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-.TH XScreenSaver 1 "" "X Version 11"
-.SH NAME
-romanboy \- Draws a 3d immersion of the real projective plane that
-smoothly deforms between the Roman surface and the Boy surface.
-.SH SYNOPSIS
-.B romanboy
-[\-display \fIhost:display.screen\fP]
-[\-install]
-[\-visual \fIvisual\fP]
-[\-window]
-[\-root]
-[\-delay \fIusecs\fP]
-[\-fps]
-[\-mode \fIdisplay-mode\fP]
-[\-wireframe]
-[\-surface]
-[\-transparent]
-[\-appearance \fIappearance\fP]
-[\-solid]
-[\-distance-bands]
-[\-direction-bands]
-[\-colors \fIcolor-scheme\fP]
-[\-onesided-colors]
-[\-twosided-colors]
-[\-distance-colors]
-[\-direction-colors]
-[\-change-colors]
-[\-view-mode \fIview-mode\fP]
-[\-walk]
-[\-turn]
-[\-no-deform]
-[\-deformation-speed \fIfloat\fP]
-[\-initial-deformation \fIfloat\fP]
-[\-roman]
-[\-boy]
-[\-surface-order \fInumber\fP]
-[\-orientation-marks]
-[\-projection \fImode\fP]
-[\-perspective]
-[\-orthographic]
-[\-speed-x \fIfloat\fP]
-[\-speed-y \fIfloat\fP]
-[\-speed-z \fIfloat\fP]
-[\-walk-direction \fIfloat\fP]
-[\-walk-speed \fIfloat\fP]
-.SH DESCRIPTION
-The \fIromanboy\fP 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 deformation (homotopy) between these two famous immersions of
-the real projective plane was constructed by François Apéry.
-.PP
-The real projective plane 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 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).
-.PP
-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, like all lines
-in the projective plane, is topologically a circle.  Points on the
-line at infinity are also used to model directions in projective
-geometry.  The origin can be visualized in different manners.  When
-using distance colors (and using static colors), the origin is the
-point that is displayed 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 orientation markers are compressed to a point.
-The line at infinity can also be visualized in different ways.  When
-using distance colors (and using static colors), the line at infinity
-is the line that is displayed as fully saturated magenta.  When
-two-sided (and static) 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.
-.PP
-Note that when the projective plane is displayed with bands, the
-orientation 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 remaining bands.  This choice, however, also implies that the band
-at infinity 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.
-.PP
-The immersed projective plane can be projected to the screen either
-perspectively or orthographically.  When using the walking modes,
-perspective projection to the screen will be used.
-.PP
-There are three display modes for the projective plane: mesh
-(wireframe), solid, or transparent.  Furthermore, the appearance of
-the projective 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.
-.PP
-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.
-.PP
-Finally, the colors with with the projective plane is drawn can be set
-to one-sided, two-sided, distance, or direction.  In one-sided mode,
-the projective plane is drawn with the same color on both "sides."  In
-two-sided mode (using static colors), the projective 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.  If changing colors are
-used in two-sided mode, changing complementary colors are used on the
-respective "sides."  In distance mode, the projective plane is
-displayed with fully saturated colors that depend on the distance of
-the points on the projective plane to the origin.  If static colors
-are used, the origin is displayed in red, while 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
-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
-projective plane is displayed as direction bands, each band will be
-displayed with a different color.
-.PP
-The rotation speed for each of the three coordinate axes around which
-the projective plane rotates can be chosen.
-.PP
-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
-projective 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.
-.PP
-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 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.
-.PP
-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
-topological 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).
-.PP
-This program is inspired by François Apéry's book "Models of the Real
-Projective Plane", Vieweg, 1987.
-.SH OPTIONS
-.I romanboy
-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 four options are mutually exclusive.  They determine how
-the projective plane is displayed.
-.TP 8
-.B \-mode random
-Display the projective plane in a random display mode (default).
-.TP 8
-.B \-mode wireframe \fP(Shortcut: \fB\-wireframe\fP)
-Display the projective plane as a wireframe mesh.
-.TP 8
-.B \-mode surface \fP(Shortcut: \fB\-surface\fP)
-Display the projective plane as a solid surface.
-.TP 8
-.B \-mode transparent \fP(Shortcut: \fB\-transparent\fP)
-Display the projective plane as a transparent surface.
-.PP
-The following four options are mutually exclusive.  They determine the
-appearance of the projective plane.
-.TP 8
-.B \-appearance random
-Display the projective plane with a random appearance (default).
-.TP 8
-.B \-appearance solid \fP(Shortcut: \fB\-solid\fP)
-Display the projective plane as a solid object.
-.TP 8
-.B \-appearance distance-bands \fP(Shortcut: \fB\-distance-bands\fP)
-Display the projective plane as see-through bands that lie at
-increasing distances from the origin.
-.PP
-.TP 8
-.B \-appearance direction-bands \fP(Shortcut: \fB\-direction-bands\fP)
-Display the projective plane as see-through bands that lie at
-increasing angles with respect to the origin.
-.PP
-The following four options are mutually exclusive.  They determine how
-to color the projective plane.
-.TP 8
-.B \-colors random
-Display the projective plane with a random color scheme (default).
-.TP 8
-.B \-colors onesided \fP(Shortcut: \fB\-onesided-colors\fP)
-Display the projective plane with a single color.
-.TP 8
-.B \-colors twosided \fP(Shortcut: \fB\-twosided-colors\fP)
-Display the projective plane with two colors: one color one "side" and
-the complementary color on the "other side."  For static colors, the
-colors are red and green.  Note that the line at infinity lies at the
-points where the red and green "sides" of the projective plane meet,
-i.e., where the orientation of the projective plane reverses.
-.TP 8
-.B \-colors distance \fP(Shortcut: \fB\-distance-colors\fP)
-Display the projective plane with fully saturated colors that depend
-on the distance of the points on the projective plane to the origin.
-For static colors, the origin is displayed in red, while 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.
-.TP 8
-.B \-colors direction \fP(Shortcut: \fB\-direction-colors\fP)
-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 projective plane is displayed as direction bands,
-each band will be displayed with a different color.
-.PP
-The following options determine whether the colors with which the
-projective plane is displayed are static or are changing dynamically.
-.TP 8
-.B \-change-colors
-Change the colors with which the projective plane is displayed
-dynamically.
-.TP 8
-.B \-no-change-colors
-Use static colors to display the projective plane (default).
-.PP
-The following three options are mutually exclusive.  They determine
-how to view the projective plane.
-.TP 8
-.B \-view-mode random
-View the projective plane in a random view mode (default).
-.TP 8
-.B \-view-mode turn \fP(Shortcut: \fB\-turn\fP)
-View the projective plane while it turns in 3d.
-.TP 8
-.B \-view-mode walk \fP(Shortcut: \fB\-walk\fP)
-View the projective plane as if walking on its surface.
-.PP
-The following options determine whether the surface is being deformed.
-.TP 8
-.B \-deform
-Deform the surface smoothly between the Roman and Boy surfaces
-(default).
-.TP 8
-.B \-no-deform
-Don't deform the surface.
-.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 options determine the initial deformation of the
-surface.  As described above, this is mostly useful if
-\fB\-no-deform\fP is specified.
-.TP 8
-.B \-initial-deformation \fIfloat\fP
-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.
-.TP 8
-.B \-roman
-This is a shortcut for \fB\-initial-deformation 0\fP.
-.TP 8
-.B \-boy
-This is a shortcut for \fB\-initial-deformation 1000\fP.
-.PP
-The following option determines the order of the surface to be
-displayed.
-.TP 8
-.B \-surface-order \fInumber\fP
-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.
-.PP
-The following options determine whether orientation marks are shown on
-the projective plane.
-.TP 8
-.B \-orientation-marks
-Display orientation marks on the projective plane.
-.TP 8
-.B \-no-orientation-marks
-Don't display orientation marks on the projective plane (default).
-.PP
-The following three options are mutually exclusive.  They determine
-how the projective plane is projected from 3d to 2d (i.e., to the
-screen).
-.TP 8
-.B \-projection random
-Project the projective plane from 3d to 2d using a random projection
-mode (default).
-.TP 8
-.B \-projection perspective \fP(Shortcut: \fB\-perspective\fP)
-Project the projective plane from 3d to 2d using a perspective
-projection.
-.TP 8
-.B \-projection orthographic \fP(Shortcut: \fB\-orthographic\fP)
-Project the projective plane from 3d to 2d using an orthographic
-projection.
-.PP
-The following three options determine the rotation speed of the
-projective plane 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.
-.TP 8
-.B \-speed-x \fIfloat\fP
-Rotation speed around the x axis (default: 1.1).
-.TP 8
-.B \-speed-y \fIfloat\fP
-Rotation speed around the y axis (default: 1.3).
-.TP 8
-.B \-speed-z \fIfloat\fP
-Rotation speed around the z axis (default: 1.5).
-.PP
-The following two options determine the walking speed and direction.
-.TP 8
-.B \-walk-direction \fIfloat\fP
-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 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.
-.TP 8
-.B \-walk-speed \fIfloat\fP
-The walking speed is measured in percent of some sensible maximum
-speed (default: 20.0).
-.SH 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
-examine 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.
-.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 2013-2020 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>, 06-jan-2020.