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-.TH XScreenSaver 1 "" "X Version 11"
-.SH NAME
-klein \- Draws a 4d Klein bottle.
-.SH SYNOPSIS
-.B klein
-[\-display \fIhost:display.screen\fP]
-[\-install]
-[\-visual \fIvisual\fP]
-[\-window]
-[\-root]
-[\-delay \fIusecs\fP]
-[\-fps]
-[\-klein-bottle \fIbottle-name\fP]
-[-figure-8]
-[-pinched-torus]
-[-lawson]
-[\-mode \fIdisplay-mode\fP]
-[\-wireframe]
-[\-surface]
-[\-transparent]
-[\-appearance \fIappearance\fP]
-[\-solid]
-[\-bands]
-[\-colors \fIcolor-scheme\fP]
-[\-onesided]
-[\-twosided]
-[\-rainbow]
-[\-depth]
-[\-change-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 \fIklein\fP program shows three different Klein bottles in 4d: the
-figure-8 Klein bottle, the pinched torus Klein bottle, or the Lawson
-Klein bottle.  You can walk on the Klein bottle, see it turn in 4d, or
-walk on it while it turns in 4d.  The figure-8 Klein bottle is well
-known in its 3d form.  The 4d form used in this program is an
-extension of the 3d form to 4d that does not intersect itself in 4d
-(which can be seen in the depth colors mode when using static colors).
-The pinched torus Klein bottle also does not intersect itself in 4d
-(which can be seen in the depth colors mode when using static colors).
-The Lawson Klein bottle, on the other hand, does intersect itself in
-4d.  Its primary use is that it has a nice appearance for walking and
-for turning in 3d.
-.PP
-The Klein bottle 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 Klein bottle.  While walking on the Klein bottle, you will
-notice that the orientation of the curling arrows changes (which it
-must because the Klein bottle is non-orientable).
-.PP
-The program projects the 4d Klein bottle to 3d using either a
-perspective or an orthographic projection.  Which of the two
-alternatives looks more appealing depends on the viewing mode and the
-Klein bottle.  For example, the Lawson Klein bottle looks nicest when
-projected perspectively.  The figure-8 Klein bottle, on the other
-hand, looks nicer while walking when projected orthographically from
-4d.  For the pinched torus Klein bottle, both projection modes give
-equally acceptable projections.
-.PP
-The projected Klein bottle can then be projected to the screen either
-perspectively or orthographically.  When using the walking modes,
-perspective projection to the screen should be used.
-.PP
-There are three display modes for the Klein bottle: mesh (wireframe),
-solid, or transparent.  Furthermore, the appearance of the Klein
-bottle can be as a solid object or as a set of see-through bands.
-Finally, the colors with with the Klein bottle is drawn can be set to
-one-sided, two-sided, rainbow, or depth.  In one-sided mode, the Klein
-bottle is drawn with the same color on both "sides."  In two-sided
-mode (using static colors), the Klein bottle is drawn with red on one
-"side" and green on the "other side."  Of course, the Klein bottle
-only has one side, so the color jumps from red to green along a curve
-on the surface of the Klein bottle.  This mode enables you to see that
-the Klein bottle is non-orientable.  If changing colors are used in
-two-sided mode, changing complementary colors are used on the
-respective "sides."  The rainbow color mode (using static colors)
-draws the Klein bottle with a color wheel of fully saturated rainbow
-colors.  If changing colors are used, the color wheel's colors change
-dynamically.  The rainbow color mode gives a very nice effect when
-combined with the see-through bands mode or with the orientation
-markers drawn.  The depth color mode draws the Klein bottle with
-colors that are chosen according to the 4d "depth" of the points.  If
-static colors are used, this mode enables you to see that the figure-8
-and pinched torus Klein bottles do not intersect themselves in 4d,
-while the Lawson Klein bottle does intersect itself.
-.PP
-The rotation speed for each of the six planes around which the Klein
-bottle 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).
-.PP
-Furthermore, in the walking modes the walking direction in the 2d base
-square of the Klein bottle and the walking speed can be chosen.
-.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 klein
-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
-which Klein bottle is displayed.
-.TP 8
-.B \-klein-bottle random
-Display a random Klein bottle (default).
-.TP 8
-.B \-klein-bottle figure-8 \fP(Shortcut: \fB\-figure-8\fP)
-Display the figure-8 Klein bottle.
-.TP 8
-.B \-klein-bottle pinched-torus \fP(Shortcut: \fB\-pinched-torus\fP)
-Display the pinched torus Klein bottle.
-.TP 8
-.B \-klein-bottle lawson \fP(Shortcut: \fB\-lawson\fP)
-Display the Lawson Klein bottle.
-.PP
-The following four options are mutually exclusive.  They determine
-how the Klein bottle is displayed.
-.TP 8
-.B \-mode random
-Display the Klein bottle in a random display mode (default).
-.TP 8
-.B \-mode wireframe \fP(Shortcut: \fB\-wireframe\fP)
-Display the Klein bottle as a wireframe mesh.
-.TP 8
-.B \-mode surface \fP(Shortcut: \fB\-surface\fP)
-Display the Klein bottle as a solid surface.
-.TP 8
-.B \-mode transparent \fP(Shortcut: \fB\-transparent\fP)
-Display the Klein bottle as a transparent surface.
-.PP
-The following three options are mutually exclusive.  They determine the
-appearance of the Klein bottle.
-.TP 8
-.B \-appearance random
-Display the Klein bottle with a random appearance (default).
-.TP 8
-.B \-appearance solid \fP(Shortcut: \fB\-solid\fP)
-Display the Klein bottle as a solid object.
-.TP 8
-.B \-appearance bands \fP(Shortcut: \fB\-bands\fP)
-Display the Klein bottle as see-through bands.
-.PP
-The following five options are mutually exclusive.  They determine
-how to color the Klein bottle.
-.TP 8
-.B \-colors random
-Display the Klein bottle with a random color scheme (default).
-.TP 8
-.B \-colors one-sided \fP(Shortcut: \fB\-onesided\fP)
-Display the Klein bottle with a single color.
-.TP 8
-.B \-colors two-sided \fP(Shortcut: \fB\-twosided\fP)
-Display the Klein bottle with two colors: one color one "side" and the
-complementary color on the "other side."  For static colors, the
-colors are red and green.
-.TP 8
-.B \-colors rainbow \fP(Shortcut: \fB\-rainbow\fP)
-Display the Klein bottle with fully saturated rainbow colors.  If the
-Klein bottle is displayed as see-through bands, each band will be
-displayed with a different color.
-.TP 8
-.B \-colors depth \fP(Shortcut: \fB\-depth\fP)
-Display the Klein bottle with colors chosen depending on the 4d
-"depth" of the points.
-.PP
-The following options determine whether the colors with which the
-Klein bottle is displayed are static or are changing dynamically.
-.TP 8
-.B \-change-colors
-Change the colors with which the Klein bottle is displayed
-dynamically.
-.TP 8
-.B \-no-change-colors
-Use static colors to display the Klein bottle (default).
-.PP
-The following four options are mutually exclusive.  They determine
-how to view the Klein bottle.
-.TP 8
-.B \-view-mode random
-View the Klein bottle in a random view mode (default).
-.TP 8
-.B \-view-mode walk \fP(Shortcut: \fB\-walk\fP)
-View the Klein bottle as if walking on its surface.
-.TP 8
-.B \-view-mode turn \fP(Shortcut: \fB\-turn\fP)
-View the Klein bottle while it turns in 4d.
-.TP 8
-.B \-view-mode walk-turn \fP(Shortcut: \fB\-walk-turn\fP)
-View the Klein bottle as if walking on its surface.  Additionally, the
-Klein bottle turns around the true 4d planes (the xy, xz, and yz
-planes).
-.PP
-The following options determine whether orientation marks are shown on
-the Klein bottle.
-.TP 8
-.B \-orientation-marks
-Display orientation marks on the Klein bottle.
-.TP 8
-.B \-no-orientation-marks
-Don't display orientation marks on the Klein bottle (default).
-.PP
-The following three options are mutually exclusive.  They determine
-how the Klein bottle is projected from 3d to 2d (i.e., to the screen).
-.TP 8
-.B \-projection-3d random
-Project the Klein bottle from 3d to 2d using a random projection mode
-(default).
-.TP 8
-.B \-projection-3d perspective \fP(Shortcut: \fB\-perspective-3d\fP)
-Project the Klein bottle from 3d to 2d using a perspective projection.
-.TP 8
-.B \-projection-3d orthographic \fP(Shortcut: \fB\-orthographic-3d\fP)
-Project the Klein bottle from 3d to 2d using an orthographic
-projection.
-.PP
-The following three options are mutually exclusive.  They determine
-how the Klein bottle is projected from 4d to 3d.
-.TP 8
-.B \-projection-4d random
-Project the Klein bottle from 4d to 3d using a random projection mode
-(default).
-.TP 8
-.B \-projection-4d perspective \fP(Shortcut: \fB\-perspective-4d\fP)
-Project the Klein bottle from 4d to 3d using a perspective projection.
-.TP 8
-.B \-projection-4d orthographic \fP(Shortcut: \fB\-orthographic-4d\fP)
-Project the Klein bottle from 4d to 3d using an orthographic
-projection.
-.PP
-The following six options determine the rotation speed of the Klein
-bottle 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 Klein
-bottle (default: 7.0).
-.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 Klein bottle by dragging the mouse while pressing the left
-mouse button.  This rotates the Klein bottle 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 Klein bottle is rotated in 4D,
-i.e., around the xy, xz, and yz planes.  To examine the Klein bottle
-at your leisure, it is best to set all speeds to 0.  Otherwise, the
-Klein bottle 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 2005-2020 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>, 11-jan-2020.