The SU(2) Spinor Map: Rotation Composition by Graphical Quaternion Triangles

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This Demonstration shows the spherical triangle congruences needed to understand the relationship between unit quaternion multiplication (i.e. the group ), spatial rotations in the adjoint representation , and a graphical quaternion multiplication construction conceived by William Rowan Hamilton [3]. Drag the triangle vertices using the 2D sliders. Use the "time" slider to see the congruences between the central triangle formed by vertices , , and the three triangles , , and . Use the "show congruence" checkbox to show or hide the moving triangles, "show labels" to show or hide labels, "show vertex vectors" to show the position vectors of the vertices , , , and "show inner triangle only" to show the quaternion arc composition triangle alone.

Contributed by: Rod Vance (a.k.a. The Wet Savanna Animal) (June 2014)
Open content licensed under CC BY-NC-SA

Details

Step 1. The Spherical Triangle Representing Quaternion Multiplication Alone

Click the checkboxes "show inner triangle only," "show vertex vectors," and "show labels." You see three unit vectors , , defining vertices on the spherical triangle . The directed arcs , , and are the unit quaternions defined by the pairs , , and of vectors, respectively. Actually, the arcs themselves contain one more bit of information than the pairs of vectors delimiting them: they define which of the two possible sections of the great circle marked out by the vectors is the arc in question.

The pair uniquely defines the arc . Indeed, given any two three-dimensional vectors , , one vector in the pair together with and uniquely determines the other vector; for example:

;

that is, one can invert the product defined by , although of course neither the scalar nor vector product on its own is invertible. The pairing is a standard way (see, e.g. [1], Chapter 1) to think of a quaternion; indeed one could say that the scalar and vector products of three-dimensional vectors from "modern" vector calculus are simply disembodied parts of the original quaternion product. With the geometry that we see in this Demonstration, it can be shown that the great circle arc closing the spherical triangle comprising great circle arcs is indeed the quaternion product .

The relationship between quaternion and rotation multiplication scaling can be intuitively understood by joining Roger Penrose [2] in his wonderful explanation of the triangle law for composition of rotations in §11.4 "How to Compose Rotations"; quaternions themselves are discussed more fully in the early sections of Chapter 11. Unit quaternions, that is, members of , were called versors by their discoverer, William Rowan Hamilton [3]. Further explanation can also be found in [4].

Step 2: Understanding Triangle Congruences

Further explanation is to be found at [4].

[1] C. Doran and A. Lasenby, Chap. 1, esp. §11.4 in Geometric Algebra for Physicists, Cambridge: Cambridge University Press, 2003.

[2] R. Penrose, Chap. 11 in The Road to Reality: A Complete Guide to the Laws of the Universe, London: Jonathan Cape, 2004.

[3] W. R. Hamilton, Chap. 1, §§8, 9 in Elements of Quaternions (W. E. Hamilton, ed.), London: Longmans, Green & Co., 1866 pp. 133–157.

[4] R. Vance, "Connected Lie Groups: The Grounding Axioms," Chap. 1 in Wet Savanna Animals: An Introduction to Lie Theory through Path Geometry (May 28, 2014). www.wetsavannaanimals.net/wordpress/connected-lie-groups-the-grounding-axioms, examples 1.3 and 1.4.

Permanent Citation

Rod Vance (a.k.a. The Wet Savanna Animal)

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