How to Effect a Composite Rotation of a Vector via Geometric (Clifford) Algebra

How to Effect a Composite Rotation of a Vector
via Geometric (Clifford) Algebra
October 14, 2017
James Smith
nitac14b@yahoo.com
https://mx.linkedin.com/in/james-smith-1b195047
Abstract
We show how to express the representation of a composite rotation
in terms that allow the rotation of a vector to be calculated conveniently
via a spreadsheet that uses formulas developed, previously, for a single
rotation. The work presented here (which includes a sample calculation)
also shows how to determine the bivector angle that produces, in a single
operation, the same rotation that is effected by the composite of two
rotations.
“Rotation of the vector v through the bivector angle M1µ1, to produce
the vector v .”
1

Contents
1 Introduction 2
2 A Brief Review of How a Rotation of a Given Vector Can be
Effected via GA 4
3 Identifying the “Representation” of a Composite Rotation 5
4 A Sample Calculation 7
5 Summary 8
6 Appendix: Identifying the Bivector Angle Sσ through which
the Vector v can be Rotated to Produce v in a Single Opera-
tion 12
1 Introduction
Suppose that we rotate some vector v through the bivector angle M1µ1 to
produce the vector that we shall call v (Fig. 1), and that we then rotate v
through the bivector angle M2µ2 to produce the vector that we shall call v .
That sequence of rotations is called the composition of the two rotations. It is
equal to the rotation through some bivector angle Sσ ([1], pp. 89-91). Geometric
Algebra (GA) is a convenient and efficient tool for manipulating rotations—single
as well as composite—as abstract symbols, but what form does a numerical
calculation of a rotation take in a concrete situation? And how can we calculate
the bivector angle Sσ ?
Those are two of the questions that we will address in this document. Our
procedure will make use of single-rotation formulas that were developed in [2].
We’ll begin with a review of how a given vector can be rotated via GA. In that
review, we’ll discuss the important concept of the representation of a rotation,
after which we’ll present an formula that can be implemented in Excel for to
calculate single rotations of a given vector.
Having finished that review, we’ll see how to express the representation of
a composite rotation in terms that can be substituted directly in the formula
for single rotations. We’ll then work a sample problem in which we’ll calculate
the results of successive rotations of a vector. We’ll also calculate the bivector
angle that produces the same rotation in a single operation. The method used
for calculating that bivector angle is presented in the Appendix.

Figure 1: Rotation of the vector v through the bivector angle M2µ2, to produce
the vector v .
Figure 2: Rotation of the vector v through the bivector angle M2µ2, to produce
the vector v .
3

Figure 3: Rotation of v through the bivector angle Sσ, to produce the vector
v in a single operation.
2 A Brief Review of How a Rotation of a Given
Vector Can be Eﬀected via GA
References [3] (pp. 280-286) and [1] (pp. 89-91) derive and explain the following
formula for ﬁnding the new vector, w , that results from the rotation of a vector
w through the angle θ with respect to a plane that is parallel to the unit bivector
Q:
Figure 4: Rotation of the vector w through the bivector angle Q1, to produce
the vector w .
4

w = e−Qθ/2
[w] eQθ/2
Notation: RQθ(w)
. (2.1) Notation: RQθ(w) is the
rotation of the vector w by the
bivector angle Qθ.
For our convenience later in this document, we will follow Reference [1] (p. 89)
in saying that the factor e−Qθ/2
represents the rotation RQθ. That factor is a
quaternion, but in GA terms it is a multivector:
e−Qθ/2
= cos
θ
2
− Q sin
θ
2
. (2.2)
As further preparation for work that we’ll do later, we’ll mention that for any
given right-handed reference system with orthonormal basis vectors â, ˆb, and ˆc,
we may express the unit bivector Q as a linear combination of the basis bivectors
âˆb, ˆbˆc, and âˆc :
Q = âˆbqab + ˆbˆcqbc + âˆcqac,
in which qab, qbc, and qac are scalars, and q2
ab + q2
bc + q2
ac = 1.
To present a convenient way of calculating rotations via Excel spreadsheets,
Ref. [2] built upon that idea to write e−Qθ/2
as
e−Qθ/2
= fo − âˆbfab + ˆbˆcfbc + âˆcfac , (2.3)
with fo = cos
θ
2
; fab = qab sin
θ
2
; fbc = qbc sin
θ
2
; and fac = qac sin
θ
2
. Similarly,
eQθ/2
= fo + âˆbfab + ˆbˆcfbc + âˆcfac . (2.4)
Using these expressions for e−Qθ/2
and eQθ/2
, and writing w as w = âwa +
ˆbwb + ˆcwc, Eq. (2.1) becomes
w = fo − âˆbfab − ˆbˆcfbc − âˆcfac âwa + ˆbwb + ˆcwc fo + âˆbfab + ˆbˆcfbc + âˆcfac .
By expanding and simplifying the right-hand, side we obtain
w = â wa f2
o − f2
ab + f2
bc − f2
ac + wb (-2fofab − 2fbcfac) + wc (-2fofac + 2fabfbc)
+ ˆb wa (2fofab − 2fbcfac) + wb f2
o − f2
ab − f2
bc + f2
ac + wc (-2fofbc − 2fabfac)
+ ˆc wa (2fofac + 2fabfbc) + wb (2fofbc − 2fabfac) + wc f2
o + f2
ab − f2
bc − f2
ac .
(2.5)
Because this result can be implemented conveniently in (for example) a
spreadsheet similar to Ref. [4], the sections that follow will show how to express
the representation of a composite rotation in the form of Eq. (2.3).
3 Identifying the “Representation” of a Com-
posite Rotation
Let’s begin by defining two unit bivectors, M1 and M2:
M1 = âˆbm1ab + ˆbˆcm1bc + âˆcm1ac;
M2 = âˆbm2ab + ˆbˆcm2bc + âˆcm2ac.
5

Now, write the rotation of a vector v by the bivector angle M1µ1 to produce
the vector v :
v = e−M1µ1/2
[v] eM1µ1/2
.
Next, we will rotate v by the bivector angle M2µ2 to produce the vector v :
v = e−M2µ2/2
[v ] eM2µ2/2
.
Combining those two equations,
v = e−M2µ2/2
e−M1µ1/2
[v] eM1µ1/2
eM2µ2/2
.
The vector v was produced from v via the composition of the rotations by
the bivector angles M1µ1 and M2µ1. The representation of that composition
is the product e−M2µ1/2
e−M1µ1/2
. We’ll rewrite the previous equation to
make that idea clearer:
v = e−M2µ2/2
e−M1µ1/2
Representation
of the composition
[v] eM1µ1/2
eM2µ2/2
.
There exists an identifiable bivector angle —we’ll call it Sσ—through which v
could have been rotated to produce v in a single operation rather than through
the composition of rotations through M1µ1 and M2µ2. (See the Appendix.)
But instead of going that route, let’s write e−M1µ1/2
and e−M2µ2/2
in a way
that will enable us to use Eq. (2.3):
e−M1µ1/2
= go − âˆbgab + ˆbˆcgbc + âˆcgac , and
e−M2µ2/2
= ho − âˆbhab + ˆbˆchbc + âˆchac ,
where go = cos
µ1
2
; gab = m1ab sin
µ1
2
; gbc = m1bc sin
µ1
2
; and gac = m1ac sin
µ1
2
,
and ho = cos
µ2
2
; hab = m2ab sin
µ2
2
; hbc = m2bc sin
µ2
2
; and hac = m2ac sin
µ2
2
.
Now, we write the representation of the the composition as
ho − âˆbhab + ˆbˆchbc + âˆchac
e−M2µ2/2
go − âˆbgab + ˆbˆcgbc + âˆcgac
e−M1µ1/2
.
After expanding that product and grouping like terms, the representation of the
composite rotation can be written in a form identical to Eq. (2.3):
Fo − âˆbFab + ˆbˆcFbc + âˆcFac , (3.1)
with
Fo = e−M2µ2/2
e−M1µ1/2
0
= hogo − habgab − hbcgbc − hacgac ,
Fab = hogab + habgo − hbcgac + hacgbc ,
Fbc = hogbc + habgac + hbcgo − hacgab , and
Fac = hogac − habgbc + hbcgab + hacgo .
(3.2)
6

Therefore, with these definitions of Fo, Fab, Fbc, and Fac, v can be
calculated from v (written as âva + ˆbvb + ˆcvc) via an equation that is analogous,
term for term, with Eq. (2.5):
v = â va F2
o − F2
ab + F2
bc − F2
ac + vb (-2FoFab − 2FbcFac) + vc (-2FoFac + 2FabFbc)
+ ˆb va (2FoFab − 2FbcFac) + vb F2
o − F2
ab − F2
bc + F2
ac + vc (-2FoFbc − 2FabFac)
+ ˆc va (2FoFac + 2FabFbc) + vb (2FoFbc − 2FabFac) + vc F2
o + F2
ab − F2
bc − F2
ac .
(3.3)
At this point, you may (and should) be objecting that I’ve gotten ahead
of myself. Please recall that Eq. (2.5) was derived starting from the “rotation”
equation ((2.1))
w = e−Qθ/2
[w] eQθ/2
.
The quantities fo, fo, fab, fbc, and fac in Eq. (2.5), for which
e−Qθ/2
= fo − âˆbfab + ˆbˆcfbc + âˆcfac , (3.4)
also meet the condition that
eQθ/2
= fo + âˆbfab + ˆbˆcfbc + âˆcfac . (3.5)
We are not justified in using Fo, Fab, Fbc, and Fac in Eq. (2.5) unless we first
prove that these composite-rotation “F’s”, for which
Fo − âˆbFab + ˆbˆcFbc + âˆcFac = e−M2µ2/2
e−M1µ1/2
, (3.6)
also meet the condition that
Fo + âˆbFab + ˆbˆcFbc + âˆcFac = eM1µ1/2
e−M2µ2/2
. (3.7)
Although more-elegant proofs may well exist, “brute force and ignorance” gets
the job done. We begin by writing eM1µ1/2
e−M2µ2/2
in a way that is analogous
to that which was presented in the text that preceded Eq. (3.1):
go + âˆbgab + ˆbˆcgbc + âˆcgac
eM1µ1/2
ho + âˆbhab + ˆbˆchbc + âˆchac
eM2µ2/2
.
Expanding, simplifying, and regrouping, we fine that eM1µ1/2
e−M2µ2/2
is indeed
equal to Fo + âˆbFab + ˆbˆcFbc + âˆcFac , as required.
4 A Sample Calculation
The vector v =
4
3
â −
4
3
ˆb +
16
3
ˆc is rotated through the bivector angle âˆbπ/2
radians to produce a new vector, v . That vector is then rotated through the
bivector angle
âˆb
√
3
+
ˆbˆc
√
3
−
âˆc
√
3
−
2π
3
to produce vector v . Calculate
7

Figure 5: Rotation of v through the bivector angle âˆbπ/2, to produce the vector
v .
a The vectors v and v , and
b The bivector angle Sσ through which v could have been rotated to produce
v in a single operation.
We begin by calculating vector v . The rotation is diagrammed in Fig. 5
As shown in Fig. 6, v =
4
3
â +
4
3
ˆb +
16
3
ˆc.
We’ll calculate v in two ways: as the rotation of v by the bivector angle
âˆb
√
3
+
ˆbˆc
√
3
−
âˆc
√
3
−
2π
3
, and as the result of the rotation by the composite of
the two individual rotations. The rotation of v by
âˆb
√
3
+
ˆbˆc
√
3
−
âˆc
√
3
−
2π
3
is
diagrammed in Fig. 7. Fig. 8 shows that v =
4
3
â +
16
3
ˆb +
4
3
ˆc.
As we can see from Fig. 9, that result agrees with that which was obtained
by calculating v in a single step, as the composition of the individual rotations.
Fig. 9 also shows that the bivector angle Sσ is ˆbˆc (−π/2), which we can also
write as ˆcˆb (π/2). That rotation is diagrammed in Fig. 10.
5 Summary
We have seen how to express the representation of a composite rotation in terms
that allow the rotation of a vector to be calculated conveniently via a spreadsheet
that used formulas developed in [2] for a single rotation. The work presented
here also shows how to determine the bivector angle that produces, in a single
operation, the same rotation that is effected by the composite of two rotations.
8

Figure 6: A spreadsheet (Ref. [5]) that uses Eq. (2.5) to calculate v as the
rotation of v through the bivector angle âˆbπ/2.
Figure 7: Rotation of v . Note the significance of the negative sign of the scalar
angle: the direction in which v is to be rotated is contrary to the orientation of
the bivector. That significance is clearer in Fig. 10.
9

Figure 8: A spreadsheet (Ref. [5]) that uses Eq. (2.5) to calculate v as the
rotation of v .
10

Figure 9: A spreadsheet (Ref. [6]) that uses Eq. (3.2) to calculate v via the
composite rotation of v.
11

Figure 10: Rotation of v by Sσ to produce v in a single operation. Note the
significance of the negative sign of the scalar angle: the direction in which v
rotated is contrary to the orientation of the bivector ˆbˆc, and contrary also to
the direction of the rotation from ˆb to ˆc.
6 Appendix: Identifying the Bivector Angle Sσ
through which the Vector v can be Rotated
to Produce v in a Single Operation
Let v be an arbitrary vector. We want to identify the bivector angle Sσ through
which the initial vector, v, can be rotated to produce the same vector v that
results from the rotation of v through the composite rotation by M1µ1, then by
M2µ2:
e−M2µ2/2
e−M1µ1/2
[v] eM1µ1/2
eM2µ2/2
= v = e−Sσ/2
[v] eSσ/2
. (6.1)
We want Eq. (6.1) to be true for all vectors v. Therefore, eSσ/2
must be equal to
eM1µ1/2
eM2µ2/2
, and e−Sσ/2
must be equal to e−M1µ1/2
eM2µ2/2
. The
second of those conditions is the same as saying that the representations of
the Sσ rotation and the composite rotation must be equal. We’ll write that
condition using the Fo’s defined in Eq. (3.2), with S expressed in terms of the
unit bivectors âˆb, ˆbˆc, and âˆc:
cos
σ
2
− âˆbSab + ˆbˆcSbc + âˆcSac
S
sin
σ
2
= Fo − âˆbFab + ˆbˆcFbc + âˆcFac .
Now, we want to identify σ and the coefficients of âˆb, ˆbˆc, and âˆc. First, we
note that both sides of the previous equation are multivectors. According to the
postulates of GA, two multivectors A1 and A2 are equal if and only if for every
grade k, A1 k = A2 k. Equating the scalar parts, we see that cos
σ
2
= Fo.
Equating the bivector parts gives âˆbSab + ˆbˆcSbc + âˆcSac sin
σ
2
= âˆbFab +
12

ˆbˆcFbc + âˆcFac. Comparing like terms, Sab = Fab/ sin
σ
2
, Sbc = Fbc/ sin
σ
2
, and
Sac = Fac/ sin
σ
2
.
Why is it correct to identify the
S’s by comparing like terms? In
simple terms, because the unit
bivectors âˆb, ˆbˆc âˆb are
orthogonal. Two linear
combinations of those bivectors
are equal if and only if the
coefficients match, term for
term.
Next, we need to find sin
σ
2
. Although we could do so via sin
σ
2
= 1 − cos2 σ
2
,
for the purposes of this discussion we will use the fact that S is, by definition, a
unit bivector. Therefore, ||S|| = 1, leading to
sin
σ
2
= âˆbFab + ˆbˆcFbc + âˆcFac
= F2
ab + F2
bc + F2
ac .
Now, the question is whether we want to use sin
σ
2
= + F2
ab + F2
bc + F2
ac, or
sin
σ
2
= − F2
ab + F2
bc + F2
ac. The truth is that we can use either: if we use
− F2
ab + F2
bc + F2
ac instead of + F2
ab + F2
bc + F2
ac, then the sign of S changes
as well, leaving the product S sin
σ
2
unaltered.
The choice having been made, we can find the scalar angle σ from the values
of sin
σ
2
and cos
σ
2
, thereby determining the bivector angle Sσ.
References
[1] A. Macdonald, Linear and Geometric Algebra (First Edition) p. 126,
CreateSpace Independent Publishing Platform (Lexington, 2012).
[2] 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.
[3] D. Hestenes, 1999, New Foundations for Classical Mechanics, (Second
Edition), Kluwer Academic Publishers (Dordrecht/Boston/London).
[4] J. A. Smith, 2017, “Rotation of a Vector about an Axis” (an Excel spread-
sheet), https://drive.google.com/file/d/0B2C4TqxB32RRNHBHV2tpSUhRTUk/view?usp=sharing.
[5] J. A. Smith, 2017, “Rotation by a given bivector angle” (an Excel spread-
sheet), https://drive.google.com/file/d/0B2C4TqxB32RRX2JfcDd5NjZiZ00/view?usp=sharing .
[6] J. A. Smith, 2017, “Composite rotation in GA” (an Excel spreadsheet),
https://drive.google.com/file/d/0B2C4TqxB32RRaktDZktjcExPeUE/view?usp=sharing .
13

How to Effect a Composite Rotation of a Vector via Geometric (Clifford) Algebra

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