File:SO(3) 60-90-60.svg
Original file (SVG file, nominally 800 × 1,000 pixels, file size: 13 KB)
Captions
←—— | ||
← | ← | ← |
g | f | g−1 |
Summary
[edit]DescriptionSO(3) 60-90-60.svg |
English: w:Conjugacy in w:SO(3) acting on a 2-sphere shown in w:stereographic projection. The rotation f is a 90° rotation, the rotation g is a 60° rotation. In words, if a point is moved along the black arrow (g−1), then blue arrow (f), and then yellow arrow (g), then the action of g f g−1 (green arrows) is obtained. If one considers f as a one-parametric subgroup, then the picture shows how g (yellow) maps orbits of f (blue) to orbits of g f g−1 (green).
There are two spherical triangles in the lower part of the image that show g(f(g−1(x))) where x is a g’s fixed point and f(g−1(x)) is a g’s fixed point. A gradient-filled shape in the upper part shows the situation where all three actions are not trivial. It is not a “true” quadrilateral: yellow and black sides are bent arcs, not line (i.e. great circle) segments. “60°” inscriptions mean that g is a 60° rotation, not that arcs have π/3 length. Only arrows along the central vertical line have π/3 lengths on the sphere. At the upper left corner there is a depiction of the same sphere in orthographic projection and a schema of the stereographic projection is present. Near side of the sphere corresponds to the lower side of the image and is seen upside-down (the main image shows the sphere from inside, whereas the orthographic projection shows its exterior view). |
Date | |
Source | Own work |
Author | Incnis Mrsi |
Other versions | , a more interesting example with the Spin(3) group. |
Licensing
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Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue |
File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 11:53, 9 October 2016 | 800 × 1,000 (13 KB) | Incnis Mrsi (talk | contribs) | some fixes, tweaks, and additional details on the orthographic inset | |
16:06, 8 October 2016 | 800 × 1,000 (12 KB) | Incnis Mrsi (talk | contribs) | backported numerous improvements from file: Spin_180,60.svg | ||
19:06, 25 January 2014 | 800 × 1,000 (11 KB) | Incnis Mrsi (talk | contribs) | {| cellpadding=12px align=center | colspan=3 align=center style="font-size:200%; color:#00FF00" |←—— |- align=center | style="font-size:200%; color:#FFFF00" |← | style="font-size:200%; color:#00CCFF" |← | style="font-size:200%; color:#0... |
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Width | 800 |
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Height | 1000 |