Geometric Algebra

Geometric Algebra > Geometric Algebra Cheat Sheet

Geometric Algebra Cheat Sheet

This is a list of terminology, notation and equations that are useful in Geometric Algebra. If you have more additions, corrections or suggestions please send me (Tora) a message in the Bivector discord or email me.

⥣ Terminology

Name	Notation	Description
Geometric Algebra	$C l^{p, q, r}, C l^{p, q}, C l^{p}$	Geometric algebra with $p$ basis vectors squaring to $+ 1$ , $q$ basis vectors squaring to $- 1$ and $r$ basis vectors squaring to $0$ . If indices are omitted they are assumed to be zero. $r > 0$ algebras are called degenerate.
Scalar	$s$	Ordinary numbers, usually real numbers.
Basis vector	$e_{i}$	Basis vectors of a geometric algebra.
Vector	$a = a_{1} e_{2} + a_{2} e_{2} + . . .$	Linear combination of basis vectors.
Basis blade	$e_{i j . . .}$	Product of basis vectors.
Blade	$B$	Product of vectors.
Grade		The grade of a basis blades is how many basis vectors it is made of. A pure grade element can be a linear combination of identically graded basis blades. A mixed grade element can be a combination of differently graded basis blades.
Bivector, Trivector, Quadvector, ...		Multivector of grade denoted by the prefix (eg. bi is grade $2$ , tri is grade $3$ ).
Pseudoscalar, Pseudovector, Pseudobivector, ...		Opposite grade element. For an $n$ dimensional algebra, a pseudovector has grade $n - 1$ , a pseudobivector has grade $n - 2$ , ...
Multivector	$A$	Sum of elements of the algebra. Can be of mixed grade in general.
Simple Bivector		Bivectors are simple if they square to scalars.
Exponential of element squaring to scalar	$e^{s B}$	Elements squaring to scalars, such as simple bivectors, have easy formulas for the exponential:
Exponential of a general element	$e^{A}$	We can often decompose general elements into commuting elements squaring to scalars using the invariant decomposition. We can then calculate the exponential of these elements and multiply them to get the exponential of the general element.
Rotor	$R$	Product of an even number of normalized vectors. Used for example for performing rotations. Often obtained by exponentiating bivectors.
Simple Rotor		Rotors are simple if they are composed of two reflections.
Logarithm of a simple rotor		The logarithm of a simple rotor is a simple bivector.

⥥ Operations

⥥ Equations