The grade-1 multi-vectors V of the conformal GA can be used to represent conformal points, dual planes, and dual spheres. Let an arbitrary 1-vector be given by

\quad V = \alpha_1\, e_1 +\alpha_2\, e_2 +\alpha_3\, e_3 +\alpha_o\, n_0 +\alpha_\infty\, n_\infty

then the basis elements of the geometric objects are:

\quad \text{point: } n_0, e_1, e_2, e_3, n_\infty, \;\; \text{point is 'normalized' when } \alpha_o =1

\quad \text{dual plane: } e_1, e_2, e_3, n_\infty, \;\; \text{i.e., }\alpha_o = 0

\quad \text{dual sphere: } n_0, e_1, e_2, e_3, n_\infty

\quad \text{Euclid. vector: } e_1, e_2, e_3,\,\, \alpha_o = \alpha_\infty = 0

We know that the point and sphere are related (a point being a zero radius sphere). We can also account for ‘imaginary’ spheres when the radius is negative. However these do not exhaust this subspace of possible 1-vectors. For example, if

\quad V = \alpha_1\, e_1 +\alpha_2\, e_2 +\alpha_3\, e_3 +\alpha_o\, n_0,\, \text{ i.e., }\alpha_\infty=0

Is there a geometric interpretation of such vectors? And does this now close the grade-1 subspace, meaning any 1-vector has to be one of these now four objects?

My motivation is to build code that quickly determines what kind of geometric object any 1-vector is, and provide numerically meaningful thresholds between the various objects. E.g., a sphere who’s radius is less than a multiple of machine precision will be ‘rounded’ to a conformal point.