Transformations

The orthogonal group is isomorphic to the conformal group , the group of angle-preserving transformations, so CGA contains the conformal transformations through . In robotics we can set aside pure reflections and the special conformal transformations, and work with subgroups built from an even number of unit vectors — subgroups of .

Every transformation below acts on every multivector by the same sandwich product,

and each is the exponential of a bivector. That is the shape of the whole page: a Lie group of versors, a Lie algebra of bivectors, and the maps between them.

Rotations

Rotations in 3D are usually the matrix Lie group , whose double cover is represented by unit quaternions. In CGA the rotors form an isomorphic group, with the bivector algebra

as its Lie algebra. The exponential and logarithmic maps are

The bivector basis elements are planes of rotation, which is the more natural description than an axis: a rotation happens in a plane, and in 3D the axis is the plane's dual.

Translations

The translation group of is the space itself under addition, . In CGA it also has a versor form, written . Unlike the rotors this is not a double cover, since is already simply connected. Its Lie algebra is

and the maps are strikingly simple — the exponential truncates after one term:

A Euclidean translation vector becomes a bivector by wedging with infinity:

This is the payoff of the conformal signature. In a Euclidean algebra translation is addition and rotation is multiplication, and the two never compose cleanly. Here both are versors, both act by sandwiching, and both are exponentials of bivectors.

Uniform scaling

Uniform scaling preserves shape, angles, orientation, parallelism and collinearity, changing only distances by an isotropic factor. Restricting to positive factors preserves handedness too. The versor is called a dilator , with the one-dimensional Lie algebra :

The scaling factor relates to the bivector by . Note that scaling is always with respect to the origin — the hyperbolic functions here, against the trigonometric ones for rotors, reflect the mixed signature of the plane.

Rigid transformations: motors

Rigid transformations are the group robotics uses most, traditionally the special Euclidean group ; dual quaternions represent its double cover . In CGA the group is

and its elements are called motors. The canonical decomposition is a translator composed with a rotor:

A motor is a single object that carries a full rigid motion, composes by multiplication, inverts by reversion, and transforms every primitive in the algebra by the same sandwich product. Interpolating between two motors follows a screw path, which is the natural rigid-body interpolation rather than an interpolation of position and orientation separately.

Why this matters in practice

Collecting the pattern: rotations, translations and scalings are all versors, all exponentials of bivectors, all applied identically, and all composable by multiplication. A controller written in terms of motors does not need to convert between rotation matrices, quaternions and translation vectors, and an optimization over motors has a well-defined tangent space at every point.

Next: robot modeling, where these are applied to kinematic chains.