Geometric Algebra: Imaginary Numbers Are Real T IMOTHY F. H AVEL L ECTURE #4 1 of 22 Feb. 1, 2002 G EOMETRIC A LGEBRA : Imaginary Numbers Are Real Timothy F. Havel (Nuclear Engineering) LECTURE #4 J. W. Gibbs, On Multiple Algebra, Science Mag. 25:37-66, 1886. In its geometrical applications, multiple algebra will naturally take on one of two principal forms, according as vectors or points are taken as the elementary quantities. These forms of multiple algebra may be named vector analysis and point analysis. The former is included in the latter, since the subtraction of points gives us vectors, and in this way Grassmann’s vector analysis is included in his point analysis. On the other hand, if we represent points by vectors drawn from a common origin, and then develop those relations between such vectors representing points, which are independent of the position of the origin, we may obtain a large part, possibly all, of an algebra of points. The vector analysis, thus enlarged, is hardly to be distinguished from a point analysis, but the treatment of the subject in this way has something of a makeshift character, as opposed to the unity and
Transcript
G e o m e t r i c A l g e b r a : I m a g i n a r y N u m b e r s A r e R e a l
T
I M O T H Y
F . H
A V E L
L
E C T U R E
# 4
1 of 22Feb. 1, 2002
G
EOMETRIC
A
LGEBRA
:
Imaginary Numbers Are Real
Timothy F. Havel (Nuclear Engineering)
L
ECTURE
#4
J. W. Gibbs,
On Multiple Algebra
, Science Mag.
25:37-66
, 1886.
In its geometrical applications, multiple algebra will naturally
take on one of two principal forms, according as vectors or points
are taken as the elementary quantities. These forms of multiple
algebra may be named vector analysis and point analysis. The
former is included in the latter, since the subtraction of points
gives us vectors, and in this way Grassmann’s vector analysis is
included in his point analysis. On the other hand, if we represent
points by vectors drawn from a common origin, and then develop
those relations between such vectors representing points, which
are independent of the position of the origin, we may obtain a
large part, possibly all, of an algebra of points. The vector
analysis, thus enlarged, is hardly to be distinguished from a
point analysis, but the treatment of the subject in this way has
something of a makeshift character, as opposed to the unity and
G e o m e t r i c A l g e b r a : I m a g i n a r y N u m b e r s A r e R e a l
T
I M O T H Y
F . H
A V E L
L
E C T U R E
# 4
2 of 22Feb. 1, 2002
B
ARYCENTRIC
C
ALCULUS
August Ferdinand Möbius (1790-1868)
The
barycentric sum
of points in an -DEuclidean space is denoted by ,where is the total
weight
. Such sums can also beviewed as a
vector space
of dimension , wherein thepoints correspond to a basis (as shown above), namely
(so ). The P.-D. inner product vs. this basis induces the coordinate change
, where
is the
centroid
of the ; the coordinates of vs. this new basis are called its
affine coordinates
.
[1,0,0]
[0,1,0]
[1,1,1]
[0,0,1]
[1,1,0]
[-1,1,1]
[-2,1,1]
n 1+ pk En�{ }k 0=n n
En Wq w0 p0º wn pn+ +∫
W wkkÂ=
Rn 1+ n 1+
pk
pk pk´ º 0 1 0 º, , , ,[ ] Rn 1+�={ }k 0=n
Wq w0 º wn, ,[ ]´x y∑ n 1+( ) xkykkÂ= Wq Æ
W q c∑ q c�( )c 1–;[ ] c p0 º pn+ +( ) n 1+( )§∫ c´
pk qwkk wk pk c–( )
kÂ;[ ]
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L E C T U R E # 4
3 of 22Feb. 1, 2002
Points at InfinityIf every (unit weight!) point of can be uniquely expressed asa barycentric sum of a system of points , thissystem is called a point basis for .
Points of zero weight are the limit of a sequence of pointswhich moves off to infinity in a fixed direction as the sum of theweights goes to zero; therefore they are called points at infinity,and identified with a direction. In general, they also have amagnitude, but this depends on how the limit is taken.
If we choose our basis (or metric!) so that are (ortho)normal, the weights vs. the basis areaffine coordinates, and (unit weight) points can be viewed asvectors to an affine hyperplane in , as shown:
Enwk pkk pk{ }
k 0=n
En
p1 c– º pn c–, ,{ }c p1 º pn, , ,{ }
Rn 1+
w1
w2
q1
q 2
(q2 - q1)wq
hyperplane
wq
wc
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Line-Bound Vectors
Thus the points at infinity (and their magnitudes) can beviewed as vectors parallel to the affine hyperplane.
This interpretation in shows that the outer product ofa two points is an oriented segment of the line between them, i.e.
Since the magnitude of the bivector is twice the length of thesegment times its height above the origin, any other pair ofpoints separated by the same distance along the line generatethe same line bound vector; this can also be proven as follows:
Note that a line-bound vector is geometrically distinct from a freevector representing a point at infinity!
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Free Areal Magnitudes
The outer product of two free vectors is called a free arealmagnitude. We can write this as
.
This shows that the ordered sum of the line-bound vectorsaround a triangle yields a free areal magnitude, i.e.
This is just a discrete version of Stokes’ theorem ... with ageometric interpretation. To go from here to the continuousversion, just approximate the curve by a polygon, triangulate it,apply the discrete version to each triangle, and take the limit asthe number of sides goes to infinity.
The outer product of three points is a plane-bound area ...and so on into as many dimensions as you like!
q2 q1–( ) q3 q1–( )� q2 q3� q1 q3�– q1 q2�+=
q1Ÿ q2
q2Ÿ q3
q3Ÿ q1
(q3 - q1)Ÿ( q2 - q1)
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Forces and Torques in One
We regard a force as a free vector ; taking the outer productwith a point in a rigid body yields a line-bound vector which contains all the information needed to determine how theforce affects the body. To see this, observe that if the body ispivoted about the point , then the acceleration at each point is given (up to a constant factor) by
,
as shown in the drawing below:
This illustrates a general rule that we shall see many examplesof: The generators of motion are bivectors.
A second force applied to another point produces the sameresponse at any point only if , or
.
This in turn can be true for all only if and hence, which proves our claim above.
fp f p�
c r
a f p c–( )�( ) r c–( )∑=
cr
pa
f
f qr f p c–( )� g q c–( )�=
f p g q�–� f g–( ) c�=
c f g=f p� g q�=
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The Theory of Screws by Sir Robert Ball
A complementary interpretation of line-bound vectors is as aninfinitesimal motion. In the plane, rotation about a point withangular velocity is represented by a weighted point , and allinformation on the instantaneous motion of is in
,
where is the linear velocity of . To prove this, note thatthe derivative of the squared distance to any fixed point is
.
Now if is the unit free area, then .Also, since , , so , and:
A translation is represented by the free vector , i.e. as arotation about a point-at-infinity.
In 3-D space, an instantaneous rotation about an axis thrua point is represented by a line-bound vector, i.e. by a rotor
, and the resulting motion of a point by .Instantaneous translations are represented by a translator
, while the sum of a rotor & translator is a general screw.
2i q p v^���� ����( ) 2iq q p p c–( )���� ����( ) 2i qc p����( ) q�( )= = =
t^ Et=
ac
q c a�( ) p q c a�( ) p�
t^ a�
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Known Only by Their Effects
The analog of a translator for forces is a sum of two forces,whose line-bound vectors that are equal in magnitude, oppositein direction, and on different lines:
The sum of such a pair of forces iscalled a couple, and is the outer product of two free vectors.
This brings us to one of the deepest mysteries of geometry: Thereality of nonfactorizable elements in the algebra. For example,a general sum of forces cannot itself be written as the outerproduct of any two points or free vectors . This follows sincethe l.h.s. below is but the r.h.s. vanishes only if the points /vectors are linearly dependent:
Since , any such wrench can always be writtenas the sum of a couple and a finite force; similarly, any screw canbe written as the sum of a rotor and a translator.
cq
p
f -f
f p� f q�– f p q–( )�=
x y�0
x y x y� � � f p� g q�+( ) f p� g q�+( )� 2 f p g q� � �= =
g f g+( ) f–=
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THE REGRESSIVE PRODUCT Grassmann actually defined many kinds of geometrical mul-tiplication, including ultimately the geometric product itself.
Of particular interest was the regressive (outer) product,which may be defined via duality as
.
While the usual (progressive) outer product is a blade in thedirect sum of the nonintersecting subspaces of its factors, theregressive product is a blade in the intersection of the spanningsubspaces of its factors. In , for example,
More generally, the progressive & regressive products arerelated by the “shuffle” formula,
,
where a shuffle is a permutation of that preserves theorder of the first and last elements, is the parity ofthat permutation, and the square brackets indicates the dual ofthe outer product of the enclosed factors ( ).
X YÚ Xi 1–( ) Y i 1–( )�( )i 1–∫
E2
p q�( ) r s�( )Ú p q�( )i( )– r s�( )i( )i� p q�( )i( ) r s�( )∑= =
r p q�( )i( ) s∑ s p q�( )i( ) r∑– i r p q� �( ) s s p q� �( ) r–( )= =
a1 º ak� �( ) b1 º bl� �( )Ú
1–( )p ap 1( ) º ap n l–( ) b1 º bl, , , , ,[ ]ºshuffles p
Â=
º ap n l 1+–( ) º ap k( )� �
1 º k, ,{ }k n k– 1–( )p
n k l≥
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The Metric Connection
Now let us bring a metric in, by defining a quadratic form in thebarycentric coordinates of the points vs. abasis ; the corresponding symmetric bilinearform may be written using matrices as
,
where ( ) are the squared distances among thebasis points . Note that on the difference of a pairof basis points, e.g. , this form eval-uates to ; more generally, it gives thelength of any free vector directly. On any pair of basis points isthe form is clearly , and a general innerproduct of pairs of unit weight points is .To be convinced of this, take the vertices of a right pyramid as abasis, i.e. & for all ( ). The bary-centric coordinates w.r.t. points are Cartesian coord-inates vs. the frame with origin & orthonormal axes ,and you can show that for all free vectors .
q q0 q1 º qn[ ]´p0 p1 º pn, , ,[ ]
D r s,,,,( ) ∫ r0 r1 º rn[ ] 0 d012 2§– º d0n
2 2§–
d012 2§– 0 º d1n
2 2§–
º º º ºd0n
2 2§– d1n2 2§– º 0
s0
s1
...
sn
dij2 i j, 0 º n, ,=
p0 p1 º pn, , ,[ ]p0 p1– 1 1– 0 º 0[ ]´
D p0 p1– p0 p1–,( ) d012=
D pi p j,( ) dij2 2§–=
D r s,( ) r s– 2– 2§=
d0i2 1= dij
2 2= 0 i j, n£ £ i j<p1 º pn, ,p0 pi p0–
D u v,( ) uiviiÂ= u v,
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Distance Geometry
As a Mathematical Theory
∑ Studies mathematical spaces through their metrics.
∑ Includes the classical hyperbolic and elliptic as well asgeneral Riemannian geometries.
∑ The Euclidean case corresponds to the invariant theory ofthe group of rigid motions.
The Fundamental Theorems
I) Any rational polynomial in the Cartesian coordinates of a setof points, invariant under the rigid motions, can be rewrittenas a polynomial in:
1) The squared distances between pairs of points & .
2) The oriented volumes spanned by the sim-plices of points .
Note these invariants are not algebraically independent overthe real numbers.
II) The algebraic relations among the invariants (called syzygies)can be rewritten as a system of (in)equalities in the -pointCayley-Menger determinants:
D a b,( )a b
V a b º c, , ,( )a b º c, , ,· Ò
N
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if , the oriented volumes are related to thenonsymmetric determinants by
.
∑ In the case of three points, for example,
where . This is just Heron’s formula for the areaof a triangle in terms of its sides.
� Thus the condition is equivalent to all threetriangle inequalities among the points.
The nonnegativity of Cayley-Menger determinants cantherefore be viewed as a nonlinear generalization of thetriangle inequality.
0 D a b º c, , ,( )£ (vanishing if N dim 1)+>
21–2
------Ë ¯Ê � N
det
0 1 1 º 11 0 D a b,( ) º D a c,( )
º º º º º1 D a c,( ) D b c,( ) º 0
;= in addition,
N dim 1+=
V a b º c, , ,( )V p q º r, , ,( ) D a b º c, , , ; p q º r, , ,( )=
V2
a b c, ,( ) D a b c, ,( )= =
14--- d a b,( ) d a c,( ) d b c,( )+ +( ) d a b,( ) d a c,( ) d b c,( )–+( )
d a b,( ) d a c,( ) d b c,( )+–( ) d– a b,( ) d a c,( ) d b c,( )+ +( )
d D∫
D a b c, ,( ) 0≥
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The Quadratic Form of a Cayley-Menger Matrix
(or perhaps “Menger Meets Artin”)
∑ Any symmetric matrix is the matrix of some quadratic form,so any symmetric determinant is a Gramian of vectors insome metric vector space.
∑ For a Cayley-Menger determinant,
,
the zero’s down the diagonal shows these are null vectors,so the metric must be indefinite.
∑ The border of -1’s shows that one vector has the same innerproduct with all others, which thus lie in an affine hyperplaneM intersecting the null cone N in a “parabola”.
∑ For D Euclidean distances, the indicated sign conditions onthe principle minors shows the vectors live in an DMinkowski space (i.e. ).
D a b º c, , ,( )
det–
0 1– 1– º 1–
1– 0 D– a b,( ) 2§ º D– a c,( ) 2§º º º º º1– D– a c,( ) 2§ D– b c,( ) 2§ º 0
0≥=
n
n 2+( )V R n 1+ 1,= 1– 1 º 1, , ,[ ]
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∑ The null cone is and the hyperplane
is , where the point at infinity is
. These are depicted below.
∑ Relative to this metric, the inner product of two points is
minus half the squared distance between them, as follows:
N v V� v v∑ 0={ }∫
M v V� v e�∑ 1–={ }=
e� 1 1 0, ,[ ]∫
p q– 2–2
-----------------------p 2 1+( )
2------------------------- p 2 1–( )
2------------------------- p
1– 0 0 0 1 0
0 0 1
q 2 1+( ) 2§q 2 1–( ) 2§
q
=
-1
[1,º,1]
�parabola
hyperplane
(Euclidean)
(elliptic)
interior(hyperbolic)
exterior
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Möbius Sphere Geometry
∑ More generally, any vector on
with determines a sphere of radius
& center . The points of the sphere are exactly
, i.e. .
∑ In addition, vectors on orthogonal to a fixedsphere’s vector define alinearly dependent family ofspheres which intersectthat of orthogonally.
∑ Vectors other than with determine aEuclidean plane; these can be scaled so that and
is the ^ distance to the center of . They are the limitof a sequence of spheres whose centers go to (as above).
∑ The isometry group of is isomorphic to the conformal
group, which maps spheres & planes to the same (or );
the stabilizer of in this group is the Euclidean group.
∑ Vectors with are projectively points in Klein’s model
of hyperbolic geometry; those on the hyperplane with a given
norm form a pencil of horospheres (a la Wachter).
p P 1+( ) 2§ P 1–( ) 2§ p, ,[ ]=
M p 2 p 2 P–= 0> p
p
p ^ M N« « s V� s e�∑ 1–= s s∑ 0= s p∑ 0=, ,{ }
∑∑
M
p
p
u V� e� u e�∑ 0=
u 1=
u p∑ p
e�
V
e�e�
p p2 0<
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Doing It in the Algebra
(via Hestenes’ conformal split)
Let , be null vectors in Minkowski space with, and let be the bivector ; then the
conformal split of any other vector is
,
and hence
.
It follows that , so that
projects into a scale factor & a homogeneous part on an affine hyperplane thru the “origin” .
In the special case of a point , the scale factor is, so , and letting , we find that
,
i.e. , and hence .
e∞
e� R n 1+ 1,
e∞
e�∑ 1–= E e� e∞
�x
xE x E∑ x E�+ x e�∑( )e∞
x e∞
∑( )e�– x e� e∞
� �+= =
x xE2= =
x e�∑( )– e∞
x e∞
∑( )e�– x E�( ) E∑+ x∞
x� x+ +∫
e� x� x e�∑( )– E e� x+=
e� x�( ) e∞
∑ x e�∑( )– e∞
x^+ x e�∑( )– e∞
x
x e�∑( )–-----------------------+Ë ¯
Ê �= =
x x e�∑( )–
e∞
x x e�∑§– e∞
p M N«�p e�∑( )– 1= p
∞e∞
= p p^∫
0 p2 2 p� p∞
∑( ) p2+ 2– p e∞
∑( ) p2+= = =
p� p2 2§( )e�= p e∞ p2 2§( )e� p+ +=
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Thus the product of any two such “homogenous points” is:
This shows that the scalar part of this inner product of points isjust the Euclidean distance between the points,
.
Outer-multiplying the bivector part by , on the other hand,
,
gives us the moment and direction of the line-bound vectorgenerated by the points. Squaring this then gives the scalar
i.e. the Cayley-Menger determinant !
pq e∞
p2 2§( )e� p+ +( ) e∞
q2 2§( )e� q+ +( )=
p q∑ p2 e�e∞
( ) q2 e∞e�( )+( ) 2§ p q�+ +=
p e∞
q2 2§( )e�+( ) e∞
p2 2§( )e�+( )q+ +
p q– 2– 2§ p2 q2–( ) 2§( )E p q�+ +=
p q–( )e∞
pq2 p2q–( ) 2§( )e�+ +
1 2§–
p q∑ p q– 2– 2§=
e�
e� p q� � p q�( )e� q p–( )E+=
e� p q� � 2– q p e�� �( ) e� p q� �( )∑= =
det
e� e�∑ e� p∑ e� q∑
e� p∑ p p∑ p q∑
e� q∑ p q∑ q q∑
det
0 1– 1–
1– 0 p q– 2– 2§
1– p q– 2– 2§ 0
=
D p q,( )– p q– 2–=
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Circles in the Plane or Hyperplanes?
Consider now the case (the Euclidean plane). Then theouter product of any three independent homogeneous points
may be written as the dual of a vector, i.e. , wherethe unit pseudo-scalar of satisfies . Let us alsonormalize such that as usual, so that
.
Also, observe that the intersection of the plane with is , since and .
Thus the locus of the equation (with a homogen-eous point) can be described in the following equivalent ways,
,
thereby proving in (almost) one line the following:
Theorem: If are homogeneous points with (so that the corresponding points of
are affinely independent), a homogeneous point is linearlydependent on iff lies on the circle with center &squared radius , where is the dual of .
n 2=
p q r, , si is–=R 3 1, i2 1–=
s s e�∑ 1–=
sp q r� �( )– i
e� p q r� �( )i( )∑----------------------------------------------∫ p q r� �( )– i
e� p q r� � �( )i------------------------------------------ p q r� �( )– i
e� p q r� � �----------------------------------------= =
s e�,· ÒM N« o s s2 2§( )e�+∫ e� o∑ 1–= o2 0=
s x∑∑∑∑ 0= x
0 p q r� � x�e� p q r� � �
---------------------------------------- p q r� �( )i( ) x∑–e� p q r� � �
----------------------------------------------= =
s x∑∑∑∑ s2 2§ o x∑+ s2 x o– 2–( ) 2§= =∫
p q r, , M N«�e� p q r� � � 0π p q r,,,, ,,,, R 2
x
p q r, , x o
s2 s M� p q r� �
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An Example in 2D = 4D
The above figure shows three points on a unit circle, for which:
It follows that
,
& thus , i.e. !
p
q
rQ F
o
p q– 2 p q–( )2 p o–( ) q o–( )–( )2= =
p o–( ) eiEQ p o–( )–( )21 eiEQ–( ) 1 e i– EQ–( )= =
2 1 Q( )cos–( ) 4 Q 2§( )sin( )2= =
1 s2= =
i p q r� �( )( )2
e� p q r� � � 2------------------------------------------- 1 4§( ) p q–( )2 p r–( )2 q r–( )2
p r–( )2 q r–( )2 p r–( ) q r–( )∑( )2–---------------------------------------------------------------------------------------------=
Q 2§( )sin2 p q– 2 1 F( )cos2–= = F Q 2§=
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A Hot Night in Liouville!
As in , we can write an arbitrary isometry of as aproduct of reflections. Since is automatically invariant underthe isometries, these act on homogeneous points as
,
where and . However, we canalso write it as
�
In the case that representsa unit sphere centered on the origin , we have and
,
so that by � above
i.e. and . This reflection is thus an inver-sion in the unit sphere at ... a conformal transformation!
R n R n 1+ 1,
N
vpv 1–– v e∞ p2 2§( )e� p+ +( )v 1–
–=
J e∞ Q p( )( )2 2§( )e� Q p( )+ +( )◊=
Q: R n R nÆ J vpv 1–( ) e�∑ 0π∫
vpv 1–– p v� p v∑–( )v 1– p 2 p v∑( )v 1–
–= =
v s o s2 2§( )e�– e∞
e� 2§–∫ ∫ ∫e∞
v 1– v=
2 p v∑( )– 2 e∞
p2 2§( )e� p+ +( ) e∞
e� 2§–( )∑– p2 1–= =
vpv 1–– e
∞p2 2§( )e� p+ +( ) p2 1–( ) e
∞e� 2§–( )+=
p2e∞
e� 2§ p+ + p2 e∞
p 2– 2§( )e� p 1–+ +( )= =
Q p( ) p 1–= J p2=e∞
G e o m e t r i c A l g e b r a : I m a g i n a r y N u m b e r s A r e R e a l
T I M O T H Y F . H A V E L
L E C T U R E # 4
21 of 22Feb. 1, 2002
Now consider the reflection w.r.t. a with, so that represents an affine hyper-
plane with normal and distance to the origin ;then by our previous equation �, we get
where is thereflection of w.r.t. this hyperplane. In particular, the comp-osition of two reflections in parallel hyperplanes is
,
where is thus the vector of translation by twice thedistance between the hyperplanes. Hence represents the translation as a Lorentz transformation!
Finally, consider the action of the exponential
on homogenous points, i.e.
.
This is thus a dilation about by , which with the abovegenerates the whole conformal group by Liouville’s theorem.
G e o m e t r i c A l g e b r a : I m a g i n a r y N u m b e r s A r e R e a l
T I M O T H Y F . H A V E L
L E C T U R E # 4
22 of 22Feb. 1, 2002
Invariant Theory Revisited
In Tribute to Gian-Carlo Rota
� Projective geometry can be done at four distinct levels: (1)synthetic, (2) Grassmann algebra, (3) invariant theoretic, and(4) algebraic geometry (usually over the complex numbers).
� Analogous levels can also be identified in the Cayley-Klein(projective metric) geometries (over the real numbers).
� The extension to metric affine geometry is nontrivial, sincetransvections are not isometries. We now see that levels (1) & (3)are Möbius & distance geometry, resp.; the missing link is:
