Complete list of grade-1 geometric objects

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.

Correction and expansion to original post.

After thinking about the example 1-vectors V where \alpha_\infty = 0, it became apparent that this set is within the dual-sphere set. (I believe it represents the set of dual-spheres which include the origin point n_o.) So it appears that the set of 1-vectors is completely covered by subsets of: points, dual-planes, dual-spheres, and Euclidean vectors.

However, part of me is challenging the notion that single grade multivectors M = M_{\langle n \rangle} always have geometric non-ambiguous interpretations. To be more explicit do the following subsets

  • 1-vectors: points, dual-planes, dual-spheres, Euclidean vectors,
  • Bi-vectors: point-pairs, flat-points, dual-lines, free-vectors, tangent-vectors,
  • Tri-vectors: lines, circles, free-bivectors, tangent-bivectors, and
  • 4-vectors: planes, spheres, free-trivectors.

each follow the MECE principle, i.e., are they mutually exclusive (ME) and collectively exhaustive (CE)?

At this point I am setting aside that fact that Euclidean objects can have two distinct encodings, e.g., both spheres and dual-spheres would for all practical purposes be used to represent Euclidean spheres.