Conformal geometric algebra (CGA) is the geometric algebra constructed over the resultant space of a map from points in an n-dimensional base space R^p,q to null vectors in R^p+1,q+1. This allows operations on the base space, including reflections, rotations and translations to be represented using versors of the geometric algebra; and it is found that points, lines, planes, circles and spheres gain particularly natural and computationally amenable representations.

The effect of the mapping is that generalized (i.e. including zero curvature) k-spheres in the base space map onto (k + 2)-blades, and so that the effect of a translation (or any conformal mapping) of the base space corresponds to a rotation in the higher-dimensional space. In the algebra of this space, based on the geometric product of vectors, such transformations correspond to the algebra's characteristic sandwich operations, similar to the use of quaternions for spatial rotation in 3D, which combine very efficiently. A consequence of rotors representing transformations is that the representations of spheres, planes, circles and other geometrical objects, and equations connecting them, all transform covariantly. A geometric object (a k-sphere) can be synthesized as the wedge product of k + 2 linearly independent vectors representing points on the object; conversely, the object can be decomposed as the repeated wedge product of vectors representing k + 2 distinct points in its surface. Some intersection operations also acquire a tidy algebraic form: for example, for the Euclidean base space R³, applying the wedge product to the dual of the tetravectors representing two spheres produces the dual of the trivector representation of their circle of intersection.

As this algebraic structure lends itself directly to effective computation, it facilitates exploration of the classical methods of projective geometry and inversive geometry in a concrete, easy-to-manipulate setting. It has also been used as an efficient structure to represent and facilitate calculations in screw theory. CGA has particularly been applied in connection with the projective mapping of the everyday Euclidean space R³ into a five-dimensional vector space R^4,1, which has been investigated for applications in robotics and computer vision. It can be applied generally to any pseudo-Euclidean space – for example, Minkowski space R^3,1 to the space R^4,2.

In this article, the focus is on the algebra ${\displaystyle {\mathcal {G}}(4,1)}$ as it is this particular algebra that has been the subject of most attention over time; other cases are briefly covered in a separate section. The space containing the objects being modelled is referred to here as the base space, and the algebraic space used to model these objects as the representation or conformal space. A homogeneous subspace refers to a linear subspace of the algebraic space.

The terms for objects: point, line, circle, sphere, quasi-sphere etc. are used to mean either the geometric object in the base space, or the homogeneous subspace of the representation space that represents that object, with the latter generally being intended unless indicated otherwise. Algebraically, any nonzero null element of the homogeneous subspace will be used, with one element being referred to as normalized by some criterion.

Boldface lowercase Latin letters are used to represent position vectors from the origin to a point in the base space. Italic symbols are used for other elements of the representation space.

The base space R³ is represented by extending a basis for the displacements from a chosen origin and adding two basis vectors e₋ and e₊ orthogonal to the base space and to each other, with e₋² = −1 and e₊² = +1, creating the representation space ${\displaystyle {\mathcal {G}}(4,1)}$.

It is convenient to use two null vectors n_o and n_∞ as basis vectors in place of e₊ and e₋, where n_o = (e₋ − e₊)/2, and n_∞ = e₋ + e₊. It can be verified, where x is in the base space, that: