File:Space of rotations.png
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Summary
This schema represents two rotations in the mathematical rotation space.It has been made by me to illustrate the article "quaternions and spatial rotation".
First rotation (yellow) is of 80 degrees and is represented by a short arrow. Second rotation (red) is of 160 degrees, around another axis, and is represented by a longer arrow.
Each rotation pertains to a "sphere" of all rotations by the same angle. Here we can see the yellow (inside) and red (outside) spheres.
The small circles are oriented conforming to the "positive" right hand convention for angles
The lengths of the arrows representing the rotations are proportional to the sine of their half-angle, so the represented spheres are "slices" of the 4 dimensions hypersphere : sin (80° / 2) = 0.6428 and sin (160° / 2) = 0.9848
Any better graphist than me (read : any graphist) is welcome to redo it.
Сводка
На схеме отображено два вращения в математическом пространстве. Рисунок был сделан участником Wikipedia:User:MathsPoetry в качестве иллюстрации к статье Wikipedia:Quaternions and spatial rotation.
Первое вращение (жёлтым) на 80° представлено короткой стрелкой. Второе вращение (красным) на 160° осуществлено вокруг другой оси и представлено линией длиннее.
Каждое вращение принадлежит к "сфере" всех возможных вращений на такой же угол. Здесь мы можем видеть жёлтую (внутренняя) и красную (внешняя) сферы.
Маленькие окружности расположены в положительной правосторонней системе координат.
Длины стрелок, представляющих вращения, пропорциональны синусу половины их угла, поэтому представленные сферы - это проекции (срезы) четырёхмерной гиперсферы : sin (80° / 2) = 0.6428 and sin (160° / 2) = 0.9848.
Автор рисунка будет рад, если другой художник доработает его.
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 |
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File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 08:49, 19 April 2009 | 898 × 858 (67 KB) | wikimediacommons>MathsPoetry | * now follows right-hand convention for orientation of rotations * length of arrows representing rotations is now proporttional to sin(alpha /2) |
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