Bayro-Corrochano E., Scheuermann G. Geometric Algebra Computing: in Engineering and Computer Science

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28 D. Hestenes

because it is proportional to the Euclidean pseudoscalar I

=ie, which is invariant

because i and e are invariant. This concept and result is unique to CGA, so it merits

further discussion.

The pseudoscalar I

= ie squares to I

=−e

= 0, so it cannot be used for an

invertible duality mapping. However, we can deﬁne a conjugate pseudoscalar I

∗

≡

so that I

∗

·I

=I

∗

=−e ·e

=1. Then we can express the invariant (113)as

a scalar:

∧S

) ·(ie

) =m

·n

·S

∗

, (114)

where a dual screw = coscrew has been deﬁned by

∗

≡S

·(ie

) =S



=−in

−e

. (115)

This equivalent to the reciprocal screw ﬁrst introduced by Ball [17]. Comparison

with (97) shows that wrenches are coscrews! Of course, the dual deﬁned here should

not be confused with the dual introduced in (11). The asterisk notation has been used

for both to emphasize their conceptual commonality.

Finally, we note that for a single screw, the pitch h appears as a ratio of the

invariants (114) and (112):

h =−

S ·S

∗

S ·S

=n ·m

−1

. (116)

Screw theory is basically about the generators of displacements. The simplicity of

its formulation within CGA belies the richness and complexity of its applications in

mechanical engineering, for which the serious student must consult the literature.

As guides to the screw theory literature, I recommend two books. The ﬁrst book

[16] concerns use of modern mathematical concepts and notation, which can be

compared to the approach taken here. The second book [18] is by two long-time

practitioners of screw theory. Though it is designed as a textbook for the ill-prepared

engineering student of today, it provides a mature perspective on current status with

an authoritative entrée to the literature.

Screw theory literature going back to the nineteenth century contains many gems

that can be recovered by reformulation within CGA. Here is another worthy topic

for doctoral research. It requires more than historical study, for many of the gems

need polishing—there are unsolved problems to be addressed in the new light of

CGA.

To facilitate translations from the literature, relations between CGA and matrix

algebra are established next.

13 Conformal Split and Matrix Representation

Besides its invariance and incredible compactness, one great advantage of the CGA

formulation of rigid body mechanics in the preceding sections is the ease of relating

it to alternative formulations by a conformal split. In this section we consider the

New Tools for Computational Geometry 29

Fig. 13

conformal split in more detail, especially to clarify and facilitate connections to the

vast literature on mechanical systems.

As deﬁned in (43), one factor in the conformal split is the geometric algebra

1,1

=G(R

1,1

). The vector space R

1,1

is sometimes referred to as 2D Minkowski

space to emphasize its similarity to 4D Minkowski space R

3,1

, which is a standard

model for spacetime in relativistic physics [4, 6]. It can be generated from a null

basis {e,e

| e

= e

= 0,e · e

=−1} or from an equivalent orthonormal basis

√

(λe ∓λ

−1

), λ =0,e

=±1}.

Though the orthonormal basis is more familiar to most readers, as we have

seen already, the null basis is more signiﬁcant geometrically. It generates a basis

{1,e,e

,E} (depicted in Fig. 13) for the entire algebra G

1,1

, with the basic proper-

ties

=1,e

e =E −1,Ee=−eE =e, e

E =−Ee

. (117)

These properties have been used many times in previous sections.

The algebra of dual numbers D ={α + eβ } is an important subalgebra of G

1,1

However, it was ﬁrst proposed by Clifford as an extension of the real numbers anal-

ogous to complex numbers, with the null unit e replacing the imaginary unit i.He

introduced it as an extension of scalars in quaternions to form what he called bi-

quaternions [10, 16]. We can regard it as an extension of the algebra G

to G

⊗D.

Clifford clearly recognized the geometric signiﬁcance of this extension for incorpo-

rating the additivity and commutativity properties of translations. In terms of trans-

lation versors, these properties are expressed by



1 +



1 +



=1 +

(a +b)e =T

a+b

. (118)

Clifford’s biquaternions have been used to represent translations and screws by

many authors since. However, we have seen in the preceding section that the dual

numbers must be extended to the entire algebra G

1,1

to accommodate coscrews and

screw invariants. That has been done in the literature primarily by employing ma-

trices in the following way. The algebra G

1,1

is isomorphic to the algebra M

(R)

of real 2 ×2 matrices. That is readily established by exhibiting the isomorphism of

30 D. Hestenes

basis elements:







−





−10



E 



0 −1



, 1 





(119)

Accordingly, every multivector M in G

1,1

has a matrix representation

M explicitly

given by

M =

A(1+E)+B(e

−

)+C(e

−e

−

)+D(1−E) 

M =





, (120)

where the matrix elements are real numbers. This representation is readily gen-

eralized by allowing the matrix elements to have values in other algebras, G

particular. Thus, we arrive at the isomorphism:

4,1

⊗G

1,1

M





. (121)

Properties of this isomorphism are surprisingly rich and have been thoroughly stud-

iedin[13]. That enabled a critique of the matrix representation for the conformal

group, which contributed to developing the invariant formulation in CGA introduced

in [1].

The matrix algebra M

) has been much used in screw theory with the ele-

ments of G

interpreted as complex quaternions. More often, it has been used with

the elements of G

represented as 3×3 matrices or column vectors. The alternatives

are best explained by a representative example.

With an obvious change of notation, we can write (103) for the change in screw

coordinates induced by a shift r = x

− x

from base point P to point Q in the

form

=ev

−iω =T

=e(v

−r ×ω) −iω. (122)

This has a matrix representation

with the explicit form







1 −r×







−r ×ω



. (123)

Similarly, we can write (100) for the induced change of coscrew coordinates in the

form

=−e

f +iΓ

∗

=−e

f +i(Γ

+r ×f). (124)

Its matrix representation

∗

has the explicit form







−r× 1







+r ×f



. (125)

These four equations sufﬁce to show how any equation in the literature on screw

theory or robotics can be translated into CGA and vice versa. For example, in [18]

New Tools for Computational Geometry 31

the screws in (123) and coscrews in (125) are represented as 6-component column

vectors. This example reveals a signiﬁcant drawback of the matrix representations:

the matrices do not encode the distinction between screws and coscrews, which,

in contrast, is explicitly expressed by the distinction between e and e

in (122) and

(124). Expressed in more general terms: the matrix representations suppress the geo-

metric meaning of matrix elements, which is explicitly encoded in the algebra G

1,1

Furthermore, the matrix representation (121) implicitly forces one to adopt a con-

formal split, which means, as we have seen, that one is forced into a covariant rather

than invariant approach to geometry. Nevertheless the isomorphism G

1,1

M

(R)

is essential for relating CGA to the robotics literature.

14 Linked Rigid Bodies & Robotics

The potential for application of CGA to robotics is best illustrated by a simple ex-

ample. Figure 14 depicts a kinematic chain with three segments in a reference pose:

+a +b +c. (126)

Rotations at its three joints are speciﬁed by twistors {R

}. Rotation at the

ﬁrst joint gives

+a +b +R

−1

+a +b +c

. (127)

Subsequent or concurrent rotation at the second joint gives

+a +R



b +R

−1



−1

+a +b

. (128)

Finally, the net result of rotations at all the joints is general pose:

x =e



a +R



b +R

−1



−1



−1

321

. (129)

Fig. 14

32 D. Hestenes

A conformal split with the ﬁxed point e

gives the position vector for the endpoint:

x =x ∧E =R



a +R



b +R

−1



−1



−1

321

. (130)

Of course, the rotations need not be conﬁned to a plane (as presumed in Fig. 14

for simplicity of illustration). Restrictions on the range of motion at each joint are

encoded in the twistors. For example, a rotation R

with one degree of freedom can

be given the angular form

=exp



−



, where 0 ≤α

≤π, (131)

and unit vector n

is the direction of the joint axis for a right-handed rotation. Kine-

matics of the chain can be described a follows. Irrespective of how the joints are

characterized, the twistors satisfy equations of the form

=−

iω

. (132)

Hence, for R

,wehave

=−

iω

with ω

=ω

−1

. (133)

Finally, with ω

321

= ω

+ R

−1

+ R

−1

, for the derivative of the

end point position vector (130), we get

x =ω

×a

+ω

×b

+ω

321

×c

321

. (134)

This is only the beginning for application of CGA to robotics, but we have all the

theoretical machinery we need for any task. To incorporate dynamics we introduce

the inertia properties of each body with (95) and the applied wrenches with (97).

Moreover, CGA offers a promising approach to modeling complex interactions be-

tween bodies, such as viscoelastic coupling at joints [2].

The next phase in the development of CGA robotics is detailed applications to

speciﬁc problems. This development is already underway by attendees at this con-

ference and others in the GA community. However, I see a need for more systematic

mining of the robotics literature to incorporate established problems, results, and

methods in CGA and promote broader interaction within the engineering commu-

nity. I thought about offering suggestions for literature to consult. But the robotics

literature is so vast and variable in complexity and quality that I fear my suggestions

could be as misleading as helpful. Consequently, I add only one recent reference

[19] to those I have already mentioned.

To sum up, CGA provides a powerful mathematical framework for robotics R&D

with the twin goals of (1) simplicity and clarity in mathematical formulation, (2) ef-

ﬁciency and speed in computation.

Acknowledgement I thank Arvid Halma and Leo Dorst for assistance in preparing the docu-

ment.

New Tools for Computational Geometry 33

References

1. Hestenes, D.: Old Wine in New Bottles: A new algebraic framework for computational ge-

ometry. In: Bayro-Corrochano, E., Sobczyk, G. (eds.) Advances in Geometric Algebra with

Applications in Science and Engineering, pp. 1–14. Birkhäuser, Basel (2001)

2. Hestenes, D., Fasse, E.: Homogeneous rigid body mechanics with elastic coupling. In:

Dorst, L., Doran, C., Lasenby, J. (eds.) Applications of Geometric Algebra in Computer Sci-

ence and Engineering, pp. 197–212. Birkhäuser, Basel (2002)

3. Hestenes, D.: A uniﬁed language for mathematics and physics. In: Chisholm, J.S.R., Com-

mon, A.K. (eds.) Clifford Algebras and their Applications in Mathematica Physics, pp. 1–23.

Reidel, Dordrecht (1986)

4. Hestenes, D.: Space-Time Algebra. Gordon & Breach, New York (1966)

5. Hestenes, D., Sobczyk, G.: C

LIFFORD ALGEBRA TO GEOMETRIC CALCULUS, a uniﬁed lan-

guage for mathematics and physics. Kluwer, Dordrecht (1984). Paperback (1985). Fourth

printing 1999

6. Doran, C., Lasenby, A.: Geometric Algebra for Physicists. Cambridge University Press, Cam-

bridge (2002)

7. Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science. Morgan Kauf-

mann Publ., Elsevier, San Mateo, Amsterdam (2007/2009)

8. Li, H.: Invariant Algebras and Geometric Reasoning. World Scientiﬁc, Singapore (2008)

9. Hestenes, D.: Grassmann’s vision. In: Schubring, G. (ed.) Hermann Günther Grassmann

(1809–1877)—Visionary Scientist and Neohumanist Scholar, pp. 191–201. Kluwer, Dor-

drecht (1996)

10. Clifford, W.K.: Mathematical Papers. Macmillan, London (1882). Ed. by R. Tucker. Reprinted

by Chelsea, New York (1968)

11. Doran, C., Hestenes, D., Sommen, F., Van Acker, N.: Lie groups as spin groups. J. Math. Phys.

34, 3642–3669 (1993)

12. Lasenby, A.: Recent applications of conformal geometric algebra. In: Li, H., et al. (ed.) Com-

puter Algebra and Geometric Algebra with Applications. LNCS, vol. 3519, pp. 298–328.

Springer, Berlin (2005)

13. Hestenes, D.: The design of linear algebra and geometry. Acta Appl. Math. 23, 65–93 (1991)

14. Onishchik, A., Sulanke, R.: Projective and Cayley–Klein Geometries. Springer, Berlin (2006)

15. Hestenes, D.: New Foundations for Classical Mechanics. Kluwer, Dordrecht (1986). Paper-

back (1987). Second edition (1999)

16. Selig, J.: Geometrical Methods in Robotics. Springer, Berlin (1996)

17. Ball, R.S.: A Treatise on the Theory of Screws. Cambridge University Press, Cambridge

(1900). Reprinted in paperback (1998)

18. Davidson, J., Hunt, K.: Robots and Screw Theory: Applications of Kinematics and Statics to

Robotics. Oxford University Press, London (2004)

19. Featherstone, R., Orin, D.: Dynamics. In: Siciliano, B., Khatib, O. (eds.) Springer Handbook

of Robotics, pp. 35–65. Springer, Berlin (2008)

Tutorial: Structure-Preserving Representation

of Euclidean Motions Through Conformal

Geometric Algebra

Leo Dorst

Abstract A new and useful set of homogeneous coordinates has been discovered

for the treatment of Euclidean geometry. They render Euclidean motions not merely

linear (as the classical homogeneous coordinates do), but even turn them into or-

thogonal transformations, through a clever choice of metric in two (not one) addi-

tional dimensions.

To take full advantage of this new possibility, a good representation of orthogo-

nal transformations is required. We ﬁnd that multiple reﬂections, while classically

giving unwieldy expressions involving the dot product, become practical by intro-

ducing the more fundamental geometric product (which has the dot product merely

as its symmetric part). We obtain a sandwiching operation between products of vec-

tors as our representation of motions, which is not only easily concatenated, but also

incorporates the computational advantages of complex numbers and quaternions in

a real manner. The antisymmetric part of the geometric product produces a spanning

operation that permits the construction of lines, planes, spheres and tangents from

vectors. Since the sandwiching operation distributes over this construction, ‘objects’

are fully integrated with ‘motions’, in a structure preserving manner.

Additional techniques such as duality (permitting a universal intersection opera-

tion), and the rewriting of operators logarithmically (to obtain quantities that can be

interpolated linearly) complete the techniques of what is ultimately a very conve-

nient geometric algebra. It easily incorporates general conformal transformations,

and can be implemented to run almost as efﬁciently as classical homogeneous coor-

dinates. The resulting high-level programming language naturally integrates quan-

titative computation with the automatic administration of geometric data structures.

Rather than the usual introductions which plow through Clifford algebra before

they reach this very useful ‘conformal model’ (if they do at all), this tutorial does

the reverse. It structures the new concepts in a manner that shows how each ad-

L. Dorst (



)

Intelligent Systems Laboratory, University of Amsterdam, Science Park 107, 1098 XG,

Amsterdam, The Netherlands

e-mail: L.Dorst@uva.nl

E. Bayro-Corrochano, G. Scheuermann (eds.), Geometric Algebra Computing,

36 L. Dorst

ditional sophistication is related to what went before, and how it extends its ex-

pressive and computational power. Throughout, the concepts and techniques will

be illustrated by interactive visualization software (GAviewer, freely available at

www.geometricalgebra.net).

1 Introduction

“Doing geometry” in computer science or engineering requires at least the following

ingredients in a practical computational framework:

• descriptive primitives: such as points, lines, planes, circles, spheres, tangents

• basic constructions: connections, intersections, parametric speciﬁcation

• motions: translation, rotation, reﬂection, projection

• properties: size, location, orientation

• practical numerics: approximation, estimation, interpolation, linearization

These ingredients should interweave seamlessly. Notably, the framework should be

structure preserving, in the sense that constructions and properties of primitives

should be covariant under motions. For instance, when moving a circle determined

by three points, it should not be necessary to decompose the circle back into the

points, move those, and then reconstruct; rather, the circle should be a basic element

of computation with an associated motion operator (which should moreover prefer-

ably be identical to that for points). Also, all ingredients should be speciﬁable in a

sufﬁciently high-level programming language, which avoids coordinates as speciﬁ-

cation level though it may revert to them when executing the operations. The usual

linear algebra tools have neither of these desirable properties, not even when us-

ing homogeneous coordinates. Yet a practical computational framework exists that

can do all of the above. It is called “conformal geometric algebra” (CGA), and this

chapter brieﬂy exposes its essential structure. We will explain all elements of Fig. 1,

and more.

2 Conformal Geometric Algebra

2.1 Trick 1: Representing Euclidean Points in Minkowski Space

Let us focus on a 3D space in which we want to perform Euclidean motions. We

can consider it as a 3D vector space and use a position vector x to denote a point

X relative to an (arbitrary) origin. This is naive practice, and not very convenient,

since Euclidean motions are then not even linear transformations. A commonly used

improvement is the homogeneous model, in which the space is augmented with an

extra dimension e

, and the point X at x represented as e

+x. Now Euclidean mo-

tions are linear transformations but still not structure preserving. More is required.

Structure Preserving Motions Through CGA 37

Fig. 1 An example of the ease of CGA (from [2]). A circle C is generated from three points c

, c

as C = c

∧c

. A line is given as a 3-blade L.ThecircleC is to be rotated around the

line L, producing RCR

−1

,withR speciﬁed as R = exp(L

∗

φ/2). The rotation is interpolated in

k steps using R

1/k

. Then the whole scene is reﬂected in the plane π given by a normal vector n

and a point p on it as π =p · (n ∧e

∞

); any element X is reﬂected as X →(−1)

grade(X)

πXπ

−1

In appropriate software such as [3], these coordinate-free formulas are the literal speciﬁcation of a

computer program producing the scene

In CGA, we introduce two extra dimensions for representational purposes, thus

constructing a ﬁve-dimensional space. We introduce two basis vectors for these extra

dimensions, e

and e

∞

, and the speciﬁc metric given below. As we will see, the

null vectors in this extended space (i.e., the vectors x satisfying x · x = 0inthe

chosen metric) represent weighted points in the Euclidean space (though one usually

employs unit weight points satisfying x ·e

∞

=0). Such vectors representing points

have algebraic properties to construct other elements in a coordinate-free, invariant

manner, as explained in the paper by Hestenes [5] elsewhere in this volume.

In the present introductory paper, we mostly prefer to use an explicit expression

for such a vector x representing a point X, relating it to the “classical” Euclidean

position vector x of the point relative to the chosen origin through

x =e

+x +

x

∞

. (1)