9804

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

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)

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The relevant congruences (shown when you move the "time" slider) are those between the central triangle formed by the vertices , , and the three triangles , , and . When the "time" slider begins to slide, turns red, then rotates radians about the axis through vertex to become the central triangle , then rotates radians about the axis through vertex to become . Then turns green and slides along the arc defined by the side , showing that the two radian rotations compose to a rotation about an axis normal to the great circle containing the arc , but with the rotation angle twice the arclength of the arc . Analogous relationships hold between , , and the arc , as well as between , , and the arc . The congruences shown by the moving triangles show that the rotation mapping to followed by the rotation mapping to is the rotation mapping to .
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
Now click the "show congruence" checkbox to see triangles , , and . The spherical triangle relates to ordinary three-dimensional rotations by taking heed that a rotation of angle about an axis can be decomposed as the product of two rotations, each through a half-turn, about axes through the end points of any arc section of length of a great circle in the plane defined by the axis on the unit sphere. Thus a rotation through an angle of twice the arclength of about the axis defined by can be realized by a rotation through radians about the axis followed by a second rotation through radians about the axis . Slide the "time" slider to see the triangle turn red and rotate about onto the central triangle, then rotate about onto ; triangle then turns green and rotates along the arc to triangle to show the effect of the composition of the two rotations by radians. The triangles , , and are all congruent to the unit quaternion multiplication triangle with vertices . , , and are the images of triangle after the latter has been rotated radians about the axis defined by the unit vectors , , and , respectively. Penrose [2] describes them as the central triangle having been "reflected" in vertices , , and , respectively. So, the composition of a rotation through about axis followed by a radian rotation about maps triangle onto . The product of two rotations is again a rotation, and any rotation of the unit sphere is uniquely defined by a spherical triangle and its image under the rotation. Therefore, is the triangle after the latter has been rotated about the axis normal to the great circle containing the arc , but the magnitude of the rotation, given , by inspection is twice the arclength of the arc . The unit quaternion thus represents an rotation about its axis, but through an angle that is twice the arclength of the quaternion itself. This is the reason why an element of is often written with an angle of ; the angle is the angle of the rotation represented by the quaternion. Likewise and are the images of and under rotations about axes defined by the quaternions and , respectively. And, of course, the rotation mapping onto is the composition of the two rotations mapping first to , then to . So this rotation, represented by the quaternion , can be found graphically as the great circle arc closing the spherical triangle when we pay proper heed to the scaling factor between the unit quaternion's arclength and the rotation's angle.
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.
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