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.