An Introduction to Geometric Algebra (Part 1)

1. A little history and philosophy

Numbers and geometry are two fundamental topics from the beginning of mathematics. Thanks to genius René Descartes, one of the greatest philosophers and mathematicians in the 17th century, who invented coordinate geometry (analytical geometry), a solid bridge was established between numbers and geometry. Points are associated to a set of numbers called coordinates, lines and surfaces are to algebraic equations. It seems that Cartesian coordinate system provides a very powerful tool to handle geometric objects and describe transformations. And indeed the idea of coordinate system has influenced the whole math, including the discovery of differential calculus by Newton and Leibniz regarding derivative (algebraic quantity) as the slope of tangent line (geometric quantity), and the emergence of differential geometry of curves and surfaces (geometric object), which can be described by parameterized smooth mappings (algebraic object).

Unfortunately, one has to admit that a lot of computational complexities result from interlude of numbers, which is really costly. In geometry, we mainly concern three aspects: 1. geometric objects themselves, such as shape, size (length, area, volume), angle etc.; 2. relationships between them like intersection, disjoint; 3. transformations, say stretch and contraction, reflection and rotation, also projection and rejection. We find actually all these aspects are independent with coordinate. What does that mean? That means no matter how coordinate system is chosen, these properties are invariant. In this sense, position and orientation of coordinate system doesn’t matter. Coordinate system being either skew or orthogonal, either scaled or normal, has no effect nothing about what we care. Hence a proper coordinate system is anticipated to facilitate the computation, though it is usually hard to get one or even impossible to have one in most cases.

Persons imbedded in Cartesian philosophy may devote their whole energy to finding methods of choosing a good coordinate system. And certainly, there are ones who do not, expecting to describe geometry without coordinate, or more specifically, coordinate-free geometry. Like Josiah Willard Gibbs, another great scientist making indispensable contributions to mathematical physics in the 19th century, found out multiplication of vectors, say dot product and cross product, has interesting geometric interpretations and formally gave them proper notations, which now is well known as part of vector analysis.

Actually, J.W. Gibbs is not the first to discovery an algebra for coordinate-free geometry. William Rowan Hamilton generalized complex number when he walked around Royal Canal in Dublin, Ireland after several years’ thinking, a shape of quaternion emerged into his mind. Then he could not resist his excitement and immediately carved the famous quaternion formula into the stone of Broom Bridge. Quaternion describes rotations in 3 dimension quite well. Inspired by W.R. Hamilton, many algebras such as bicomplex, hypercomplex, biquaternion have been come up with soon, all are associated distinct geometric pictures.

At almost meantime, Herrmann Günther Grassmann, German mathematician, the inventor of multilinear algebra, suggested exterior product in replace of cross product to achieve (multi)vector multiplications in higher dimension. In Grassmann algebra, a p-dimensional object is denoted as a p-vector, associated with magnitude and direction(orientation). For example, 1-vector is as usual a line segment. 2-vector (or bivector) is an area element, with magnitude as the area and direction as the plane which it lies, instead of normal vector in Gibbs’ vector algebra.

Grassmann algebra (also known as exterior algebra) is good enough to answer a lot of geometric questions since it can describe higher dimensional object. But William Kingdon Clifford, mathematician and philosopher, hoped for a unified algebraic framework incorporating all above number systems. His research on extensive Grassmann algebra led him to geometric algebra, which is part of Clifford algebra focusing more on theoretical, rather than geometric aspect. Clifford algebra then offers many great insights in both mathematics and theoretical physics. But it seems its geometric interpretation was forgot by people. Until late 20th century when David Hestenes rediscovery it, geometric algebra resuscitate to apply in more areas than fundamental physics, including image processing and robotics. D. Hestnes said, “geometry without algebra is dumb, algebra without geometry is blind.”

We will in the following briefly look at geometric algebra, hoping for an interesting journey.

2. Inspiration for geometric algebra

Let’s see what happens if multiply two vectors in \mathbb R^3. Let \{e_1, e_2,e_3\} be a set of basis for \mathbb R^3 and two vectors x=x^1e_1+x^2e_2+x^3e_3,\ y=y^1e_1+y^2e_2+y^3e_3, only associative law and distributive law assumed.

\begin{aligned} xy&=(x^1e_1+x^2e_2+x^3e_3)(y^1e_1+y^2e_2+y^3e_3)\\&=x^1y^1e_1e_1+x^1y^2e_1e_2+x^1y^3e_1e_3\\&+x^2y^1e_2e_1+x^2y^2e_2e_2+x^2y^3e_2e_3\\&+x^3y^1e_3e_1+x^3y^2e_3e_2+x^3y^3e_3e_3 \end{aligned}

To endow the product with geometric meaning, we wish to have e_ie_i=1,\ e_ie_j=-e_je_i\ (i,j=1,2,3,\ j\neq i). Then the product becomes

\begin{aligned}xy=&\underbrace{x^1y^1+x^2y^2+x^3y^3}_{x\cdot y}\\+&\underbrace{(\overbrace{x^1y^2-x^2y^1}^{(x\times y)_3})e_1e_2+(\overbrace{x^2y^3-x^3y^2}^{(x\times y)_1})e_2e_3+(\overbrace{x^3y^1-x^1y^3}^{(x\times y)_2})e_3e_1}_{x\times y}\end{aligned}

It’s readily seen that the product, composing of dot product and cross product, also applies to 2 dimensional case. Readers may find out that we add a non-scalar e_ie_j to a scalar. Does that make any sense? What is the result? Actually we have seen similar scenarios before, say a+b\mathbf{i} in complex number, w+x\mathbf{i}+y\mathbf{j}+z\mathbf{k} in quaternion number, and a_0+a_1x+a_2x^2 in quadratic polynomial. The sum here, called formal sum, is just a collector bringing things together. The meaningful part is not the sum itself, instead, each component in the sum. Dot product is usually explained as projection of a vector on the other, but we will see later this is not proper. And cross product can be seen as normal vector of parallelogram with two vectors as sides and its magnitude is the area. However, cross product has no definition for vectors in higher dimension, as which Grassman thought Gibbs’ cross product is flawed. And then he defined anticommutative wedge product, which contributes to

xy=x\cdot y+x\wedge y

Note we also have

\displaystyle{x\cdot y=\frac{xy+yx}{2},\qquad x\wedge y=\frac{xy-yx}{2}}

Equipped with wedge product, we can try to compute triple product with z. Of course, the product with vector is expected to subsume dot product and wedge product.

\begin{aligned}xyz&=(x\cdot y)z+(x\wedge y)z\\&=(x\cdot y)z+(x\wedge y)\cdot z+(x\wedge y)\wedge z\\&=x(y\cdot z)+x(y\wedge z)\\&=(y\cdot z)x+x\cdot(y\wedge z)+x\wedge(y\wedge z)\\&=(y\cdot z)x-(z\cdot x)y+(x\cdot y)z+x\wedge y\wedge z\end{aligned}

The geometric image shows that the sum of any two of the first three terms is perpendicular to the third, and the volume of parallelepiped equals the magnitude of the last term. Similarly, it’s readily to show (show it)

\displaystyle{x\cdot(y\wedge z)=\frac{x(y\wedge z)-(y\wedge z)x}{2}},\qquad x\wedge(y\wedge z)=\frac{x(y\wedge z)+(y\wedge z)x}{2}

3. Some jargons and formulae

Since now we have a first touch on geometric algebra, it’s time to formally describe it. Yes, we don’t define it in this post because it obscures the visual image. Basically a geometric algebra of dimension n, denoted by \mathscr G(V,n), or simply \mathscr G(n) is a vector space V endowed with an associative bilinear product, called geometric product. The element in \mathscr G(n) is called multivector, composing of scalar, vector, area element, etc. A r-vector is a r dimensional geometric object, which is linear combination of r-blades, written as A_r, that each lies in the same subspace as that of r-dimension analogy of parallelepiped with r linearly independent vectors as sides. Formally,

\mathscr G(V,n)=\bigoplus\limits_{r=0}^n\bigwedge^r(V)

Evidently, a multivector A can be resolved into different grades,

\displaystyle{A=\sum_{r=0}^n\langle A\rangle_r=\lambda+v+\sum_{i}A_{2,i} +\cdots +\sum_j A_{n,j}}

where \langle A \rangle_r=\sum_iA_{r,i} takes the degree r component of A, \lambda is scalar, v is vector, and A_r=u_1\wedge \cdots \wedge u_r for r linearly independent vectors u_1,\ldots, u_r. Geometric product with a vector a can be explicitly written as

\displaystyle aA_r=a\cdot A_r+a\wedge A_r,\qquad A_ra=A_r\cdot a+A_r\wedge a

The first part of the product is called inner product and the second part outer product, defined as the following,

\displaystyle a\cdot A_r:=\langle aA_r\rangle_{r-1}=\frac{aA_r-(-1)^rA_ra}{2},\\ a\wedge A_r:=\langle aA_r\rangle_{r+1}=\frac{aA_r+(-1)^rA_ra}{2}

Note that we don’t mention basis, hence coordinate of vector, at all, which embody the property of coordinate-free of geometric algebra.

4. Properties of geometric algebra

We will list a collection of properties in geometric algebra. Readers will see that each property is supported by geometric fact which is very intuitive. By convention, we use lower letters to denote vectors and upper letters for blades.

  1. a^2=a\cdot a,\ a\wedge a = 0
  2. a\cdot A_r=(-1)^{r-1}A_ra,\ a\wedge A_r=(-1)^rA_ra
  3. aA_r=(a^T+a^\perp)A_r=a^T\cdot A_r+a^\perp\wedge A_r
  4. a\cdot(b\wedge c)=a\cdot b\ -a\cdot c\ b
  5. a\cdot b\cdot A_r=(a\wedge b)\cdot A_r

For the limitedness of time, I have to omit the proof and explanation for now. I will make it up in a few days. As we can have seen, inner product with a vector contracts the subspace to one perpendicular to it, while outer product with a vector extends the subspace to one containing it. It is should be emphasized that geometric product can be defined on any two multivectors, which we will discuss it in the following post. Geometric product itself doesn’t have clear geometric insight, yet it collects different geometric facts together.