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projectiveplane(6)	      XScreenSaver manual	    projectiveplane(6)

NAME
       projectiveplane - Draws a 4d embedding of the real projective plane.

SYNOPSIS
       projectiveplane [-display host:display.screen] [-install] [-visual vis-
       ual] [-window]  [-root]	[-delay	 usecs]	 [-fps]	 [-mode	 display-mode]
       [-wireframe]   [-surface]   [-transparent]   [-appearance   appearance]
       [-solid]	[-distance-bands]  [-direction-bands]  [-colors	 color-scheme]
       [-twosided-colors]  [-distance-colors] [-direction-colors] [-depth-col-
       ors] [-view-mode	view-mode] [-walk] [-turn] [-walk-turn]	[-orientation-
       marks]	[-projection-3d	  mode]	 [-perspective-3d]  [-orthographic-3d]
       [-projection-4d mode] [-perspective-4d]	[-orthographic-4d]  [-speed-wx
       float] [-speed-wy float]	[-speed-wz float] [-speed-xy float] [-speed-xz
       float] [-speed-yz float]	[-walk-direction float]	[-walk-speed float]

DESCRIPTION
       The projectiveplane 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  em-
       bedding	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  differ-
       ent colors, i.e., different 4d depths.

       The  real  projective  plane is a non-orientable	surface.  To make this
       apparent, the two-sided color mode can be used.	Alternatively,	orien-
       tation  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).

       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 sin-
       gled 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 cir-
       cle.  Points on the line	at infinity are	also used to model  directions
       in projective geometry.	The origin can be visualized in	different man-
       ners.  When using distance colors, the origin is	the point that is dis-
       played  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	orien-
       tation  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 ma-
       genta.  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.

       Note that when the projective plane is displayed	with bands, the	orien-
       tation 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 remain-
       ing bands.  This	choice,	however, also implies that the band at	infin-
       ity  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.

       The  program projects the 4d projective plane to	3d using either	a per-
       spective	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 pro-
       jections	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.

       The projected projective	plane can then be projected to the screen  ei-
       ther  perspectively or orthographically.	 When using the	walking	modes,
       perspective projection to the screen will be used.

       There are three display modes for the  projective  plane:  mesh	(wire-
       frame), solid, or transparent.  Furthermore, the	appearance of the pro-
       jective 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 in-
       creasing	distances from the origin, or  direction  bands,  i.e.,	 bands
       that lie	at increasing angles with respect to the origin.

       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.

       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-ori-
       entable.	 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 differ-
       ent  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	direc-
       tions 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.

       The  rotation speed for each of the six planes around which the projec-
       tive plane rotates can be chosen.  For the walk-and-turn	mode, only the
       rotation	 speeds	around the true	4d planes are used (the	xy, xz,	and yz
       planes).

       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 projec-
       tive 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.

       This program is somewhat	inspired by Thomas Banchoff's book "Beyond the
       Third Dimension:	Geometry, Computer Graphics, and  Higher  Dimensions",
       Scientific American Library, 1990.

OPTIONS
       projectiveplane accepts the following options:

       -window Draw on a newly-created window.	This is	the default.

       -root   Draw on the root	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 four options are mutually exclusive.  They	determine  how
       the projective plane is displayed.

       -mode random
	       Display	the  projective	 plane	in  a random display mode (de-
	       fault).

       -mode wireframe (Shortcut: -wireframe)
	       Display the projective plane as a wireframe mesh.

       -mode surface (Shortcut:	-surface)
	       Display the projective plane as a solid surface.

       -mode transparent (Shortcut: -transparent)
	       Display the projective plane as a transparent surface.

       The following three options are mutually	exclusive.  They determine the
       appearance of the projective plane.

       -appearance random
	       Display	the  projective	 plane	with  a	random appearance (de-
	       fault).

       -appearance solid (Shortcut: -solid)
	       Display the projective plane as a solid object.

       -appearance distance-bands (Shortcut: -distance-bands)
	       Display the projective plane as see-through bands that  lie  at
	       increasing distances from the origin.

       -appearance direction-bands (Shortcut: -direction-bands)
	       Display	the  projective	plane as see-through bands that	lie at
	       increasing angles with respect to the origin.

       The following four options are mutually exclusive.  They	determine  how
       to color	the projective plane.

       -colors random
	       Display	the  projective	 plane with a random color scheme (de-
	       fault).

       -colors twosided	(Shortcut: -twosided-colors)
	       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  pro-
	       jective	plane meet, i.e., where	the orientation	of the projec-
	       tive plane reverses.

       -colors distance	(Shortcut: -distance-colors)
	       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	infin-
	       ity  is	displayed in magenta.  If the projective plane is dis-
	       played as distance bands, each band will	be  displayed  with  a
	       different color.

       -colors direction (Shortcut: -direction-colors)
	       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 projec-
	       tive plane is displayed as direction bands, each	band  will  be
	       displayed with a	different color.

       -colors depth (Shortcut:	-depth)
	       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.

       The  following four options are mutually	exclusive.  They determine how
       to view the projective plane.

       -view-mode random
	       View the	projective plane in a random view mode (default).

       -view-mode turn (Shortcut: -turn)
	       View the	projective plane while it turns	in 4d.

       -view-mode walk (Shortcut: -walk)
	       View the	projective plane as if walking on its surface.

       -view-mode walk-turn (Shortcut: -walk-turn)
	       View the	projective plane as if walking on its surface.	 Addi-
	       tionally,  the projective plane turns around the	true 4d	planes
	       (the xy,	xz, and	yz planes).

       The following options determine whether orientation marks are shown  on
       the projective plane.

       -orientation-marks
	       Display orientation marks on the	projective plane.

       -no-orientation-marks
	       Don't  display  orientation  marks on the projective plane (de-
	       fault).

       The following three options are mutually	exclusive.  They determine how
       the projective plane is projected from 3d to 2d (i.e., to the screen).

       -projection-3d random
	       Project	the projective plane from 3d to	2d using a random pro-
	       jection mode (default).

       -projection-3d perspective (Shortcut: -perspective-3d)
	       Project the projective plane from 3d to 2d using	a  perspective
	       projection.

       -projection-3d orthographic (Shortcut: -orthographic-3d)
	       Project	the  projective	 plane	from  3d to 2d using an	ortho-
	       graphic projection.

       The following three options are mutually	exclusive.  They determine how
       the projective plane is projected from 4d to 3d.

       -projection-4d random
	       Project	the projective plane from 4d to	3d using a random pro-
	       jection mode (default).

       -projection-4d perspective (Shortcut: -perspective-4d)
	       Project the projective plane from 4d to 3d using	a  perspective
	       projection.

       -projection-4d orthographic (Shortcut: -orthographic-4d)
	       Project	the  projective	 plane	from  4d to 3d using an	ortho-
	       graphic projection.

       The following six options determine the rotation	speed of  the  projec-
       tive  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  visualiza-
       tion.   Therefore,  in  walk-and-turn mode the speeds you have selected
       are divided by 5	internally.

       -speed-wx float
	       Rotation	speed around the wx plane (default: 1.1).

       -speed-wy float
	       Rotation	speed around the wy plane (default: 1.3).

       -speed-wz float
	       Rotation	speed around the wz plane (default: 1.5).

       -speed-xy float
	       Rotation	speed around the xy plane (default: 1.7).

       -speed-xz float
	       Rotation	speed around the xz plane (default: 1.9).

       -speed-yz float
	       Rotation	speed around the yz plane (default: 2.1).

       The following two options determine the walking speed and direction.

       -walk-direction float
	       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 dis-
	       tance 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.

       -walk-speed float
	       The walking speed is measured in	percent	of some	sensible maxi-
	       mum speed (default: 20.0).

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.

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.

SEE ALSO
       X(1), xscreensaver(1)

COPYRIGHT
       Copyright  (C)  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  no-
       tice 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.

AUTHOR
       Carsten Steger <carsten@mirsanmir.org>, 03-oct-2014.

X Version 11		      5.36 (10-Oct-2016)	    projectiveplane(6)

NAME | SYNOPSIS | DESCRIPTION | OPTIONS | INTERACTION | ENVIRONMENT | SEE ALSO | COPYRIGHT | AUTHOR

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