From: Janne Pesonen
Newsgroups: sci.math.num-analysis,sci.math.symbolic,sci.physics.research,sci.math.research
Subject: Re: Clifford Algebra and Quaternions
Date: Fri, 06 Feb 1998 10:58:23 +0200
Cassiano Durand wrote:
> Given points, lines and planes in the usual representation,
> how can one express distances (and angles) using the
> geometric algebra (Clifford) approach?
The book 'New Foundations for Classical Mechanics' by David Hestenes
contains answers to your questions. It includes 104 page introduction
to geometric algebra, and chapter 2.6 is devoted to analytical geometry
(rest of the book is classical mechanics, as the title implies).
More advanced (and much more abstract) text is "Clifford algebra to
Geometric Calculus" by Hestenes and Garret Sobczyk, which includes a
chapter on differential geometry (but nothing on basic analytical
geometry).
There is a homepage of geometric algebra group in Cavendish laboratory,
Cambridge university. The address is
http://www.mrao.cam.ac.uk/~clifford/
Regards,
Janne Pesonen
-----------------------------------------------------------------------------
http://fkmarilyn.pc.helsinki.fi/Janne/index.html
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==============================================================================
From: "Richard J. Petti"
Subject: Re: Clifford Algebra and Quaternions
Date: Tue, 10 Feb 1998 03:08:29 GMT
Newsgroups: sci.math.num-analysis,sci.math,sci.math.symbolic,sci.physics.research
Cassiano Durand wrote in article
<34D9F2F6.EE2C522B@alcor.cs.purdue.edu>...
> Greetings,
>
>
> Given points, lines and planes in the usual representation,
> how can one express distances (and angles) using the
> geometric algebra (Clifford) approach?
>
> I know that one can represent points, planes and points
> as dual quaternions. For instance, the point P=(x,y,z)
> is represented by the dual quaternion
> P1 + e P2
> where P1=(1,0,0,0) (this is a shortcut representation for
> the quaternion 1 + 0i + 0j + 0k) and P2=(0,x,y,z).
> The plane ax+by+cz+d=0, subject to a^2+b^2+c^2=1
> is represented by the dual quaternion
> E1 + e E2
> where E1=(0,a,b,c) e E2=(d,0,0,0). The line with tangent vector
> t=(t1,t2,t3) and moment m=(m1,m2,m3) (i.e. Plucker coordinates)
> is represented by
> L1 + e L2
> where L1=(0,t1,t2,t3) e L2=(0,m1,m2,m3).
>
> Using this representation we can define incidence between
> these objects, like a point on a point, a point on a plane, a
> line on a plane, etc., but we would like to know how to
> express distances explicitally using this approach.
> Any help regarding this matter is greatly appreciated
>
> Thank you
> Cassiano Durand
>
> ______________________________________________________
> Cassiano Durand phone: 765-494-6002
> Computer Science Department fax: 765-494-0739
> Purdue University http://www.cs.purdue.edu/people/crbd
> West Lafayette, IN 47907-1398
> _______________________________________________________
>
>
Attached is an introduction to the geometric interpretations of
Clifford algebra operations, implemented with Macsyma. This
demonstration will be included in future releases of Macsyma but
is not included in the current releases Pc Macsyma 2.3 and
Unix Macsyma 421.
Macsyma's ATENSOR package also implements Clifford algebra,
including the quaternions, which are a special case.
Richard J. Petti
Macsyma Inc.
20 Academy Street
Arlington, MA 02174
tel: 781-646-4550
fax: 781-646-3161
email: petti@macsyma.com
URL: http://www.macsyma.com
------------------------------------------------------------------
Introduction to Spin Groups Using Macsyma
This demonstration contains basic concepts needed to make geometric sense
of spinors
in physics. We recommend that after exploring this demonstration, you
explore the
demo "spinors" which completes the basic geometric interpretation of
spinors.
1. Clifford Algebras
Clifford Algebras are the most natural places to carry on multilinear
algebra for
vector spaces that has a symmetric bilinear form (such as a metric tensor).
Consider a vector space V of dimension N with a symmetric bilinear form
b : V x V -> Reals. V and b define a Clifford algebra C(V, b) as the
quotient of
the universal tensor algebra T(V) with the ideal generatd by the relations
{ u @ v + v @ u - 2*b(u,v) | u, v in V },
where "@" denotes Clifford multiplication.
1.1 General Clifford Algebras implemented in Macsyma
Clifford algebras are implemented in Macsyma in the ATENSOR package. In
the
ATENSOR package,
- the symbols v[1], v[2] , ... } denote a basis of V,
- Macsyma's dot operator "." denotes the Clifford multiplication,
- a function denoted "sf" denotes the symmetric bilinear form.
(c1) init_atensor('clifford)
C:\mac23a\Macsyma2\tensor\atensor.fas being loaded.
(d1) clifford
This form simplifies toa scalar in a Clifford algebra.
(c2) v[1] . v[2] + v[2] . v[1]
(d2) v . v + v . v
2 1 1 2
(c3) atensimp(%)
(d3) 2 sf(v , v )
1 2
(c4) v[1] . v[1]
(d4) v . v
1 1
(c5) atensimp(%)
(d5) sf(v , v )
1 1
1.2 The Clifford Algebra of 3D Euclidean Space
Specialize to the case where dimension of V = 3 and the form b is positive
definite
(and its matrix is the 3 x 3 identity matrix in the basis { v[1], v[2],
v[3] }).
(c6) init_atensor('clifford, 3)
(d6) clifford
Verify the defining relations of the Clifford algebra. Here are all three
basic
two-forms in C(V,b).
(c7) [ v[2] . v[3] + v[3] . v[2] ; v[3] . v[1] + v[1] . v[3] ;
v[1] . v[2] + v[2] . v[1] ]
[ v . v + v . v ]
[ 3 2 2 3 ]
[ ]
(d7) [ v . v + v . v ]
[ 3 1 1 3 ]
[ ]
[ v . v + v . v ]
[ 2 1 1 2 ]
(c8) atensimp(%)
[ 0 ]
[ ]
(d8) [ 0 ]
[ ]
[ 0 ]
Express this as an equation.
(c9) ( %th(2) = %,
apply('matrix,map("[",flatten(args(matrixmap("=",lhs(%%),rhs(%%)))))))
[ v . v + v . v = 0 ]
[ 3 2 2 3 ]
[ ]
(d9) [ v . v + v . v = 0 ]
[ 3 1 1 3 ]
[ ]
[ v . v + v . v = 0 ]
[ 2 1 1 2 ]
Here are all three squares of the basis elements of V.
(c10) [ v[1] . v[1] ; v[2] . v[2] ; v[3] . v[3] ]
[ v . v ]
[ 1 1 ]
[ ]
(d10) [ v . v ]
[ 2 2 ]
[ ]
[ v . v ]
[ 3 3 ]
(c11) atensimp(%)
[ 1 ]
[ ]
(d11) [ 1 ]
[ ]
[ 1 ]
Expressing this as equations, we see that each v[i] is its own inverse in
the
Clifford algebra.
(c12) ( %th(2) = %,
apply('matrix,map("[",flatten(args(matrixmap("=",lhs(%%),rhs(%%)))))))
[ v . v = 1 ]
[ 1 1 ]
[ ]
(d12) [ v . v = 1 ]
[ 2 2 ]
[ ]
[ v . v = 1 ]
[ 3 3 ]
1.3 The Most Common Clifford Algebras in Physics
The two most common spin groups in physics are the Pauli algebra and the
Dirac
algebra over the field of real numbers.
- The Pauli algebra is Spin(V, b) for dim(V) = 3 and b positive definite.
(This
is the algebra described above). The Pauli algebra is used most
frequently in
the non-relativistic theory of electrons and other fermions.
- The Diract algebra is Spin(V, b) for dim(V) = 4 and b with signature (+
- - - )
or (+ + + - ). The Dirac algebra is used most frequently in the
relativistic
theory of electrons and other fermions.
Macsyma's atenor package has an implementation of the Dirac algebra ,using
the
signature ( + + + - ).
(c13) init_atensor(dirac)
(d13) dirac
Work in a basis of V denoted by v[1] , v[2] , v[3] , v[4].
Here are some typical simplifications.
(c14) atensimp( v[3] . v[1] )
(d14) - v . v
1 3
(c15) atensimp( v[2] . v[2] )
(d15) 1
(c16) atensimp( v[4] . v[4] )
(d16) - 1
2. Geometric Interpretations of Clifford Products
The Clifford product generalizes the metric tensor, which measures lengths
of
vectors and the angles between vectors. We continue in 3D Euclidean space
so
Macsyma can return explicit results, but the constructions work in any
dimension
with a bilinear form of any signature.
(c17) init_atensor('clifford, 3)
(d17) clifford
Declare some symbols to represent scalar coefficients.
(c18) declare([a,b,c,d,e], scalar)$
2.1 Areas of Parallelograms and Volumes of Parallelopipeds
The Clifford product yields the area of the parallelogram determined by two
vectors
u and w. This generalizes that fact that the metric tensor gives the
lengths of
vectors. Start with a simple case: find the area of the parallelogram
determined by
d*v[1] and e*v[2] . The parallelogram is represented in Clifford algebra as
an
antisymmetric Clifford product.
(c19) atensimp( (d*v[1]) . (e*v[2]) - (e*v[2]) . (d*v[1]) )/2
(d19) (v . v ) d e
1 2
The squared area of the parallelogram is given by
(c20) atensimp( aten_reverse(%) . %)
2 2
(d20) d e
Now for a more complicated case. Consider a parallelogram defined by two
vectors
c*v[1] and w = d*v[1]+e*v[2] .
(c21) (u : c*v[1], w : d*v[1]+e*v[2],
display(u,w))$
u = v c
1
w = v e + v d
2 1
The parallelogram is represented by an antisymmetric Clifford product of
the two
vectors u, w.
(c22) (parallelogram : atensimp(u.w - w.u)/2,
display(parallelogram))$
parallelogram = (v . v ) c e
1 2
The squared area of the parallelogram is given by the Clifford product
below.
(c23) area_squared : atensimp(aten_reverse(parallelogram) . parallelogram)
2 2
(d23) c e
If the angle between u and w is theta, and the length of w is d, then we
have for
the squared area
(c24) trigsimp(subst([d=r*cos(theta), e=r*sin(theta)], %))
C:\mac23a\Macsyma2\share\trigsimp.fas being loaded.
2 2 2
(d24) c r sin (theta)
This construction generalizes to give volumes of parallelopipeds of any
dimension.
2.2 Reflections in one direction
Conjugate a vector x in V by v[1] and take its negative.
That is, compute (- v) . x . (v^^-1)
(c25) (x :a*v[1]+b*v[2]+c*v[3],
display(x))$
x = v c + v b + v a
3 2 1
Conjugate a vector x in V by v[1] and take its negative.
That is, compute (- v) . x . (v^^-1)
(c26) - v[1] . x . v[1]
(d26) - (v . v . v ) c - (v . v . v ) b - (v . v . v ) a
1 3 1 1 2 1 1 1 1
The component of w in the direction of v[1] is reversed, and the component
of u
orthogonal to v[1] is unaffected.
(c27) atensimp(%)
(d27) v c + v b - v a
3 2 1
2.3 Rotations in a Plane
Given vectors u=c*v[1] and w = d*v[1]+e*v[2] in the v[1],v[2] plane,
conjugation
by u and w defines a rotation in the v[1],v[2] plane. Use that u^^-1 =
v[1]/c^2
and w^^-1 = w/(d^2+e^2) .
(c28) atensimp( (-w) . (- u) . x . (u/c^2) . w/(d^2+e^2) )
2 2 2 2
v c e v b e v a e 2 v b d e 2 v a d e v c d
3 2 1 1 2 3
(d28) ------- - ------- - ------- - ---------- + ---------- + -------
2 2 2 2 2 2 2 2 2 2 2 2
e + d e + d e + d e + d e + d e + d
2 2
v b d v a d
2 1
+ ------- + -------
2 2 2 2
e + d e + d
Suppose that the angle between v[1] and w is theta, so
w = cos(theta)*v[1] + sin(theta)*v[2] .
(c29) subst([d=cos(theta), e= sin(theta)] , %)
2 2
v c sin (theta) v b sin (theta)
3 2
(d29) ------------------------- - -------------------------
2 2 2 2
sin (theta) + cos (theta) sin (theta) + cos (theta)
2
v a sin (theta) 2 v b cos(theta) sin(theta)
1 1
- ------------------------- - ----------------------------
2 2 2 2
sin (theta) + cos (theta) sin (theta) + cos (theta)
2
2 v a cos(theta) sin(theta) v c cos (theta)
2 3
+ ---------------------------- + -------------------------
2 2 2 2
sin (theta) + cos (theta) sin (theta) + cos (theta)
2 2
v b cos (theta) v a cos (theta)
2 1
+ ------------------------- + -------------------------
2 2 2 2
sin (theta) + cos (theta) sin (theta) + cos (theta)
Simplifying, we see that the component of u = a*v[1]+b*v[2]+c*v[3] which
lies in
the v[1],v[2] plane is rotated by an angle of 2*theta, while the component
of u
which is orthogonal to the v[1],v[2] plane is unaffected.
(c30) ratsimp(trigreduce(%),v[1],v[2])
C:\mac23a\Macsyma2\library1\trgred.fas being loaded.
C:\mac23a\Macsyma2\library1\binoml.fas being loaded.
(d30) v (a cos(2 theta) - b sin(2 theta))
1
+ v (a sin(2 theta) + b cos(2 theta)) + v c
2 3
2.4 Angles between k-Planes
Given two 2-planes, the Clifford product tells us the cosine of the angle
between
these planes. This generalizes that fact that the metric tensor tells us
the cosine
of the angle between two vectors.
Consider the two planes v[1]^v[3] = (v[1].v[3]-v[3].v[1])/2
and w^v[3] = (w.v[3]-v[3].w)/2 .
(c31) (plane1 : (v[1].v[3]-v[3].v[1])/2,
plane2 : atensimp(w.v[3]-v[3].w)/2,
display(plane1, plane2) )$
v . v - v . v
1 3 3 1
plane1 = -----------------
2
2 (v . v ) e + 2 (v . v ) d
2 3 1 3
plane2 = -----------------------------
2
The scalar part of the following Clifford product gives the cosine of the
minimum
angle between the planes (which is orthogonal to all the directions shared
by the
two planes) times the areas of the two planes.
(c32) atensimp( aten_reverse(plane2) . plane1 )
(d32) d - (v . v ) e
1 2
The area of v[1]^v[2] =1. We need the squared area of (w^v[3]), which is
(c33) atensimp( aten_reverse(plane2) . plane2 )
2 2
(d33) e + d
The normalized Clifford product is
(c34) %th(2)/ sqrt(%)
d - (v . v ) e
1 2
(d34) ---------------
2 2
sqrt(e + d )
If the angle between v[1] and w is theta and the length of w is 1, then we
get
the expression below, which has scalar part cos(theta).
(c35) trigsimp(subst([d=cos(theta), e=sin(theta)], %))
(d35) cos(theta) - (v . v ) sin(theta)
1 2
Although this example uses 2-planes, this construction works for the angle
between to k-planes for any positive integer k.
3. Spin Groups
3.1 Definition of Spin(V, b)
The units - that is, invertible elements - of a Clifford algebra C(V, b)
form a
multiplicative group. The spin group Spin(V, b) is the subgroup of "even"
units u
in C(V, b) such that aten_reverse(u) . u = 1. This last condition is a
scalar
normalization, which picks out a sphere of unit radius in the group of
units.
3.2 Spin(V, b) is the Universal Covering Group of SO(V, b)
Each element of Spin(V, b) can be used to conjugate vectors in V. Each
element of
Spin(V, b) thereby defines a rotation in the special orthogonal group SO(V,
b).
For each element s Spin(V, b), s and -s define the same rotation. In fact,
there is
a homomorphism from Spin(V, b) to SO(V, b) which has kernel { 1, -1}. That
is,
Spin(V, b) forms a double covering group of SO(V, b). (In the special case
where
dim(V)=2, this is an infinite cyclic covering.)
It turns out that Spin(V,b) is simply connected, so Spin(V, b) is the
universal
covering group of SO(V, b).
3.3 Lie Algebra of Spin(V, b)
The Lie algebra of Spin(V, b) is, geometrically, the collection of
infinitesimal
spin rotations. They are represented by the antisymmetric rank 2 elements
in
C(V, b).
Compute the Lie bracket of the infinitesimal rotation in the x[1],x[2]
plane and
the rotation in the v[2],v[3] plane.
(c36) plane1: (v[1].v[2] - v[2].v[1])/(2*sqrt(2))
v . v - v . v
1 2 2 1
(d36) -----------------
2 sqrt(2)
(c37) plane2: (v[2].v[3] - v[3].v[2])/(2*sqrt(2))
v . v - v . v
2 3 3 2
(d37) -----------------
2 sqrt(2)
The Lie bracket is the commutator using Clifford multiplication. This
result is
equivalent to v[1]^v[3] .
(c38) atensimp( plane1 . plane2 - plane2 . plane1 )
(d38) v . v
1 3
4. Geometrical Signficance of Spinors in Field Theories in Physics
Why do physical field theories use spinors, which are elements of
representation
spaces of spin groups? Physicists are accustomed to keeping track of
rotational
frames of reference, which the rotation group SO(V, b) does. Suppose that,
in
addition, you need to track closed paths through SO(V, b) to determine
which paths
are continuously deformable to a point. Tracking the final frame in SO(V,
b) is
not adequate for this purpose; you must know the path taken through SO(V,
b).
The spin group is the rotation group augmented by precisely the additional
information
needed to determine whether a closed path in SO(V, b) is continuously
deformable to
a point using only the endpoint of the path. That is what universal
covering
groups do in general.
A Thought Experiment
Here is a thought experiment which illustrates what is meant by tracking
whether a
path on SO(V, b) is continuously deformable to a point, where dim(V)=3.
Imagine a huge piece of stiff black material with a small sphere excised.
In
the center of the spherical void is a much smaller sphere of stiff green
material. The hard green sphere is connected to the outer black material
along
the z-axis by two vertical shafts of compliant red rubber.
Assign to the outer black material a frames of reference (ordered set of
three
basis vectors) to track its orientation. Assign the same initial frame of
reference to the inner green sphere.
(c39) block([plotnum0:13, plotnum1:11, plot_size: 150, plot_phi:45,
r1: 4., r2: 12., r3:1.,
plot_style: 'outer_space, color_function:'color_red,
title: "Rigid Sphere in a Void in a Rigid Medium with Compliant
Connecting Posts"],
declare([r1,r2,r3], special),
combine_plots(
plotsurf([r3*[cos(phi),sin(phi),z],r3*[cos(phi),sin(phi),-z]],
phi,0,2*%pi,z,1.01*sqrt(r1^2-r3^2),0.99*sqrt(r2^2-r3^2)),
plotnum0: 20, plotnum1:20, color_function: 'color_green,
plotsurf([r1*[cos(phi)*sin(theta),sin(phi)*sin(theta), cos(theta)]],
phi,
0,2*%pi,theta,0,%pi),
color_function: 'color_gray,
plotsurf([r2*[cos(phi)*sin(theta),sin(phi)*sin(theta), cos(theta)]],
phi,%pi/2,2*%pi,theta,%pi/36,35/36*%pi) ))$
C:\mac23a\Macsyma2\library2\pltfuncs.fas being loaded.
(Plot not shown)
Imagine that by radio control we can remotely rotate the inner sphere. As
we
rotate the sphere, the frame of reference of the inner sphere differs from
that
of the outer material. We can record the rotation as either an element of
SO(V, b) or Spin(V, b).
Now suppose that we rotate the inner sphere in one plane through 360
degrees.
The compliant red material is highly deformed. The element of SO(V, b)
which
represents this configuration is the identity element, since the frame of
reference of the inner sphere is identical to the frame of the outer black
material. The element of Spin(V, b) which represents this configuration is
the
antipodal element, which in spin representations is the number -1. (We
represent
the compliant rubber in orange to indicate that it is deformed by a 360
degree
rotation of the inner sphere.)
(c40) block([plotnum0:13, plotnum1:11, plot_size: 150, plot_phi:45,
r1: 4., r2: 12., r3:1.,
plot_style: 'outer_space, color_function:'color_orange,
title: "Rigid Sphere in a Void in a Rigid Medium with Compliant
Connecting Posts"],
declare([r1,r2,r3], special),
combine_plots(
plotsurf([r3*[cos(phi+z/(r2-r1)),sin(phi+z/(r2-r1)),z],
r3*[cos(phi-z/(r2-r1)),sin(phi-z/(r2-r1)),-z]],
phi,0,2*%pi,z,1.01*sqrt(r1^2-r3^2),0.99*sqrt(r2^2-r3^2)),
plotnum0: 20, plotnum1:20, color_function: 'color_green,
plotsurf([r1*[cos(phi)*sin(theta),sin(phi)*sin(theta), cos(theta)]],
phi,0,2*%pi,theta,0,%pi),
color_function: 'color_gray,
plotsurf([r2*[cos(phi)*sin(theta),sin(phi)*sin(theta), cos(theta)]],
phi,%pi/2,2*%pi,theta,%pi/36,35/36*%pi) ))$
(Plot not shown)
Next suppose that we rotate the inner sphere in one plane through 720
degrees.
The compliant red material can snap back to its initial undeformed
configuration
without tearing. The element of SO(V, b) which represents this
configuration is
the identity element. The element of Spin(V, b) which represents this
configuration is the identity element. (We represent the compliant rubber
as
yellow to indicate that it is deformed by a 720 degree rotation of the
inner
sphere.)
(c41) block([plotnum0:13, plotnum1:11, plot_size: 150, plot_phi:45,
r1: 4., r2: 12., r3:1.,
plot_style: 'outer_space, color_function:'color_yellow,
title: "Rigid Sphere in a Void in a Rigid Medium with Compliant
Connecting Posts"],
declare([r1,r2,r3], special),
combine_plots(
plotsurf([r3*[cos(phi+2.*z/(r2-r1)),sin(phi+2.*z/(r2-r1)),z],
r3*[cos(phi-2.*z/(r2-r1)),sin(phi-2.*z/(r2-r1)),-z]],
phi,0,2*%pi,z,1.01*sqrt(r1^2-r3^2),0.99*sqrt(r2^2-r3^2)),
plotnum0: 20, plotnum1:20, color_function: 'color_green,
plotsurf([r1*[cos(phi)*sin(theta),sin(phi)*sin(theta), cos(theta)]],
phi,0,2*%pi,theta,0,%pi),
color_function: 'color_gray,
plotsurf([r2*[cos(phi)*sin(theta),sin(phi)*sin(theta), cos(theta)]],
phi,%pi/2,2*%pi,theta,%pi/36,35/36*%pi) ))$
(Plot not shown)
Spinor field theories are tracking entanglement relations between two
rotational
frames of reference connected by a path in SO(V, b).
Do DEMO(DIRACMAT1); for representations of the Pauli and Dirac algebras in
Macsyma.
Clean up
(c42) remvalue(area_squared, parallelogram, plane1, plane2, u, w, x)$
==============================================================================
From: Janne Pesonen
Newsgroups: sci.math.num-analysis
Subject: Re: Clifford, anyone?
Date: Thu, 05 Feb 1998 22:19:48 +0200
Richard Remski wrote:
> I've been told a lot recently about Clifford or so-called "geometric
> algebra" ... that it provides a clean way to perform effectively
> algebraic
> matrix operations without bothering with all the matrix solution
> hassles.
> Thus far what little I've seen about it seems focused in
> space-time/astrophysical applications. I don't know squat about them,
> but
> would be interested in learning the math. Any sources?
These sources might interest you:
David Hestenes: New Foundations for Classical Mechanics (Reidel, 1986);
includes 104 page introduction to geometric algebra
D. Hestenes & G. Sobczyk: Clifford algebra to Geometric Calculus
(Reidel, 1984)
T. G. Vold: An introduction to geometric algebra with an application in
rigid body mechanics. Am. J. Phys. 61 (1993) 491
T. G. Vold: An introduction to geometric calculus and its application
electrodynamics. Am. J. Phys. 61 (1993) 505
S. Gull, A. Lasenby & C. Doran: Imaginary Numbers are not Real.-the
Geometric Algebra of Spacetime. Found. Phys. 23 (1993) 1175.
Regards,
Janne Pesonen
http://fkmarilyn.pc.helsinki.fi/Janne/index.html
==============================================================================
From: suter@fawlty8.eng.monash.edu.au (Mr D. Suter)
Newsgroups: sci.math.num-analysis
Subject: Re: Clifford, anyone?
Date: 8 Jan 1998 01:54:12 GMT
In article <690qho$sb2$1@nnrp1.rcsntx.swbell.net> "Richard Remski" writes:
>Dumb question:
>
>I've been told a lot recently about Clifford or so-called "geometric
>algebra" ... that it provides a clean way to perform effectively algebraic
>matrix operations without bothering with all the matrix solution hassles.
>Thus far what little I've seen about it seems focused in
>space-time/astrophysical applications. I don't know squat about them, but
>would be interested in learning the math. Any sources?
>
>Anyone believe that Clifford algebra has applications for FEA or other
>numerical methods?
>
You may want to follow (if you haven't already seen) the links
http://www.mrao.cam.ac.uk/~clifford/
The cambridge group are also heavily involved with space-time/astrophysical applications
but they have a couple of publications on their proposed use in such
areas as computer vision and robotics (my main areas of interest).
In a nutshell, clifford algebras are generalisations in the line
Reals -> complex -> quaternions ->......
In the applications I've seen, there hasn't been much call for higher
algebras than quaternions (and these basically to represent 3-D
rotations - so in some sense that's old hat). Moreover, though
I may be corrected on this - my impression has been that the
main claims have been to have brought an elegance to the formulation
that may suggest insights in problem re-formulation and/or solution -
and even than the gains appear to be minimal. I haven't seen
any advances developed for FEA or other general numerical methods.
I do find the stuff interesting though and hold some hopes that
greater and more general payoffs can be found in my areas (computer
vision and robotics) and in general numerical methods so
don't take my somewhat negative comments to mean I suggest it's all a dead
end. I am saying that if you expect easily digestible and ready for application
insights, tools etc, then I don't think they type you are looking for
exists. I'd be delighted to be corrected on this - anyone think
I've missed some works/links or grossly mis-understood what I've seen?
regards
d.s.
==============================================================================
From: Janne Pesonen
Newsgroups: sci.math
Subject: Re: seek succor re tensors and differential forms
Date: Fri, 16 Oct 1998 13:14:11 +0200
To: nweissma@ic.sunysb.edu
nweissma@ic.sunysb.edu wrote:
> i'm enrolled in "tensors and manifolds", having an impossible time seeing
> the forest in the trees. can someone offer effective pedagogy re tensors,
> differential forms (n-forms), minkowski space, "space-like", "time-like"
> and the like??
Visit these two pages:
1) http://modelingnts.la.asu.edu/GC_R&D.html by David Hestenes (the
founder of modern geometric algebra)
Following .pdf documents should answer ALL your questions:
i) Synopsis of geometric algebra
(http://modelingnts.la.asu.edu/pdf/NFMPchapt1.pdf)
ii) Geometric calculus
(http://modelingnts.la.asu.edu/pdf/NFMPchapt2.pdf)
iii) Differential forms in geometric calculus
(http://modelingnts.la.asu.edu/pdf/DIF_FORM.pdf)
iv) Spacetime calculus
(http://modelingnts.la.asu.edu/pdf/SpaceTimeCalc.pdf)
2) http://www.mrao.cam.ac.uk/~clifford/
yours,
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Janne Pesonen
http://fkmarilyn.pc.helsinki.fi/Janne/index.html
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