"Rotation of a Rotation" via Geometric (Clifford) Algebra
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We solve Problem 7.2.3 from Alan Macdonald's excellent textbook, "Linear and Geometric Algebra" to develop a formula for the "rotation of a rotation".
![The “Rotation of a Rotation” in Geometric
Algebra
October 25, 2017
James Smith
nitac14b@yahoo.com
https://mx.linkedin.com/in/james-smith-1b195047
References [1]-[4] discuss, in detail, how to rotate vectors and planes via
Geometric Algebra (GA). Here, we solve a problem from Ref. [1] , p. 127, which
reads (paraphrasing),
(Rotating rotations). Let M1 and M2 be unit bivectors. Consider
a rotation by the bivector angle M1µ1. Now, “rotate the rotation”:
that is, rotate M1µ1 by the bivector angle M2µ2 to obtain the
new bivector rotation angle e−M2µ2/2
[M1µ1] eM2µ2/2
. Show that
this rotated rotation is represented by Z = Z ZZ −1
, where Z =
e−M1µ1/2
represents the original rotation, Z = e−M2µ2/2
represents
its rotation, and Z −1
= eM2µ2/2
.
Hint: The unit bivector M1 is the product ef of orthonormal vectors.
Thus
e−M2µ2/2
[M1] eM2µ2/2
= e−M2µ2/2
(e) eM2µ2/2
e−M2µ2/2
(f) eM2µ2/2
is also a unit bivector.
Proof
Let’s use the symbol M3 to represent the rotated bivector e−M2µ2/2
[M1] eM2µ2/2
.
Then, the new bivector rotation angle e−M2µ2/2
[M1µ1] eM2µ2/2
is M3µ1. Be-
cause M3 is a unit bivector (per the “hint”), the representation of the rotated
rotation is
Z = e−M3µ1/2
= cos
µ1
2
− M3 sin
µ1
2
.](https://image.slidesharecdn.com/ejerciciolaga7-171025233753/85/Rotation-of-a-Rotation-via-Geometric-Clifford-Algebra-1-320.jpg)
![Now, we’ll note that cos
µ1
2
= e−M2µ2/2
cos
µ1
2
eM2µ2/2
. Therefore,
Z = e−M2µ2/2
cos
µ1
2
eM2µ2/2
− e−M2µ2/2
[M1] eM2µ2/2
M3
sin
µ1
2
= e−M2µ2/2
cos
µ1
2
− M1 sin
µ1
2
=e−M1µ1/2
eM2µ2/2
= Z ZZ −1
.
References
[1] A. Macdonald, Linear and Geometric Algebra (First Edition) p. 126,
CreateSpace Independent Publishing Platform (Lexington, 2012).
[2] D. Hestenes, 1999, New Foundations for Classical Mechanics, (Second
Edition), Kluwer Academic Publishers (Dordrecht/Boston/London).
[3] J. A. Smith, 2017, “How to Effect a Desired Rotation of a
Vector about a Given Axis via Geometric (Clifford) Algebra”
http://vixra.org/abs/1708.0462.
[4] J. A. Smith, 2017, “How to Effect a Composite Rotation of a Vector via
Geometric (Clifford) Algebra”, http://vixra.org/abs/1710.0147.
2](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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"Rotation of a Rotation" via Geometric (Clifford) Algebra
- 1. The “Rotation of a Rotation” in Geometric Algebra October 25, 2017 James Smith nitac14b@yahoo.com https://mx.linkedin.com/in/james-smith-1b195047 References [1]-[4] discuss, in detail, how to rotate vectors and planes via Geometric Algebra (GA). Here, we solve a problem from Ref. [1] , p. 127, which reads (paraphrasing), (Rotating rotations). Let M1 and M2 be unit bivectors. Consider a rotation by the bivector angle M1µ1. Now, “rotate the rotation”: that is, rotate M1µ1 by the bivector angle M2µ2 to obtain the new bivector rotation angle e−M2µ2/2 [M1µ1] eM2µ2/2 . Show that this rotated rotation is represented by Z = Z ZZ −1 , where Z = e−M1µ1/2 represents the original rotation, Z = e−M2µ2/2 represents its rotation, and Z −1 = eM2µ2/2 . Hint: The unit bivector M1 is the product ef of orthonormal vectors. Thus e−M2µ2/2 [M1] eM2µ2/2 = e−M2µ2/2 (e) eM2µ2/2 e−M2µ2/2 (f) eM2µ2/2 is also a unit bivector. Proof Let’s use the symbol M3 to represent the rotated bivector e−M2µ2/2 [M1] eM2µ2/2 . Then, the new bivector rotation angle e−M2µ2/2 [M1µ1] eM2µ2/2 is M3µ1. Be- cause M3 is a unit bivector (per the “hint”), the representation of the rotated rotation is Z = e−M3µ1/2 = cos µ1 2 − M3 sin µ1 2 .
- 2. Now, we’ll note that cos µ1 2 = e−M2µ2/2 cos µ1 2 eM2µ2/2 . Therefore, Z = e−M2µ2/2 cos µ1 2 eM2µ2/2 − e−M2µ2/2 [M1] eM2µ2/2 M3 sin µ1 2 = e−M2µ2/2 cos µ1 2 − M1 sin µ1 2 =e−M1µ1/2 eM2µ2/2 = Z ZZ −1 . References [1] A. Macdonald, Linear and Geometric Algebra (First Edition) p. 126, CreateSpace Independent Publishing Platform (Lexington, 2012). [2] D. Hestenes, 1999, New Foundations for Classical Mechanics, (Second Edition), Kluwer Academic Publishers (Dordrecht/Boston/London). [3] J. A. Smith, 2017, “How to Effect a Desired Rotation of a Vector about a Given Axis via Geometric (Clifford) Algebra” http://vixra.org/abs/1708.0462. [4] J. A. Smith, 2017, “How to Effect a Composite Rotation of a Vector via Geometric (Clifford) Algebra”, http://vixra.org/abs/1710.0147. 2