet 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
+red to white to cyan to black and back to red on one side of the
+surface and from green to white to violet to black and back to green
+on the other side.  The colors are aligned with the meridians of the
+sphere in this mode. If random mode is selected, the color scheme is
+changed each time an eversion has been completed.
+.PP
+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.
+.PP
+This program is inspired by the following paper: Adam Bednorz, Witold
+Bednorz: "Analytic sphere eversion using ruled surfaces", Differential
+Geometry and its Applications 64:59-79, 2019.
+.SH OPTIONS
+.I sphereeversion
+accepts the following options:
+.TP 8
+.B \-window
+Draw on a newly-created window.  This is the default.
+.TP 8
+.B \-root
+Draw on the root window.
+.TP 8
+.B \-install
+Install a private colormap for the window.
+.TP 8
+.B \-visual \fIvisual\fP
+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.
+.TP 8
+.B \-delay \fImicroseconds\fP
+How much of a delay should be introduced between steps of the
+animation.  Default 10000, or 1/100th second.
+.TP 8
+.B \-fps
+Display the current frame rate, CPU load, and polygon count.
+.PP
+The following three options are mutually exclusive.  They determine
+how the deformed sphere is displayed.
+.TP 8
+.B \-mode random
+Display the sphere in a random display mode (default).
+.TP 8
+.B \-mode surface \fP(Shortcut: \fB\-surface\fP)
+Display the sphere as a solid surface.
+.TP 8
+.B \-mode transparent \fP(Shortcut: \fB\-transparent\fP)
+Display the sphere as a transparent surface.
+.PP
+The following four options are mutually exclusive.  They determine the
+appearance of the deformed sphere.
+.TP 8
+.B \-appearance random
+Display the sphere with a random appearance (default).
+.TP 8
+.B \-appearance solid \fP(Shortcut: \fB\-solid\fP)
+Display the sphere as a solid object.
+.TP 8
+.B \-appearance parallel-bands \fP(Shortcut: \fB\-parallel-bands\fP)
+Display the sphere as see-through bands that lie along the parallels
+of the sphere.
+.TP 8
+.B \-appearance meridian-bands \fP(Shortcut: \fB\-meridian-bands\fP)
+Display the sphere as see-through bands that lie along the meridians
+of the sphere.
+.PP
+The following three options are mutually exclusive.  They determine
+whether a graticule is displayed on top of the sphere.
+.TP 8
+.B \-graticule random
+Randomly choose whether to display a graticule (default).
+.TP 8
+.B \-graticule on
+Display a graticule.
+.TP 8
+.B \-graticule off
+Do not display a graticule.
+.PP
+The following option determines the order of the surface to be
+displayed.
+.TP 8
+.B \-surface-order \fIorder\fP
+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.
+.PP
+The following four options are mutually exclusive.  They determine how
+to color the deformed sphere.
+.TP 8
+.B \-colors random
+Display the sphere with a random color scheme (default).
+.TP 8
+.B \-colors twosided \fP(Shortcut: \fB\-twosided-colors\fP)
+Display the sphere with two colors: red on one side and green on the
+other side.
+.TP 8
+.B \-colors parallel \fP(Shortcut: \fB\-parallel-colors\fP)
+Display the sphere with colors that run from red to cyan on one side
+of the surface and from green to violet 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.
+.TP 8
+.B \-colors meridian \fP(Shortcut: \fB\-meridian-colors\fP)
+Display the sphere with colors that run from red to white to cyan to
+black and back to red on one side of the surface and from green to
+white to violet to black and back to green 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.
+.PP
+The following option determines the deformation speed.
+.TP 8
+.B \-deformation-speed \fIfloat\fP
+The deformation speed is measured in percent of some sensible maximum
+speed (default: 10.0).
+.PP
+The following three options are mutually exclusive.  They determine
+how the deformed sphere is projected from 3d to 2d (i.e., to the
+screen).
+.TP 8
+.B \-projection random
+Project the sphere from 3d to 2d using a random projection mode
+(default).
+.TP 8
+.B \-projection perspective \fP(Shortcut: \fB\-perspective\fP)
+Project the sphere from 3d to 2d using a perspective projection.
+.TP 8
+.B \-projection orthographic \fP(Shortcut: \fB\-orthographic\fP)
+Project the sphere from 3d to 2d using an orthographic projection.
+.PP
+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.
+.TP 8
+.B \-speed-x \fIfloat\fP
+Rotation speed around the x axis (default: 0.0).
+.TP 8
+.B \-speed-y \fIfloat\fP
+Rotation speed around the y axis (default: 0.0).
+.TP 8
+.B \-speed-z \fIfloat\fP
+Rotation speed around the z axis (default: 0.0).
+.SH 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.
+.SH ENVIRONMENT
+.PP
+.TP 8
+.B DISPLAY
+to get the default host and display number.
+.TP 8
+.B XENVIRONMENT
+to get the name of a resource file that overrides the global resources
+stored in the RESOURCE_MANAGER property.
+.SH SEE ALSO
+.BR X (1),
+.BR xscreensaver (1)
+.SH COPYRIGHT
+Copyright \(co 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.
+.SH AUTHOR
+Carsten Steger <carsten@mirsanmir.org>, 01-jun-2020.