--- /dev/null
+.TH XScreenSaver 1 "" "X Version 11"
+.SH NAME
+projectiveplane - Draws a 4d embedding of the real projective plane.
+.SH SYNOPSIS
+.B projectiveplane
+[\-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]
+[\-twosided-colors]
+[\-distance-colors]
+[\-direction-colors]
+[\-depth-colors]
+[\-view-mode \fIview-mode\fP]
+[\-walk]
+[\-turn]
+[\-walk-turn]
+[\-orientation-marks]
+[\-projection-3d \fImode\fP]
+[\-perspective-3d]
+[\-orthographic-3d]
+[\-projection-4d \fImode\fP]
+[\-perspective-4d]
+[\-orthographic-4d]
+[\-speed-wx \fIfloat\fP]
+[\-speed-wy \fIfloat\fP]
+[\-speed-wz \fIfloat\fP]
+[\-speed-xy \fIfloat\fP]
+[\-speed-xz \fIfloat\fP]
+[\-speed-yz \fIfloat\fP]
+[\-walk-direction \fIfloat\fP]
+[\-walk-speed \fIfloat\fP]
+.SH DESCRIPTION
+The \fIprojectiveplane\fP program shows a 4d embedding of the real
+projective plane. You can walk on the projective plane, see it turn
+in 4d, or walk on it while it turns in 4d. The fact that the surface
+is an embedding of the real projective plane in 4d can be seen in the
+depth colors mode: set all rotation speeds to 0 and the projection
+mode to 4d orthographic projection. In its default orientation, the
+embedding of the real projective plane will then project to the Roman
+surface, which has three lines of self-intersection. However, at the
+three lines of self-intersection the parts of the surface that
+intersect have different colors, i.e., different 4d depths.
+.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 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, 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 disc. 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, the line at infinity is the line that is
+displayed as fully saturated magenta. When two-sided 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 disc 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 program projects the 4d projective plane to 3d using either a
+perspective or an orthographic projection. Which of the two
+alternatives looks more appealing is up to you. However, two famous
+surfaces are obtained if orthographic 4d projection is used: The Roman
+surface and the cross cap. If the projective plane is rotated in 4d,
+the result of the projection for certain rotations is a Roman surface
+and for certain rotations it is a cross cap. The easiest way to see
+this is to set all rotation speeds to 0 and the rotation speed around
+the yz plane to a value different from 0. However, for any 4d
+rotation speeds, the projections will generally cycle between the
+Roman surface and the cross cap. The difference is where the origin
+and the line at infinity will lie with respect to the
+self-intersections in the projections to 3d.
+.PP
+The projected projective plane can then 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 two-sided, distance, direction, or depth. In two-sided mode, 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. 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. The origin is displayed in
+red, 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. Finally, in depth mode the
+projective plane with colors chosen depending on the 4d "depth" (i.e.,
+the w coordinate) of the points on the projective plane at its default
+orientation in 4d. As discussed above, this mode enables you to see
+that the projective plane does not intersect itself in 4d.
+.PP
+The rotation speed for each of the six planes around which the
+projective plane rotates can be chosen. For the walk-and-turn more,
+only the rotation speeds around the true 4d planes are used (the xy,
+xz, and yz planes).
+.PP
+Furthermore, in the walking modes 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
+This program is somewhat inspired by Thomas Banchoff's book "Beyond
+the Third Dimension: Geometry, Computer Graphics, and Higher
+Dimensions", Scientific American Library, 1990.
+.SH OPTIONS
+.I projectiveplane
+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 three 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 twosided \fP(Shortcut: \fB\-twosided-colors\fP)
+Display the projective plane with two colors: red on one "side" and
+green on the "other side." 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.
+The origin is displayed in red, 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.
+.TP 8
+.B \-colors depth \fP(Shortcut: \fB\-depth\fP)
+Display the projective plane with colors chosen depending on the 4d
+"depth" (i.e., the w coordinate) of the points on the projective plane
+at its default orientation in 4d.
+.PP
+The following four 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 4d.
+.TP 8
+.B \-view-mode walk \fP(Shortcut: \fB\-walk\fP)
+View the projective plane as if walking on its surface.
+.TP 8
+.B \-view-mode walk-turn \fP(Shortcut: \fB\-walk-turn\fP)
+View the projective plane as if walking on its surface. Additionally,
+the projective plane turns around the true 4d planes (the xy, xz, and
+yz planes).
+.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-3d random
+Project the projective plane from 3d to 2d using a random projection
+mode (default).
+.TP 8
+.B \-projection-3d perspective \fP(Shortcut: \fB\-perspective-3d\fP)
+Project the projective plane from 3d to 2d using a perspective
+projection.
+.TP 8
+.B \-projection-3d orthographic \fP(Shortcut: \fB\-orthographic-3d\fP)
+Project the projective plane from 3d to 2d using an orthographic
+projection.
+.PP
+The following three options are mutually exclusive. They determine
+how the projective plane is projected from 4d to 3d.
+.TP 8
+.B \-projection-4d random
+Project the projective plane from 4d to 3d using a random projection
+mode (default).
+.TP 8
+.B \-projection-4d perspective \fP(Shortcut: \fB\-perspective-4d\fP)
+Project the projective plane from 4d to 3d using a perspective
+projection.
+.TP 8
+.B \-projection-4d orthographic \fP(Shortcut: \fB\-orthographic-4d\fP)
+Project the projective plane from 4d to 3d using an orthographic
+projection.
+.PP
+The following six options determine the rotation speed of the
+projective plane around the six possible hyperplanes. 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. In walk-and-turn mode, the 3d rotation
+speeds are ignored (i.e., the wx, wy, and wz speeds). In
+walk-and-turn mode, smaller speeds must be used than in the turn mode
+to achieve a nice visualization. Therefore, in walk-and-turn mode the
+speeds you have selected are divided by 5 internally.
+.TP 8
+.B \-speed-wx \fIfloat\fP
+Rotation speed around the wx plane (default: 1.1).
+.TP 8
+.B \-speed-wy \fIfloat\fP
+Rotation speed around the wy plane (default: 1.3).
+.TP 8
+.B \-speed-wz \fIfloat\fP
+Rotation speed around the wz plane (default: 1.5).
+.TP 8
+.B \-speed-xy \fIfloat\fP
+Rotation speed around the xy plane (default: 1.7).
+.TP 8
+.B \-speed-xz \fIfloat\fP
+Rotation speed around the xz plane (default: 1.9).
+.TP 8
+.B \-speed-yz \fIfloat\fP
+Rotation speed around the yz plane (default: 2.1).
+.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, i.e.,
+around the wx, wy, and wz planes. If you press the shift key while
+dragging the mouse with the left button pressed the projective plane
+is rotated in 4D, i.e., around the xy, xz, and yz planes. 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 two walk modes.
+.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 2005-2014 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>, 03-jan-2014.
projects / xscreensaver / blobdiff