1sphereeversion(6x) XScreenSaver manual sphereeversion(6x)
2
6 sphereeversion - Displays a sphere eversion.
7
9 sphereeversion [--display host:display.screen] [--install] [--visual
10 visual] [--window] [--root] [--window-id number] [--delay usecs]
11 [--fps] [--eversion-method method] [--analytic] [--corrugations]
12 [--mode display-mode] [--surface] [--transparent] [--appearance appear‐
13 ance] [--solid] [--parallel-bands] [--meridian-bands] [--graticule
14 mode] [--colors color-scheme] [--twosided-colors] [--parallel-colors]
15 [--meridian-colors] [--earth-colors] [--deformation-speed float]
16 [--projection mode] [--perspective] [--orthographic] [--surface-order
17 order] [--lunes-1] [--lunes-2] [--lunes-4] [--lunes-8] [--hemi‐
18 spheres-1] [--hemispheres-2] [--speed-x float] [--speed-y float]
19 [--speed-z float]
20
22 The sphereeversion program shows a sphere eversion, i.e., a smooth de‐
23 formation (homotopy) that turns a sphere inside out. During the ever‐
24 sion, the deformed sphere is allowed to intersect itself transversally.
25 However, no creases or pinch points are allowed to occur.
26 The sphere can be deformed with two eversion methods: analytic or cor‐
28 rugations. The analytic sphere eversion method is described in the fol‐
29 lowing paper: Adam Bednorz, Witold Bednorz: "Analytic sphere eversion
30 using ruled surfaces", Differential Geometry and its Applications
31 64:59-79, 2019. The corrugations sphere eversion method is described in
32 the video "Outside In" by the Geometry Center (Bill Thurston, Silvio
33 Levy, Delle Maxwell, Tamara Munzner, Nathaniel Thurston, David Ben-Zvi,
34 Matt Headrick, et al.), 1994, and the accompanying booklet: Silvio
35 Levy: "Making Waves - A Guide to the Ideas Behind Outside In", A K Pe‐
36 ters, Wellesley, MA, 1995. See also the section "Brief Description of
37 the Corrugations Sphere Eversion Method" below.
38 The deformed sphere can be projected to the screen either perspectively
40 or orthographically.
41 There are three display modes for the sphere: solid, transparent, or
43 random. If random mode is selected, the mode is changed each time an
44 eversion has been completed.
45 The appearance of the sphere can be as a solid object, as a set of see-
47 through bands, or random. The bands can be parallel bands or meridian
48 bands, i.e., bands that run along the parallels (lines of latitude) or
49 bands that run along the meridians (lines of longitude) of the sphere.
50 If random mode is selected, the appearance is changed each time an ev‐
51 ersion has been completed.
52 For the analytic sphere eversion, it is also possible to display a
54 graticule (i.e., a coordinate grid consisting of parallel and meridian
55 lines) on top of the surface. The graticule mode can be set to on,
56 off, or random. If random mode is selected, the graticule mode is
57 changed each time an eversion has been completed.
58 The colors with with the sphere is drawn can be set to two-sided, par‐
60 allel, meridian, earth, or random. In two-sided mode, the sphere is
61 drawn with red on one side and green on the other side (analytic ever‐
62 sion) or with gold on one side and purple on the other side (corruga‐
63 tions eversion). In parallel mode, the sphere is displayed with colors
64 that run from blue to white to orange on one side of the surface and
65 from magenta to black to green on the other side. The colors are
66 aligned with the parallels of the sphere in this mode. In meridian
67 mode, the the sphere is displayed with colors that run from blue to
68 white to orange to black and back to blue on one side of the surface
69 and from magenta to white to green to black and back to magenta on the
70 other side. The colors are aligned with the meridians of the sphere in
71 this mode. In earth mode, the sphere is drawn with a texture of earth
72 by day on one side and with a texture of earth by night on the other
73 side. Initially, the earth by day is on the outside and the earth by
74 night on the inside. After the first eversion, the earth by night will
75 be on the outside. All points of the earth on the inside and outside
76 are at the same positions on the sphere. Since an eversion transforms
77 the sphere into its inverse, the earth by night will appear with all
78 continents mirror reversed. If random mode is selected, the color
79 scheme is changed each time an eversion has been completed.
80 By default, the sphere is rotated to a new viewing position each time
82 an eversion has been completed. In addition, it is possible to rotate
83 the sphere while it is deforming. The rotation speed for each of the
84 three coordinate axes around which the sphere rotates can be chosen ar‐
85 bitrarily. For best effects, however, it is suggested to rotate only
86 around the z axis while the sphere is deforming.
87 For the analytic sphere eversion, it is possible to define a surface
89 order of the sphere eversion as random or as a value between 2 and 5.
90 This determines the the complexity of the deformation. For higher sur‐
91 face orders, some z-fighting might occur around the central stage of
92 the eversion, which might lead to some irregular flickering of the dis‐
93 played surface if it is displayed as a solid object. For odd surface
94 orders, z-fighting will occur very close to the central stage of the
95 eversion since the deformed sphere is a doubly covered Boy surface (for
96 surface order 3) or a doubly covered generalized Boy surface (for sur‐
97 face order 5) in this case. If you find this distracting, you should
98 set the surface order to 2. If a random surface order is selected, the
99 surface order is changed each time an eversion has been completed.
100
102 The corrugations sphere eversion method is described in detail in the
103 video and booklet mentioned above. Briefly, the method works as fol‐
104 lows: Imagine the sphere cut into eight spherical lunes (spherical
105 biangles). Now imagine each lune to be a belt. The ends of the belt
106 (which correspond to the north and south poles of the sphere) are
107 pushed past each other. This creates a loop in the belt. If the belt
108 were straightened out, it would contain a 360 degree rotation. This
109 rotation can be removed by rotating each end of the belt by 180 de‐
110 grees. Finally, the belt is pushed to the opposite side of the sphere,
111 which causes the side of the belt that initially was inside the sphere
112 to appear on the outside.
113 The method described so far only works for a single lune (belt) and not
115 for the entire sphere. To make it work for the entire sphere, corruga‐
116 tions (i.e., waves) must be added to the sphere. This happens in the
117 first phase of the eversion. Then, the method described above is ap‐
118 plied to the eight lunes. Finally, the corrugations are removed to ob‐
119 tain the everted sphere.
120 To see the eversion for a single lune, the option --lunes-1 can be
122 used. Using this option, the eversion, as described above, is easier
123 to understand. It is also possible to display two lunes using
124 --lunes-2 and four lunes using --lunes-4. Using fewer than eight lunes
125 reduces the visual complexity of the eversion and may help to under‐
126 stand the method.
127 Furthermore, it is possible to display only one hemisphere using the
129 option --hemispheres-1. This allows to see what is happening in the
130 center of the sphere during the eversion. Note that the north and
131 south half of the sphere move in a symmetric fashion during the ever‐
132 sion. Hence, the eversion is actually composed of 16 semi-lunes
133 (spherical triangles from the equator to the poles) that all deform in
134 the same manner. By specifying --lunes-1 --hemispheres-1, the deforma‐
135 tion of one semi-lune can be observed.
136 Note that the options described above are only intended for educational
138 purposes. They are not used if none of them are explicitly specified.
139
142 sphereeversion accepts the following options:
143 --window
145 Draw on a newly-created window. This is the default.
146 --root Draw on the root window.
148 --window-id number
150 Draw on the specified window.
151 --install
153 Install a private colormap for the window.
154 --visual visual
156 Specify which visual to use. Legal values are the name of a
157 visual class, or the id number (decimal or hex) of a specific
158 visual.
159 --delay microseconds
161 How much of a delay should be introduced between steps of the
162 animation. Default 10000, or 1/100th second.
163 --fps Display the current frame rate, CPU load, and polygon count.
165 The following three options are mutually exclusive. They determine
167 which sphere eversion method is used.
168 --eversion-method random
170 Use a random sphere eversion method (default).
171 --eversion-method analytic (Shortcut: --analytic)
173 Use the analytic sphere eversion method.
174 --eversion-method corrugations (Shortcut: --corrugations)
176 Use the corrugations sphere eversion method.
177 The following three options are mutually exclusive. They determine how
179 the deformed sphere is displayed.
180 --mode random
182 Display the sphere in a random display mode (default).
183 --mode surface (Shortcut: --surface)
185 Display the sphere as a solid surface.
186 --mode transparent (Shortcut: --transparent)
188 Display the sphere as a transparent surface.
189 The following four options are mutually exclusive. They determine the
191 appearance of the deformed sphere.
192 --appearance random
194 Display the sphere with a random appearance (default).
195 --appearance solid (Shortcut: --solid)
197 Display the sphere as a solid object.
198 --appearance parallel-bands (Shortcut: --parallel-bands)
200 Display the sphere as see-through bands that lie along the par‐
201 allels of the sphere.
202 --appearance meridian-bands (Shortcut: --meridian-bands)
204 Display the sphere as see-through bands that lie along the
205 meridians of the sphere.
206 The following three options are mutually exclusive. They determine
208 whether a graticule is displayed on top of the sphere. These options
209 only have an effect if the analytic sphere eversion method is selected.
210 --graticule random
212 Randomly choose whether to display a graticule (default).
213 --graticule on
215 Display a graticule.
216 --graticule off
218 Do not display a graticule.
219 The following five options are mutually exclusive. They determine how
221 to color the deformed sphere.
222 --colors random
224 Display the sphere with a random color scheme (default).
225 --colors twosided (Shortcut: --twosided-colors)
227 Display the sphere with two colors: red on one side and green
228 on the other side (analytic eversion) or gold on one side and
229 purple on the other side (corrugations eversion).
230 --colors parallel (Shortcut: --parallel-colors)
232 Display the sphere with colors that run from from blue to white
233 to orange on one side of the surface and from magenta to black
234 to green on the other side. The colors are aligned with the
235 parallels of the sphere. If the sphere is displayed as paral‐
236 lel bands, each band will be displayed with a different color.
237 --colors meridian (Shortcut: --meridian-colors)
239 Display the sphere with colors that run from from blue to white
240 to orange to black and back to blue on one side of the surface
241 and from magenta to white to green to black and back to magenta
242 on the other side. The colors are aligned with the meridians
243 of the sphere. If the sphere is displayed as meridian bands,
244 each band will be displayed with a different color.
245 --colors earth (Shortcut: --earth-colors)
247 Display the sphere with a texture of earth by day on one side
248 and with a texture of earth by night on the other side. Ini‐
249 tially, the earth by day is on the outside and the earth by
250 night on the inside. After the first eversion, the earth by
251 night will be on the outside. All points of the earth on the
252 inside and outside are at the same positions on the sphere.
253 Since an eversion transforms the sphere into its inverse, the
254 earth by night will appear with all continents mirror reversed.
255 The following option determines the deformation speed.
257 --deformation-speed float
259 The deformation speed is measured in percent of some sensible
260 maximum speed (default: 10.0).
261 The following three options are mutually exclusive. They determine how
263 the deformed sphere is projected from 3d to 2d (i.e., to the screen).
264 --projection random
266 Project the sphere from 3d to 2d using a random projection mode
267 (default).
268 --projection perspective (Shortcut: --perspective)
270 Project the sphere from 3d to 2d using a perspective projec‐
271 tion.
272 --projection orthographic (Shortcut: --orthographic)
274 Project the sphere from 3d to 2d using an orthographic projec‐
275 tion.
276 The following option determines the order of the surface to be dis‐
278 played. This option only has an effect if the analytic sphere eversion
279 method is selected.
280 --surface-order order
282 The surface order can be set to random or to a value between 2
283 and 5 (default: random). This determines the the complexity of
284 the deformation.
285 The following four options are mutually exclusive. They determine how
287 many lunes of the sphere are displayed. These options only have an ef‐
288 fect if the corrugations sphere eversion method is selected.
289 --lunes-1
291 Display one of the eight lunes that form the sphere.
292 --lunes-2
294 Display two of the eight lunes that form the sphere.
295 --lunes-4
297 Display four of the eight lunes that form the sphere.
298 --lunes-8
300 Display all eight lunes that form the sphere (default).
301 The following two options are mutually exclusive. They determine how
303 many hemispheres of the sphere are displayed. These options only have
304 an effect if the corrugations sphere eversion method is selected.
305 --hemispheres-1
307 Display only one hemisphere of the sphere.
308 --hemispheres-2
310 Display both hemispheres of the sphere (default).
311 The following three options determine the rotation speed of the de‐
313 formed sphere around the three possible axes. The rotation speed is
314 measured in degrees per frame. The speeds should be set to relatively
315 small values, e.g., less than 4 in magnitude.
316 --speed-x float
318 Rotation speed around the x axis (default: 0.0).
319 --speed-y float
321 Rotation speed around the y axis (default: 0.0).
322 --speed-z float
324 Rotation speed around the z axis (default: 0.0).
325
327 If you run this program in standalone mode, you can rotate the deformed
328 sphere by dragging the mouse while pressing the left mouse button.
329 This rotates the sphere in 3d. To examine the deformed sphere at your
330 leisure, it is best to set all speeds to 0. Otherwise, the deformed
331 sphere will rotate while the left mouse button is not pressed.
332
334 DISPLAY to get the default host and display number.
335 XENVIRONMENT
337 to get the name of a resource file that overrides the global
338 resources stored in the RESOURCE_MANAGER property.
339 XSCREENSAVER_WINDOW
341 The window ID to use with --root.
342
344 X(1), xscreensaver(1),
345 https://profs.etsmtl.ca/mmcguffin/eversion/,
346 http://www.geom.uiuc.edu/docs/outreach/oi/software.html
347
349 Copyright © 2020 by Carsten Steger. Permission to use, copy, modify,
350 distribute, and sell this software and its documentation for any pur‐
351 pose is hereby granted without fee, provided that the above copyright
352 notice appear in all copies and that both that copyright notice and
353 this permission notice appear in supporting documentation. No repre‐
354 sentations are made about the suitability of this software for any pur‐
355 pose. It is provided "as is" without express or implied warranty.
356 Parts of the code in this program are based on the program "sphereEver‐
358 sion 0.4" by Michael J. McGuffin, which, in turn, is based on the pro‐
359 gram "Evert" developed by Nathaniel Thurston at the Geometry Center.
360 The modified code is used with permission.
361
363 Carsten Steger <carsten@mirsanmir.org>, 01-jun-2020.
364X Version 11 6.08-1.fc39 (17-Oct-2023) sphereeversion(6x)