1sphereeversion(6x) XScreenSaver manual sphereeversion(6x)
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6 sphereeversion - Displays a sphere eversion.
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9 sphereeversion [-display host:display.screen] [-install] [-visual vis‐
10 ual] [-window] [-root] [-delay usecs] [-fps] [-eversion-method method]
11 [-analytic] [-corrugations] [-mode display-mode] [-surface] [-transpar‐
12 ent] [-appearance appearance] [-solid] [-parallel-bands] [-meridian-
13 bands] [-graticule mode] [-colors color-scheme] [-twosided-colors]
14 [-parallel-colors] [-meridian-colors] [-earth-colors] [-deformation-
15 speed float] [-projection mode] [-perspective] [-orthographic] [-sur‐
16 face-order order] [-lunes-1] [-lunes-2] [-lunes-4] [-lunes-8] [-hemi‐
17 spheres-1] [-hemispheres-2] [-speed-x float] [-speed-y float] [-speed-z
18 float]
19
21 The sphereeversion program shows a sphere eversion, i.e., a smooth de‐
22 formation (homotopy) that turns a sphere inside out. During the ever‐
23 sion, the deformed sphere is allowed to intersect itself transversally.
24 However, no creases or pinch points are allowed to occur.
25 The sphere can be deformed with two eversion methods: analytic or cor‐
27 rugations. The analytic sphere eversion method is described in the fol‐
28 lowing paper: Adam Bednorz, Witold Bednorz: "Analytic sphere eversion
29 using ruled surfaces", Differential Geometry and its Applications
30 64:59-79, 2019. The corrugations sphere eversion method is described in
31 the video "Outside In" by the Geometry Center (Bill Thurston, Silvio
32 Levy, Delle Maxwell, Tamara Munzner, Nathaniel Thurston, David Ben-Zvi,
33 Matt Headrick, et al.), 1994, and the accompanying booklet: Silvio
34 Levy: "Making Waves - A Guide to the Ideas Behind Outside In", A K Pe‐
35 ters, Wellesley, MA, 1995. See also the section "Brief Description of
36 the Corrugations Sphere Eversion Method" below.
37 The deformed sphere can be projected to the screen either perspectively
39 or orthographically.
40 There are three display modes for the sphere: solid, transparent, or
42 random. If random mode is selected, the mode is changed each time an
43 eversion has been completed.
44 The appearance of the sphere can be as a solid object, as a set of see-
46 through bands, or random. The bands can be parallel bands or meridian
47 bands, i.e., bands that run along the parallels (lines of latitude) or
48 bands that run along the meridians (lines of longitude) of the sphere.
49 If random mode is selected, the appearance is changed each time an ev‐
50 ersion has been completed.
51 For the analytic sphere eversion, it is also possible to display a
53 graticule (i.e., a coordinate grid consisting of parallel and meridian
54 lines) on top of the surface. The graticule mode can be set to on,
55 off, or random. If random mode is selected, the graticule mode is
56 changed each time an eversion has been completed.
57 The colors with with the sphere is drawn can be set to two-sided, par‐
59 allel, meridian, earth, or random. In two-sided mode, the sphere is
60 drawn with red on one side and green on the other side (analytic ever‐
61 sion) or with gold on one side and purple on the other side (corruga‐
62 tions eversion). In parallel mode, the sphere is displayed with colors
63 that run from blue to white to orange on one side of the surface and
64 from magenta to black to green on the other side. The colors are
65 aligned with the parallels of the sphere in this mode. In meridian
66 mode, the the sphere is displayed with colors that run from blue to
67 white to orange to black and back to blue on one side of the surface
68 and from magenta to white to green to black and back to magenta on the
69 other side. The colors are aligned with the meridians of the sphere in
70 this mode. In earth mode, the sphere is drawn with a texture of earth
71 by day on one side and with a texture of earth by night on the other
72 side. Initially, the earth by day is on the outside and the earth by
73 night on the inside. After the first eversion, the earth by night will
74 be on the outside. All points of the earth on the inside and outside
75 are at the same positions on the sphere. Since an eversion transforms
76 the sphere into its inverse, the earth by night will appear with all
77 continents mirror reversed. If random mode is selected, the color
78 scheme is changed each time an eversion has been completed.
79 By default, the sphere is rotated to a new viewing position each time
81 an eversion has been completed. In addition, it is possible to rotate
82 the sphere while it is deforming. The rotation speed for each of the
83 three coordinate axes around which the sphere rotates can be chosen ar‐
84 bitrarily. For best effects, however, it is suggested to rotate only
85 around the z axis while the sphere is deforming.
86 For the analytic sphere eversion, it is possible to define a surface
88 order of the sphere eversion as random or as a value between 2 and 5.
89 This determines the the complexity of the deformation. For higher sur‐
90 face orders, some z-fighting might occur around the central stage of
91 the eversion, which might lead to some irregular flickering of the dis‐
92 played surface if it is displayed as a solid object. For odd surface
93 orders, z-fighting will occur very close to the central stage of the
94 eversion since the deformed sphere is a doubly covered Boy surface (for
95 surface order 3) or a doubly covered generalized Boy surface (for sur‐
96 face order 5) in this case. If you find this distracting, you should
97 set the surface order to 2. If a random surface order is selected, the
98 surface order is changed each time an eversion has been completed.
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101 The corrugations sphere eversion method is described in detail in the
102 video and booklet mentioned above. Briefly, the method works as fol‐
103 lows: Imagine the sphere cut into eight spherical lunes (spherical
104 biangles). Now imagine each lune to be a belt. The ends of the belt
105 (which correspond to the north and south poles of the sphere) are
106 pushed past each other. This creates a loop in the belt. If the belt
107 were straightened out, it would contain a 360 degree rotation. This
108 rotation can be removed by rotating each end of the belt by 180 de‐
109 grees. Finally, the belt is pushed to the opposite side of the sphere,
110 which causes the side of the belt that initially was inside the sphere
111 to appear on the outside.
112 The method described so far only works for a single lune (belt) and not
114 for the entire sphere. To make it work for the entire sphere, corruga‐
115 tions (i.e., waves) must be added to the sphere. This happens in the
116 first phase of the eversion. Then, the method described above is ap‐
117 plied to the eight lunes. Finally, the corrugations are removed to ob‐
118 tain the everted sphere.
119 To see the eversion for a single lune, the option -lunes-1 can be used.
121 Using this option, the eversion, as described above, is easier to un‐
122 derstand. It is also possible to display two lunes using -lunes-2 and
123 four lunes using -lunes-4. Using fewer than eight lunes reduces the
124 visual complexity of the eversion and may help to understand the
125 method.
126 Furthermore, it is possible to display only one hemisphere using the
128 option -hemispheres-1. This allows to see what is happening in the
129 center of the sphere during the eversion. Note that the north and
130 south half of the sphere move in a symmetric fashion during the ever‐
131 sion. Hence, the eversion is actually composed of 16 semi-lunes
132 (spherical triangles from the equator to the poles) that all deform in
133 the same manner. By specifying -lunes-1 -hemispheres-1, the deforma‐
134 tion of one semi-lune can be observed.
135 Note that the options described above are only intended for educational
137 purposes. They are not used if none of them are explicitly specified.
138
141 sphereeversion accepts the following options:
142 -window Draw on a newly-created window. This is the default.
144 -root Draw on the root window.
146 -install
148 Install a private colormap for the window.
149 -visual visual
151 Specify which visual to use. Legal values are the name of a
152 visual class, or the id number (decimal or hex) of a specific
153 visual.
154 -delay microseconds
156 How much of a delay should be introduced between steps of the
157 animation. Default 10000, or 1/100th second.
158 -fps Display the current frame rate, CPU load, and polygon count.
160 The following three options are mutually exclusive. They determine
162 which sphere eversion method is used.
163 -eversion-method random
165 Use a random sphere eversion method (default).
166 -eversion-method analytic (Shortcut: -analytic)
168 Use the analytic sphere eversion method.
169 -eversion-method corrugations (Shortcut: -corrugations)
171 Use the corrugations sphere eversion method.
172 The following three options are mutually exclusive. They determine how
174 the deformed sphere is displayed.
175 -mode random
177 Display the sphere in a random display mode (default).
178 -mode surface (Shortcut: -surface)
180 Display the sphere as a solid surface.
181 -mode transparent (Shortcut: -transparent)
183 Display the sphere as a transparent surface.
184 The following four options are mutually exclusive. They determine the
186 appearance of the deformed sphere.
187 -appearance random
189 Display the sphere with a random appearance (default).
190 -appearance solid (Shortcut: -solid)
192 Display the sphere as a solid object.
193 -appearance parallel-bands (Shortcut: -parallel-bands)
195 Display the sphere as see-through bands that lie along the par‐
196 allels of the sphere.
197 -appearance meridian-bands (Shortcut: -meridian-bands)
199 Display the sphere as see-through bands that lie along the
200 meridians of the sphere.
201 The following three options are mutually exclusive. They determine
203 whether a graticule is displayed on top of the sphere. These options
204 only have an effect if the analytic sphere eversion method is selected.
205 -graticule random
207 Randomly choose whether to display a graticule (default).
208 -graticule on
210 Display a graticule.
211 -graticule off
213 Do not display a graticule.
214 The following five options are mutually exclusive. They determine how
216 to color the deformed sphere.
217 -colors random
219 Display the sphere with a random color scheme (default).
220 -colors twosided (Shortcut: -twosided-colors)
222 Display the sphere with two colors: red on one side and green
223 on the other side (analytic eversion) or gold on one side and
224 purple on the other side (corrugations eversion).
225 -colors parallel (Shortcut: -parallel-colors)
227 Display the sphere with colors that run from from blue to white
228 to orange on one side of the surface and from magenta to black
229 to green on the other side. The colors are aligned with the
230 parallels of the sphere. If the sphere is displayed as paral‐
231 lel bands, each band will be displayed with a different color.
232 -colors meridian (Shortcut: -meridian-colors)
234 Display the sphere with colors that run from from blue to white
235 to orange to black and back to blue on one side of the surface
236 and from magenta to white to green to black and back to magenta
237 on the other side. The colors are aligned with the meridians
238 of the sphere. If the sphere is displayed as meridian bands,
239 each band will be displayed with a different color.
240 -colors earth (Shortcut: -earth-colors)
242 Display the sphere with a texture of earth by day on one side
243 and with a texture of earth by night on the other side. Ini‐
244 tially, the earth by day is on the outside and the earth by
245 night on the inside. After the first eversion, the earth by
246 night will be on the outside. All points of the earth on the
247 inside and outside are at the same positions on the sphere.
248 Since an eversion transforms the sphere into its inverse, the
249 earth by night will appear with all continents mirror reversed.
250 The following option determines the deformation speed.
252 -deformation-speed float
254 The deformation speed is measured in percent of some sensible
255 maximum speed (default: 10.0).
256 The following three options are mutually exclusive. They determine how
258 the deformed sphere is projected from 3d to 2d (i.e., to the screen).
259 -projection random
261 Project the sphere from 3d to 2d using a random projection mode
262 (default).
263 -projection perspective (Shortcut: -perspective)
265 Project the sphere from 3d to 2d using a perspective projec‐
266 tion.
267 -projection orthographic (Shortcut: -orthographic)
269 Project the sphere from 3d to 2d using an orthographic projec‐
270 tion.
271 The following option determines the order of the surface to be dis‐
273 played. This option only has an effect if the analytic sphere eversion
274 method is selected.
275 -surface-order order
277 The surface order can be set to random or to a value between 2
278 and 5 (default: random). This determines the the complexity of
279 the deformation.
280 The following four options are mutually exclusive. They determine how
282 many lunes of the sphere are displayed. These options only have an ef‐
283 fect if the corrugations sphere eversion method is selected.
284 -lunes-1
286 Display one of the eight lunes that form the sphere.
287 -lunes-2
289 Display two of the eight lunes that form the sphere.
290 -lunes-4
292 Display four of the eight lunes that form the sphere.
293 -lunes-8
295 Display all eight lunes that form the sphere (default).
296 The following two options are mutually exclusive. They determine how
298 many hemispheres of the sphere are displayed. These options only have
299 an effect if the corrugations sphere eversion method is selected.
300 -hemispheres-1
302 Display only one hemisphere of the sphere.
303 -hemispheres-2
305 Display both hemispheres of the sphere (default).
306 The following three options determine the rotation speed of the de‐
308 formed sphere around the three possible axes. The rotation speed is
309 measured in degrees per frame. The speeds should be set to relatively
310 small values, e.g., less than 4 in magnitude.
311 -speed-x float
313 Rotation speed around the x axis (default: 0.0).
314 -speed-y float
316 Rotation speed around the y axis (default: 0.0).
317 -speed-z float
319 Rotation speed around the z axis (default: 0.0).
320
322 If you run this program in standalone mode, you can rotate the deformed
323 sphere by dragging the mouse while pressing the left mouse button.
324 This rotates the sphere in 3d. To examine the deformed sphere at your
325 leisure, it is best to set all speeds to 0. Otherwise, the deformed
326 sphere will rotate while the left mouse button is not pressed.
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329 DISPLAY to get the default host and display number.
330 XENVIRONMENT
332 to get the name of a resource file that overrides the global
333 resources stored in the RESOURCE_MANAGER property.
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336 X(1), xscreensaver(1),
337 https://profs.etsmtl.ca/mmcguffin/eversion/,
338 http://www.geom.uiuc.edu/docs/outreach/oi/software.html
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341 Copyright © 2020 by Carsten Steger. Permission to use, copy, modify,
342 distribute, and sell this software and its documentation for any pur‐
343 pose is hereby granted without fee, provided that the above copyright
344 notice appear in all copies and that both that copyright notice and
345 this permission notice appear in supporting documentation. No repre‐
346 sentations are made about the suitability of this software for any pur‐
347 pose. It is provided "as is" without express or implied warranty.
348 Parts of the code in this program are based on the program "sphereEver‐
350 sion 0.4" by Michael J. McGuffin, which, in turn, is based on the pro‐
351 gram "Evert" developed by Nathaniel Thurston at the Geometry Center.
352 The modified code is used with permission.
353
355 Carsten Steger <carsten@mirsanmir.org>, 01-jun-2020.
356X Version 11 6.04-1.fc36 (06-Jun-2022) sphereeversion(6x)