A Geometric Algebra Invitation to Space-Time Physics, Robotics and Molecular Geometry

By Carlile Lavor, Sebastià Xambó-Descamps and Isiah Zaplana

This book offers a gentle introduction to key elements of Geometric Algebra, along with their applications in Physics, Robotics and Molecular Geometry. Major applications covered are the physics of space-time, including Maxwell electromagnetism and the Dirac equation; robotics, including formulations for the forward and inverse kinematics and an overview of the singularity problem for serial robots; and molecular geometry, with 3D-protein structure calculations using NMR data. The book is primarily intended for graduate students and advanced undergraduates in related fields, but can also benefit professionals in search of a pedagogical presentation of these subjects.

• Language: English
• Publisher: Springer
• Release date: Jul 12, 2018
• ISBN: 9783319906652

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© The Author(s), under exclusive licence to Springer International Publishing AG, part of Springer Nature 2018

Carlile Lavor, Sebastià Xambó-Descamps and Isiah ZaplanaA Geometric Algebra Invitation to Space-Time Physics, Robotics and Molecular GeometrySpringerBriefs in Mathematicshttps://doi.org/10.1007/978-3-319-90665-2_1

1. Low Dimensional Geometric Algebras

Carlile Lavor¹ , Sebastià Xambó-Descamps² and Isiah Zaplana³

(1)

Department of Applied Maths (IMECC-UNICAMP), University of Campinas, Campinas, SP, Brazil

(2)

Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Barcelona, Spain

(3)

Institut d’Org. i Control de Sist. Ind., Universitat Politècnica de Catalunya, Barcelona, Barcelona, Spain

The concept of geometric algebra (GA) arises out of the desire to multiply vectors with the usual rules of multiplying numbers, including the usual rules for taking inverses. From that point of view, the construction of GA is an instance of a powerful mechanism used in mathematics that may be described as creating virtue out of necessity. In general, this mechanism comes to the rescue when the need arises to extend a given structure in order to include desirable features that are not present in that structure.

As an aside, and for the purpose of illustrating how the mechanism works in more familiar grounds, let us consider the successive extensions of the notion of number:

$$\displaystyle \begin{aligned} \mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}\subset \mathbb{R}\subset \mathbb{C}. \end{aligned}$$

In the natural numbers,

$$\mathbb {N}=\{1,2,3,\dotsc \}$$

, the difference x = a − b ( $$a,b\in \mathbb {N}$$ ), which by definition satisfies a = b + x, is defined only when a > b. The need to be able to subtract any two numbers leads to the introduction of 0 and the negative numbers − a (for all $$a\in \mathbb {N}$$ ). The extension

$$\mathbb {Z}=\{\dots ,-3,-2,-1,0,1,2,3,\dotsc \}$$

of $$\mathbb {N}$$ is the set of integers. The order, addition, and subtraction can be extended in a natural way from $$\mathbb {N}$$ to $$\mathbb {Z}$$ and after that the equation a = b + x has a unique solution x for any $$a,b\in \mathbb {Z}$$ . In other words, the difference x = a − b of any two integers is always well defined. For $$a,b\in \mathbb {N}$$ , for example, x = a − b if a > b, 0 if a = b, − (b − a) if a < b. The bottom line is that $$\mathbb {Z}$$ implements the order, addition, and subtraction of formal differences a − b of natural numbers a and b, with the constraints that a − b = a′− b′ (respectively a − b < a′− b′) if and only if a + b′ = a′ + b (respectively a + b′ < a′ + b) in $$\mathbb {N}$$ .

Now the division x = a∕b (b≠0) is possible in $$\mathbb {Z}$$ precisely when b is a divisor of a. In fact, to say that the equation a = bx can be solved for $$x\in \mathbb {Z}$$ just says that a is a multiple of b. The wish to overcome this limitation of the integers leads to the introduction of fractions or rational numbers,

$$\mathbb {Q} =\{a/b : a,b\in \mathbb {Z},\; b\ne 0\}$$

. The bottom line here is that $$\mathbb {Q}$$ implements the order, addition, and subtraction of formal quotients a∕b of integers a and b (b≠0), with the constraints that a∕b = a′∕b′ (respectively a∕b < a′∕b′) if and only if ab′ = a′b (respectively ab′ < a′b) in $$\mathbb {Z}$$ . In particular, the equation a = bx ( $$a,b\in \mathbb {Z}$$ , b≠0) can be solved uniquely for x in $$\mathbb {Q}$$ : x = a∕b.

The real numbers $$\mathbb {R}$$ can be introduced as the natural extension of $$\mathbb {Q}$$ that makes possible to take the least upper bound of upper bounded sets, and $$\mathbb {C}$$ is the natural extension of $$\mathbb {R}$$ in which − 1 has a square root: $$i=\sqrt {-1}$$ . Operationally, the number i is manipulated so that i ² = −1.

At this point it is worthwhile to remark that the mechanism is more fertile than what it might appear at a first sight. As a rule, the new structures obtained to overcome some limitations of more primitive ones have a richness that goes far beyond the original specifications, both by having interesting unexpected features and for its capacity to suggest other potentially useful structures through analogy and generalization. We will see this at work throughout this brief, notably in the case of geometric algebra, and also in a number of scattered comments.

Here it should be sufficient to recall a couple of examples. Given any positive number $$a\in \mathbb {R}$$ , it turns out that it has a unique positive real nth root $$r=\sqrt [n]{a}$$ for any $$n\in \mathbb {N}$$ (this means that r n = a), a fact that is not true in $$\mathbb {Q}$$ , as reminded by the old Pythagoric story telling us that $$\sqrt {2}$$ cannot be a rational number (see E.1.1, p. 31).

In the same vein, when we accepted that there is i such that i ² = −1, and thus extending $$\mathbb {R}$$ to $$\mathbb {C}$$ , how could we suspect that for any non-zero $$z\in \mathbb {C}$$ , and $$n\in \mathbb {N}$$ , there are exactly n numbers $$\xi \in \mathbb {C}$$ such that ξ n = z (nth roots of z)? For example, (see E.1.2, p. 32), the nth roots of 1 are

$$e^{2\pi i k/n}=\cos {}(2\pi k/n)+i\sin {}(2\pi k/n)$$

( $$0\leqslant k<n$$ ).

In the case of GA, among the unexpected properties beyond its specification (which is the wish to multiply vectors as if they were numbers), we will find that it is capable of representing in a coordinate-free way both geometrical concepts and geometric operations on them. Moreover, these two roles are naturally related in a way that will be made precise in due time and which we call geometric covariance.

The aim of this chapter is to introduce and study some of the concrete geometric algebras that will be used in the remaining chapters. These include the geometric algebras $$\mathcal {G}_2$$ and $$\mathcal {G}_3$$ of the Euclidean plane E 2 and the Euclidean space E 3 (Sections 2 and 3, respectively) and the geometric algebra $$\mathcal {G}_{1,3}$$ of the Minkowski space E 1,3. To pave the way to later chapters, in the Euclidean cases we also provide details about how the $$\mathcal {G}_2$$ and $$\mathcal {G}_3$$ encode geometric notions and geometric transformations.

Convention. If f is a map and x an object (say a linear map and a vector), we allow ourselves to (optionally) write fx, instead of f(x), to denote the image of x by f. This device, which is a common practice in functional programming languages, is useful to increase the readability of expressions in contexts where no confusion can arise about the nature of f and x.

1.1 Linear Algebra Background

We assume that the reader is familiar with some basic notions of linear algebra. For reference convenience, here is a summary of what we need in the sequel.

By a vector space we mean a real vector space (also called an $$\mathbb {R}$$ -vector space). The elements of $$\mathbb {R}$$ are called scalars and will be denoted by Greek letters: α, β, ….

In each concrete case of geometric algebra, the starting point is a vector space E of finite dimension n (with $$n\leqslant 4$$ in this chapter). Its elements are denoted by boldface italic characters (e, u, v, x, y, …).

The vector subspace of E spanned by vectors x 1, …, x k (that is, the set of all linear combinations λ 1 x 1 + ⋯ + λ k x k,

$$\lambda _1,\dotsc ,\lambda _k\in \mathbb {R}$$

) will be denoted by 〈x 1, …, x k〉.

The symbol e will stand for a basis e 1, …, e n of E. In principle it is arbitrary, but often it will be assumed to have specific properties that in each case will be declared explicitly.

If E′ is another vector space, a map f : E → E′ is said to be linear if f(λx) = λfx and f(x + y) = fx + fy for all x, y ∈ E and $$\lambda \in \mathbb {R}$$ .

1.1.1 (Construction of linear maps)

The main device to construct linear maps is the following observation: If we are given any vectors

$$\boldsymbol {e}^{\prime }_1,\dotsc ,\boldsymbol {e}^{\prime }_n\in E^{\prime }$$

, then there is a unique linear map f : E → E′ such that $$f\boldsymbol {e}_j = \boldsymbol {e}^{\prime }_j$$ for j = 1, …, n. □

Metrics

The vector space E on which geometric algebra is grounded is supposed to be equipped with a metric. By this we understand a non-degenerate (or regular) symmetric bilinear form

$$q:E\times E\to \mathbb {R}$$

. Recall that the non-degenerate condition means that for any given vector x≠0 we can find a vector y such that q(x, y)≠0 or, equivalently, q(x, y) = 0 for all y implies that x = 0. Instead of q(x, x), which is the quadratic form associated with q, we will simply write q(x). Note that q(λx) = λ ² q(x).

1.1.2 (Polarization identity)

The quadratic form determines the metric:

$$\displaystyle \begin{aligned} 2q(\boldsymbol{x},\boldsymbol{y})=q(\boldsymbol{x}+\boldsymbol{y})-q(\boldsymbol{x})-q(\boldsymbol{y}) \ \mathrm{\;for all\ \;} \boldsymbol{x},\boldsymbol{y}\in E. \end{aligned}$$

Proof

Use the bilinear property and the symmetry of q:

$$\displaystyle \begin{aligned} \hspace{4.5em}q(\boldsymbol{x}+\boldsymbol{y})&amp;=q(\boldsymbol{x}+\boldsymbol{y},\boldsymbol{x}+\boldsymbol{y})=q(\boldsymbol{x},\boldsymbol{x})+q(\boldsymbol{x},\boldsymbol{y})+q(\boldsymbol{y},\boldsymbol{x})+q(\boldsymbol{y},\boldsymbol{y})\\ &amp;=q(\boldsymbol{x})+q(\boldsymbol{y})+2q(\boldsymbol{x},\boldsymbol{y}).\end{aligned} $$

□

Two vectors x, y ∈ E are said to be orthogonal precisely when q(x, y) = 0. The basis e is orthogonal if q(e j, e k) = 0 for all j≠k. As is well known, and easy to proof, orthogonal basis exist for any q (E.1.3, p. 32).

A q-isometry of E (or just isometry if q is understood from the context) is a linear map f : E → E such that q(fv, fv′) = q(v, v′) for all v, v′∈ E. Using the polarization identity 1.1.2, we see that f is an isometry if and only if q(fv) = q(v) for all vectors v. With the operation of composition, the set of q-isometries forms a group, with the identity map Id as its neutral element. It is called the orthogonal group of q and is denoted by Oq.

Note that an isometry maps orthogonal vectors to orthogonal vectors.

Euclidean Spaces

Many authors take $$\mathbb {R}^n$$ as a model for the n-dimensional Euclidean vector space, but we prefer to denote it E n (or E if the dimension is clear from the context) to stress that no basis is assigned a preferred role. So E n is a real vector space of dimension n endowed with an Euclidean metric q, which means that q(v) = q(v, v) > 0 for any non-zero vector v (note that q(0) = 0 follows from the bilinearity of q). We also say that q is positive definite.

The length (or norm) of a vector v is denoted by |v| and is defined by the formula

$$\displaystyle \begin{aligned} |\boldsymbol{v}|=\sqrt{q(\boldsymbol{v})}. \end{aligned} $$

(1.1)

Thus |v| > 0 for v≠0 and q(v) = |v|² for any v.

A vector u such that q(u) = 1 is said to be a unit vector. For any non-zero vector v, ±v∕|v| are the only unit vectors of the form λv ( $$\lambda \in \mathbb {R}$$ ), and v∕|v| is said to be the normalization of v.

Notice that if we normalize the vectors of an orthogonal basis we get an orthogonal basis of unit vectors. Such bases are said to be orthonormal.

The angle α = α(v, v′) ∈ [0, π] between two non-zero vectors v and v′ is the real number α defined by the relation

$$\displaystyle \begin{aligned} \cos \alpha = q(\boldsymbol{v},\boldsymbol{v}')/|\boldsymbol{v}||\boldsymbol{v}'|. \end{aligned} $$

(1.2)

By the Cauchy-Schwarz inequality (see E.1.4, p. 32), this is well defined. Moreover, α ∈ (0, π) when v and v′ are linearly independent and α = 0 (α = π) when v′ = λv with λ > 0 (λ < 0) . Note also that α = π∕2 if and only if q(v, v′) = 0, that is, if and only if v and v′ are orthogonal (in the Euclidean case, the term perpendicular may be used instead).

The isometry group of E n (orthogonal group) will be denoted On.

Algebras

By an algebra we understand a non-zero vector space $$\mathcal {A}$$ endowed with a bilinear product

$$\mathcal {A}\times \mathcal {A}\to \mathcal {A}$$

, (x, y)↦x ∗ y. Unless declared explicitly otherwise, we also assume that the product is associative, (x ∗ y) ∗ z = x ∗ (y ∗ z), and unital (that is, there is $$1_{\mathcal {A}}\in \mathcal {A}$$ such that $$1_{\mathcal {A}}\ne 0_{\mathcal {A}}$$ and

$$1_{\mathcal {A}}*x=x*1_{\mathcal {A}}=x$$

for all $$x\in \mathcal {A}$$ ).

1.1.3 (Example: The matrix algebra)

For any positive integer n, the vector space $$\mathbb {R}(n)$$ of n × n real matrices is an algebra with the usual matrix product. Its unit is the matrix I n that has 1 in the main diagonal and 0 elsewhere (identity matrix).

Later we will use the following observation about $$\mathbb {R}(2)$$ . Let

../images/455429_1_En_1_Chapter/455429_1_En_1_Equd_HTML.png

Then it is easily checked that

$$\{I_2,{{\mathfrak {e}}}_1,{{\mathfrak {e}}}_2,{{\mathfrak {e}}}_1{{\mathfrak {e}}}_2\}$$

is a basis of $$\mathbb {R}(2)$$ and that the following relations hold:

$${{\mathfrak {e}}}_1^2={{\mathfrak {e}}}_2^2=I_2$$

,

$${{\mathfrak {e}}}_1{{\mathfrak {e}}}_2+{{\mathfrak {e}}}_2{{\mathfrak {e}}}_1=0$$

.

The map $$\mathbb {R}\to \mathcal {A}$$ , $$\lambda \mapsto \lambda 1_{\mathcal {A}}$$ , allows us to regard $$\mathbb {R}$$ as embedded in $$\mathcal {A}$$ , and so we will not distinguish between $$\lambda \in \mathbb {R}$$ and $$\lambda 1_{\mathcal {A}}\in \mathcal {A}$$ .

Exterior Powers and Exterior Algebra

The exterior powers and the exterior algebra of E, ∧k E and ∧E, were discovered by H. Grassmann [37–39]. They do not depend on the metric q, but we have postponed its recall because they are a little more abstract, and this should not hide that they have a clear geometric meaning and are quite manageable in practice. In any case, we will use the exterior algebra to determine the (graded) linear structure of geometric algebras and other related concepts.

Let E k ( $$k\in \mathbb {N}$$ ) be the kth Cartesian power of E. It is the vector space whose elements are k-tuples of vectors (x 1, …, x k). The exterior power ∧k E ( $$1\leqslant k\leqslant n$$ ) is a vector space endowed with a skew-symmetric multilinear map

$$\displaystyle \begin{aligned} \wedge: E^k\to {{\wedge}}^kE,\quad (\boldsymbol{x}_1,\dotsc,\boldsymbol{x}_k)\mapsto \boldsymbol{x}_1\wedge\cdots\wedge\boldsymbol{x}_k. \end{aligned}$$

Recall that a map is multilinear if it is linear in each of its variables, for an arbitrary value of the remaining variables, and that it is skew-symmetric if it changes sign when any two consecutive variables are swapped. The elements of ∧k E are called k-vectors and a k-blade is a non-zero k-vector of the form x 1 ∧⋯ ∧x k (k-blades are also called decomposable k-vectors). It is to be thought as the oriented k-volume determined by the vectors x 1, …, x k (oriented area and volume for k = 2 and k = 3). Algebraically, the skew-symmetric condition is reflected by the fact that x 1 ∧⋯ ∧x k vanishes if and only if x 1, …, x k are linearly dependent. The restriction $$k\leqslant n$$ arises from the fact that there are no non-zero k-volumes for k > n, a point that can also be expressed by declaring that ∧k E = {0} for k > n. For k = 1, ∧¹ E = E and ∧ : E →∧¹ E is the identity, while $${{\wedge }}^0E=\mathbb {R}$$ by convention.

1.1.4 (Universal property)

The fundamental property of ∧k E is that for any skew-symmetric k-multilinear map f : E k → F (where F is any vector space) there exists a unique linear map f ∧ : ∧k E → F such that

$$\displaystyle \begin{aligned}f(\boldsymbol{x}_1,\dotsc,\boldsymbol{x}_k)=f^{\wedge}(\boldsymbol{x}_1\wedge\cdots\wedge\boldsymbol{x}_k).\end{aligned}$$

The exterior algebra (or Grassmann algebra) associated with E, (∧E, ∧), is the direct sum of the exterior powers ∧k E of E ( $$0\leqslant k\leqslant n$$ ),

$$\displaystyle \begin{aligned} {\wedge} E= {{\oplus}}_{k=0}^{n}\ {\wedge}^{k}E =\mathbb{R}\oplus E\oplus {\wedge}^{2}E\oplus\cdots \oplus{\wedge}^{n}E, \end{aligned}$$

endowed with the exterior product ∧ whose basic computational rule is

$$\displaystyle \begin{aligned} (\boldsymbol{x}_1\wedge\cdots\wedge\boldsymbol{x}_k)\boldsymbol{\wedge}(\boldsymbol{y}_1\wedge\cdots\wedge\boldsymbol{y}_{k'})= \boldsymbol{x}_1\wedge\cdots\wedge\boldsymbol{x}_k\wedge \boldsymbol{y}_1\wedge\cdots\wedge\boldsymbol{y}_{k'}. \end{aligned}$$

So it is a graded algebra, as

$$x\boldsymbol {\wedge } x'\in {{\wedge }}^{k+k'}E$$

when x ∈∧k E and $$x'\in {{\wedge }}^{k'}E$$ . The exterior product is skew-commutative (or supercommutative):

$$\displaystyle \begin{aligned} x\wedge x'=(-1)^{kk'}x'\wedge x, \end{aligned}$$

if x ∈∧k E, $$x'\in {{\wedge }}^{k'}E$$ . On account of the associativity of the exterior product, the distinction between ∧ and ∧ is unnecessary and it will not be done in what follows.

The elements of ∧E are called multivectors. Given a multivector x ∈∧E, there is a unique decomposition x = x 0 + x 1 + ⋯ + x n with x k ∈∧k E (k = 0, 1, …, n) and we say that x k is the grade k component of x.

1.1.5 (The parity involution)

If $$x=\sum x_k$$ (x k ∈∧k E) is a multivector, we define

$$\hat {x}=\sum (-1)^{k}x_k$$

. This gives a linear map ∧E →∧E, $$x\mapsto \hat {x}$$ , that is an