New Advances in
Geometric Algebra
Chris Doran
Astrophysics Group
Cavendish Laboratory
[e-mail]
www.mrao.cam.ac.uk/~clifford
Outline
" History
" Introduction to geometric algebra (GA)
" Rotors, bivectors and rotations
" Interpolating rotors
" Computer vision
" Camera auto-calibration
" Conformal Geometry
28/10/2003 Geometric Computation 2001 2
What is Geometric Algebra?
" Geometric Algebra is a universal Language for
physics based on the mathematics of Clifford
Algebra
" Provides a new product for vectors
" Generalizes complex numbers to arbitrary
dimensions
" Treats points, lines, planes, etc. in a single algebra
" Simplifies the treatment of rotations
" Unites Euclidean, affine, projective, spherical,
hyperbolic and conformal geometry
28/10/2003 Geometric Computation 2001 3
History
" Foundations of geometric algebra (GA) were laid
in the 19th Century
" Two key figures: Hamilton and Grassmann
" Clifford unified their work into a single algebra
" Underused (associated with quaternions)
" Rediscovered by Pauli and Dirac for quantum
theory of spin
" Developed by mathematicians in the 50s and 60s
" Reintroduced to physics in the 70s
28/10/2003 Geometric Computation 2001 4
Hamilton
Introduced his quaternion
algebra in 1844
i2 =ð j2 =ð k2 =ð ijk =ð ?ð1
Generalises complex
arithmetic to 3 (4?)
dimensions
Very useful for rotations,
but confusion over the
status of vectors
28/10/2003 Geometric Computation 2001 5
Grassmann
German schoolteacher 1809-1877
Published the Lineale
Ausdehnungslehre in 1844
Introduced the outer product
a Tð b =ð ?ðb Tð a
Encodes a plane segment
b
a
28/10/2003 Geometric Computation 2001 6
2D Outer Product
a Tð a =ð ?ða Tð a =ð 0
" Antisymmetry implies
" Introduce basis áðe1,e2âð
a =ð a1e1 +ð a2e2 b =ð b1e1 +ð b2e2
" Form product
a Tð b =ð a1b2e1 Tð e2 +ð a2b1e2 Tð e1
=ð Å»ða1b2 ?ð b2a1Þðe1 Tð e2
" Returns area of the plane + orientation.
" Result is a bivector
" Extends (antisymmetry) to arbitrary vectors
28/10/2003 Geometric Computation 2001 7
Complex Numbers
" Have a product for vectors in 2D
" Length given by aaDð
" Suggests forming
abDð =ð Å»ða1 +ð a2iÞðÅ»ðb1 ?ð b2iÞð
=ð Å»ða1b1 +ð a2b2Þð ?ð Å»ða1b2 ?ð b2a1Þði
" Complex multiplication forms the inner and outer
products of the underlying vectors!
" Clifford generalised this idea
28/10/2003 Geometric Computation 2001 8
Clifford
W.K. Clifford 1845-1879
Introduced the geometric product
ab =ð a 6ð b +ð a Tð b
Product of two vectors returns the
sum of a scalar and a bivector.
Think of this sum as like the real
and imaginary parts of a complex
number.
28/10/2003 Geometric Computation 2001 9
Properties
" Geometric product is associative and distributive
aÅ»ðbcÞð =ð Å»ðabÞðc =ð abc
aÅ»ðb +ð cÞð =ð ab +ð ac
" Square of any vector is a scalar. This implies
Å»ða +ð bÞð2 =ð Å»ða +ð bÞðÅ»ða +ð bÞð =ð a2 +ð b2 +ð ab +ð ba
" Can define inner (scalar) and outer (exterior)
products
1 1
a 6ð b =ð Å»ðab +ð baÞð a Tð b =ð Å»ðab ?ð baÞð
2 2
28/10/2003 Geometric Computation 2001 10
2D Algebra
" Orthonormal basis is 2D
áðe1,e2âð
e1 6ð e1 =ð e2 6ð e2 =ð 1 e1 6ð e2 =ð 0
" Parallel vectors commute
e1e1 =ð e1 6ð e1 +ð e1 Tð e1 =ð 1
" Orthogonal vectors anticommute since
e1e2 =ð e1 6ð e2 +ð e1 Tð e2 =ð ?ðe2 Tð e1 =ð ?ðe2e1
" Unit bivector has negative square
Å»ðe1 Tð e2Þð2 =ð Å»ðe1e2ÞðÅ»ðe1e2Þð =ð e1e2Å»ð?ðe2e1Þð
=ð ?ðe1e1 =ð ?ð1
28/10/2003 Geometric Computation 2001 11
2D Basis
Build into a basis for the algebra
1 e1, e2 e1e2 =ð I
1 2 1
scalar vectors bivector
The even grade objects form the complex numbers.
Map between vectors and complex numbers:
y
X,z
x +ð Iy =ð e1Å»ðxe1 +ð ye2Þð =ð e1X
zDð =ð x ?ð Iy =ð Xe1
x
28/10/2003 Geometric Computation 2001 12
2D Rotations
" In 2D vectors can be rotated using complex phase
rotations
v =ð exp idð u
Å»ð Þð
v, y
dð u, x
u =ð e1x
v =ð e1y
y =ð e1v =ð e1 exp Idð e1x =ð e1 cos dð +ð I sin dð e1x
Å»ð Þð Å»ð Å»ð Þð Å»ð ÞðÞð
e1Ie1 =ð e1Å»ðe1e2Þðe1 =ð e2e1 =ð ?ðI
" But
" Rotation
y =ð exp ?ðIdð x =ð xexp Idð
Å»ð Þð Å»ð Þð
28/10/2003 Geometric Computation 2001 13
3 Dimensions
" Now introduce a third vector
e3
áðe1,e2, e3âð
" These all anticommute
e2
e1
e1e2 =ð ?ðe2e1 etc.
" Have 3 bivectors now
áðe1e2, e2e3, e3e1âð
e1e2
e3e1
e2e3
28/10/2003 Geometric Computation 2001 14
Bivector Products
" Various new products to form in 3D
" Product of a vector and a bivector
e1 e1e2 =ð e2 e1 e2e3 =ð e1e2e3 =ð I
Å»ð Þð Å»ð Þð
" Product of two perpendicular bivectors:
Å»ðe2e3ÞðÅ»ðe3e1Þð =ð e2e3e3e1 =ð e2e1 =ð ?ðe1e2
" Set
i =ð e2e3, j =ð ?ðe3e1, k =ð e1e2
" Recover quaternion relations
i2 =ð j2 =ð k2 =ð ijk =ð ?ð1
28/10/2003 Geometric Computation 2001 15
3D Pseudoscalar
I =ð e1e2e3
" 3D Pseudoscalar defined by
" Represents a directed volume element
" Has negative square
I2 =ð e1e2e3e1e2e3 =ð e2e3e2e3 =ð ?ð1
" Commutes with all vectors
e1I =ð e1e1e2e3 =ð ?ðe1e2e1e3 =ð e1e2e3e1 =ð Ie1
" Interchanges vectors and planes
e1I =ð e2e3
e2e3
Ie2e3 =ð ?ðe1
e1
28/10/2003 Geometric Computation 2001 16
3D Basis
Different grades correspond to different geometric
objects
Grade 0 Grade 1 Grade 2 Grade 3
Scalar Vector Bivector Trivector
1 e1, e2,e3 e1e2, e2e3,e3e1 I
eiej =ð Nðij +ð OðijkIek
Generators satisfy Pauli relations
a ×ð b =ð ?ðI a Tð b
Recover vector (cross) product
28/10/2003 Geometric Computation 2001 17
Reflections
" Build rotations from reflections
" Good example of geometric product  arises in
operations
b
aRð =ð Å»ða 6ð nÞðn
a
aóð =ð a ?ð Å»ða 6ð nÞðn
n
Image of reflection is
b =ð aóð ?ð aRð =ð a ?ð 2Å»ða 6ð nÞðn
=ð a ?ð Å»ðan +ð naÞðn =ð ?ðnan
28/10/2003 Geometric Computation 2001 18
Rotations
" Two rotations form a reflection
a ¸ð ?ðmÅ»ð?ðnanÞðm =ð mnanm
R =ð mn
" Define the rotor
" This is a geometric product! Rotations given by
a ¸ð RaR!ð R!ð =ð nm
" Works in spaces of any dimension or signature
" Works for all grades of multivectors
A Ìð RAR!ð
" More efficient than matrix multiplication
28/10/2003 Geometric Computation 2001 19
3D Rotations
" Rotors even grade (scalar + bivector in 3D)
RR!ð =ð mnnm =ð 1
" Normalised:
" Reduces d.o.f. from 4 to 3  enough for a rotation.
" In 3D a rotor is a normalised, even element
R =ð Jð +ð B RR!ð =ð Jð2 ?ð B2 =ð 1
R =ð exp ?ðB/2
Å»ð Þð
" Can also write
" Rotation in plane B with orientation of B
" In terms of an axis R =ð exp ?ðSðIn/2
Å»ð Þð
28/10/2003 Geometric Computation 2001 20
Group Manifold
" Rotors are elements of a 4D space, normalised to 1
" They lie on a 3-sphere
" This is the group manifold
" Tangent space is 3D
" Can use Euler angles
R =ð exp ?ðe1e2dð/2 exp ?ðe2e3Sð/2 exp ?ðe1e2fð/2
Å»ð Þð Å»ð Þð Å»ð Þð
" Rotors R and  R define the same rotation
" Rotation group manifold is more complicated
28/10/2003 Geometric Computation 2001 21
Lie Groups
" Every rotor can be written as
exp ?ðB/2
Å»ð Þð
" Rotors form a continuous (Lie) group
" Bivectors form a Lie algebra under the
commutator product
" All finite Lie groups are rotor groups
" All finite Lie algebras are bivector algebras
" (Infinite case not fully clear, yet)
" In conformal case (later) starting point of screw
theory (Clifford, 1870s)!
28/10/2003 Geometric Computation 2001 22
Rotor Interpolation
" How do we interpolate between 2 rotations?
" Form path between rotors
RÅ»ð0Þð =ð R0
RÅ»ðVðÞð =ð R0 exp VðB
Å»ð Þð
RÅ»ð1Þð =ð R1
exp B =ð R!ðR1
Å»ð Þð
" Find B from
0
" This path is invariant. If points transformed, path
transforms the same way
" Midpoint simply R 1/2 =ð R0 exp ?ðB/2
Å»ð Þð Å»ð Þð
" Works for all Lie groups
28/10/2003 Geometric Computation 2001 23
Interpolation 2
" For rotors in 3D can do even better!
R1
" View rotors as unit vectors in 4D
" Path is a circle in a plane
Sð
" Use simple trig to get SLERP
R0
1
RÅ»ðVðÞð =ð sin 1 ?ð Vð Sð R0 +ð sin VðSð R1
Å»ð Å»ðÅ»ð Þð Þð Å»ð Þð Þð
sinÅ»ðSðÞð
" For midpoint add the rotors and normalise!
sin Sð/2
Å»ð Þð
RÅ»ð1/2Þð =ð R0 +ð R1
Å»ð Þð
sin Sð
Å»ð Þð
28/10/2003 Geometric Computation 2001 24
Interpolation - Applications
" Rigid body dynamics  interpolating a discrete
series of configurations
" Elasticity of a rod or shell
" Framing a curve (either Frenet or parallel
transport)
" Calculus  Relax the norm and write
RAR!ð =ð fðAfð?ð1
" È belongs to a linear space. Has a natural
calculus. Very useful in computer vision.
28/10/2003 Geometric Computation 2001 25
Rotor Calculus
" Can simplify problems with rotations by relaxing
the normalisation and using unnormalised rotors
a ¸ð fðafð!ð/_ð_ð =ð fðfð!ð
" È belongs to a linear space  has an associated
calculus
" Start with obvious result
/ðfð fðA =ð A
Öð ×ð
" Now write a rotation as
RAR!ð =ð fðAfð?ð1
28/10/2003 Geometric Computation 2001 26
Rotor Calculus 2
" Key result (not hard to prove)
/ðfð fð?ð1A =ð ?ðfð?ð1Afð?ð1
Öð ×ð
" Complete the derivation
/ðfð fðafð?ð1b =ð afð?ð1b ?ð fð?ð1bfðafð?ð1
Öð ×ð
" Pre-multiply by
fð
fð/ðfð fðafð?ð1b =ð fðafð?ð1b ?ð bfðafð?ð1 =ð 2 RaR!ð Tð b
Öð ×ð Å»ð Þð
" Note the geometric product
" Result is a bivector
" Useful in many applications
28/10/2003 Geometric Computation 2001 27
Computer Vision
" Interested in reconstructing moving scenes from
fixed cameras. Various problems:
 Self calibration - the camera matrix. (Camera
parameters can change easily)
 Localization (Establish camera geometry from point
matches).
 Reconstruction - have to deal with occlusion, etc.
" Standard techniques make heavy use of linear
algebra (fundamental matrix, epipolar geometry)
" Rotor treatment provides better alternatives
28/10/2003 Geometric Computation 2001 28
Camera Auto-calibration
" Different camera views of the same scene
" Given point matches, recover relative translation
and rotation between cameras
" Applications
 Gait analysis
 Motion analysis
 Neurogeometry
 3D reconstruction
28/10/2003 Geometric Computation 2001 29
Known Range Data
" Start with simplified problem
X =ð Xiei Yvð =ð Yiei Y =ð Yifi
X ?ð t =ð Y =ð RYvðR!ð
" Minimise least square error
n
2
S =ð Xk ?ð t ?ð RYkvðR!ð
>ð
k=ð1
" Get this from a Bayesian argument with Gaussian
distribution
2
1
P Xi pð exp ?ð Xi ?ð ei 6ð X
Å»ð Þð Å»ð Þð
2að2
28/10/2003 Geometric Computation 2001 30
Known Range Data 2
" Minimise S wrt t and R
n
1
t =ð Xk ?ð RYkR!ð
Å»ð Þð
>ð
n
k=ð1
n
fð/ðfðS =ð ?ð4 >ðk=ð1 RYkvðR!ð Tð Xk ?ð t =ð 0
Å»ð Þð
" Substitute for t write as
eJð Tð eKð FJðKð =ð 0 FJðKð =ð >ðn eJð 6ð Ykvð eKð 6ð Xk
k=ð1
FJðKð
" Solve with an SVD of
" Solution is well known, though derivation extends
easily to arbitrary camera numbers
28/10/2003 Geometric Computation 2001 31
Camera Vision
" Usually only know projective vectors on camera
plane
a
n +ð OA =ð Vða n
n2 =ð 1
" Choose scale
A
a Tð n
a ?ð a 6ð nn
OA =ð =ð n
O
a 6ð n a 6ð n
" Represent line with bivector
a1 e1e3 +ð a2 e2e3
a Tð n
A =ð =ð
Homogenous
a 6ð n a3 a3
coordinates
28/10/2003 Geometric Computation 2001 32
Noise
" Assume camera matrix is known
" 2 main sources of noise:
 Pixelisation noise (finite resolution)
 Random noise (camera motion, etc.)
" Assume the noise is all Gaussian, and marginalize
over unknown range data
" Get likelihood function
vð 2
n
t Tð xk Tð Rxk R!ð
S =ð
>ð
2
xk Tð RxkvðR!ð
k=ð1
28/10/2003 Geometric Computation 2001 33
Interpretation
" For 2 cameras result has a
simple interpretation
" Minimise the line distance
Line distance
" Problem  the set-up is
scale invariant
" Set
t2 =ð 1
" Enforce this as a
constraint
28/10/2003 Geometric Computation 2001 34
Solution
" Include a Lagrange multiplier
n
Ixk Tð Å»ðRxkvðR!ðÞð
t 6ð nk 2 ?ð Vð t2 ?ð 1 nk =ð
Å»ð Þð Å»ð Þð
>ð
xk Tð Å»ðRxkvðR!ðÞð
k=ð1
" Quadratic in t so minimising gives
>ðn a 6ð nk nk =ð Vðt
k=ð1
F a =ð >ðn a 6ð nk nk
Å»ð Þð
" Define
k=ð1
" Minimise the lowest eigenvalue of F wrt R
" Numerically efficient and stable
28/10/2003 Geometric Computation 2001 35
n Cameras
" Previous scheme extends easily to arbitrary
camera numbers
" Marginalization integrals more complicated, but
can just use sum of individual line distances
" Include Lagrange multipliers for all constraints
n
S =ð t 6ð nk +ð t 6ð nk +ð t 6ð nk
Å»ð Þð2 Å»ð Þð2 Å»ð Þð2
>ð
12 23 31
12 23 31
k=ð1
?ð 2a 6ð t12 +ð t23 +ð t31 ?ð Vð t2 +ð t2 +ð t2 ?ð 1
Å»ð Þð Å»ð Þð
12 23 31
Scale
Consistency
28/10/2003 Geometric Computation 2001 36
Solution
" Now get 3 symmetric equations
>ðn t12 6ð nk nk =ð Vðt12 +ð a
etc
12 12
k=ð1
" With two constraints
t12 +ð t23 +ð t31 =ð 0 t2 +ð t2 +ð t2 =ð 1
12 23 31
" Re-express as a matrix problem
2F12 +ð Å»ðF23 +ð F31Þð/2 3 F23 ?ð F31 /2
Å»ð Þð
M =ð
3 F ?ð F /2 3 F +ð F
Å»ð Þð Å»ð Þð
23 31 23 31
Ma =ð 3Vða
28/10/2003 Geometric Computation 2001 37
Applications
" Single person walking on model of the
Millennium bridge
28/10/2003 Geometric Computation 2001 38
Applications 2
" 3 people falling in and out of phase
28/10/2003 Geometric Computation 2001 39
Prospects
" Expand to 7 cameras
" Sports motion analysis studies (running, golf,
tennis, hockey & )
" Realistic models to help deal with occlusions,
point crossing, etc.
" Without an internal model, reconstruction can
easily go wrong (example)
" Rigid models make heavy use of rotors for joints
" Include elasticity models for limbs?
28/10/2003 Geometric Computation 2001 40
Projective Geometry
" Remove the origin by introducing an extra dimension
" Points with vectors
" Lines with bivectors
a Tð b
" Planes with trivectors
" Homogeneous coordinates
b
a
" For 3-d space use a 4-d algebra
" Contains 6 bivectors, for lines in space
" Very useful in computer vision and graphics
28/10/2003 Geometric Computation 2001 41
Projective Line
" Simplest case. Suppose x is a real number and
x1, x2
introduce homogeneous coordinates
" Can translate and invert using
xvð a b x1
xvð
1
1
xvð =ð =ð
vð
x2
xvð c c x2
2
" This is a linear transformation, but not a rotor
group.
" Use the group relation
SLÅ»ð2, RÞð pð spinÅ»ð2, 1Þð
28/10/2003 Geometric Computation 2001 42
Mixed Signature
" Group relation tells us to consider space G2,1
" This is a mixed signature space, but no need to
modify any of our axioms.
" Basis set áðe1,e, 
âð e2 =ð e2 =ð 1 
2 =ð ?ð1
1
" Everything as before, but now have null vectors
n =ð e +ð 
n =ð e ?ð 

#ð
n2 =ð n2 =ð 0 n 6ð n =ð 2
#ð #ð
" Also have bivectors with positive square
Å»ðe
e
Þð2 =ð ?ðee

=ð +ð1
28/10/2003 Geometric Computation 2001 43
Map to R(2,1)
" Projective points under SL(2,R) have an invariant
inner product
a,b =ð a1b2 ?ð a2b1
Öð ×ð
" Analogy with 2-spinors tells us to form the matrix
x1 x1x2 ?ðx2
1
x2,?ðx1 =ð
Å»ð Þð
x2 x2 ?ðx1x2
2
" This has vector equivalent
1 1
x1x2e1 +ð x2n ?ð x2n
#ð
1 2
2 2
28/10/2003 Geometric Computation 2001 44
Conformal Geometry
" Projective geometry disposes with the origin as a
special point
" But infinity is not dealt with so easily
" Deal with this by first mapping a line to a circle
2x 1 ?ð x2
x ¸ð ,
1 +ð x2 1 +ð x2
" Introduce a projective dimension, with negative
signature, form
2x 1 ?ð x2 ,1 ið x, 1 ?ð x2 , 1 +ð x2
,
2 2
1 +ð x2 1 +ð x2
28/10/2003 Geometric Computation 2001 45
Null Vectors
" Scale is irrelevant, so have constructed the vector
X =ð 2xe1 +ð x2n ?ð n
#ð
" Simple to check that
Points as null vectors
X2 =ð 0
" In a 3d space conformal line is intersection of null
cone with a plane
" Result is a hyperbola
" Picture
28/10/2003 Geometric Computation 2001 46
Conformal Plane
" Similar trick works for the plane
" Map to complex numbers and go to homogeneous
complex coordinates
" Use group relation SLÅ»ð2, CÞð pð spinÅ»ð3,1Þð
" This time get null vectors in a 4d space
" Points in n dimensions represented as null vectors
in (n+1,1) space. This is the conformal rep
X =ð 2x +ð x2n ?ð n
#ð
28/10/2003 Geometric Computation 2001 47
Euclidean Geometry
" Inner product between vectors gives
A 6ð B =ð Å»ða2n +ð 2a ?ð nÞð 6ð Å»ðb2n +ð 2b ?ð nÞð
#ð #ð
=ð ?ð2Å»ða ?ð bÞð2
" Returns the Euclidean distance!
" This means that reflections and rotations in
conformal space preserve the Euclidean distance
" Provides a rotor formulation of translations,
inversions, etc.
28/10/2003 Geometric Computation 2001 48
Inversions
" Can carry out further Euclidean transforms
x Ìð x?ð1 =ð x/x2
" For conformal vector, effect is
X Ìð Xvð =ð x?ð2 n +ð 2x ?ð x2n
Å»ð #ð Þð
" Need to swap the null vectors
" Do this with a reflection
?ðene =ð n ?ð ene =ð n
#ð #ð
" So an inversion reduces to a reflection
X Ìð Xvð ið ?ðeXe
28/10/2003 Geometric Computation 2001 49
Translation
" Similar is true for translations. Define the rotor
na
R =ð exp na/2 =ð 1 +ð
Å»ð Þð
2
" Find that
na na
RXR!ð =ð 1 +ð x2n +ð 2x ?ð n 1 ?ð
#ð
Å»ð Þð
2 2
=ð x +ð a n +ð 2 x +ð a ?ð n
Å»ð Þð2 Å»ð Þð #ð
xvð =ð x +ð a
" So set
" Have
RXR!ð =ð xvð n +ð 2xvð ?ð n
Å»ð Þð2 #ð
" Translations performed by rotors!
28/10/2003 Geometric Computation 2001 50
Lines and Circles
" Lines are encoded as circles through 3 points
" Equation through A, B, C is simply
X Tð A Tð B Tð C =ð 0
" If one is the point at infinity, the line is straight
" Find centre with
c =ð TnT
" A reflection of infinity
" Also have inner product, gives
angle between circles
28/10/2003 Geometric Computation 2001 51
Spheres
" Similarly, 4 points define a sphere
X Tð A Tð B Tð C Tð D =ð 0
" A 4-vector in a 5-d algebra
" Introduce dual
a =ð I A Tð B Tð C Tð D
" A positive norm vector
a2
=ð _ð2
Å»ða 6ð nÞð2
Points are spheres of zero radius
28/10/2003 Geometric Computation 2001 52
Intersection and Normals
" Projective notions of meet and join extend to circles
and spheres
" Form the trivector
A 6ð Å»ðIBÞð
" Extends easily to all objects
" Circles, spheres etc are
oriented
" Multivectors carry round intrinsic definition of the
normal to a surface
" Useful for surface propagation
28/10/2003 Geometric Computation 2001 53
Examples
Thanks to Anthony
Lasenby
Conformal rep very
useful for spherical
propagation and
refection
28/10/2003 Geometric Computation 2001 54
Examples 2
" A 3D Example
" Plenty more of these!
" Currently implemented
in Maple
" Conformal algebra will
form basis for robust
algorithms
28/10/2003 Geometric Computation 2001 55
Summary
" Conformal geometry unifies Euclidean, spherical and
hyperbolic geometry
" Wealth of new techniques for CG industry
" Natural objects for data streaming
" Theorems proved in one geometry have immediate
counterparts in others
" Completes 19th century geometry
" Only realised in last 10 years!
28/10/2003 Geometric Computation 2001 56
Further Details
" All papers on Cambridge GA group website:
www.mrao.cam.ac.uk/~clifford
" Applications of GA to computer science and
engineering discussed at AGACSE 2001
conference. Next week!
www.mrao.cam.ac.uk/agacse2001
" A 1 day course on GA will be presented at
SIGGRAPH 2001 in Los Angeles
www.siggraph.org/s2001
" IMA Conference in Cambridge, 9th Sept 2002
28/10/2003 Geometric Computation 2001 57
Applications
" Computer graphics. Build robust algorithms with
control over data types
" Dynamics. Algebra of screw theory automatically
incorporated
" Surface evolution. Represent evolution and
reflections from a surface easily
" Robotics. Combine rotations and translations in a
single rotor
" Relativity. Forms the basis of twistor theory.
28/10/2003 Geometric Computation 2001 58