The use of hyperreals/surreals for geometric computations came up in the thread about infinite-dimensional polytopes, but since geometry involving infinite/infinitesimal quantities (as an inherent part of the coefficient field) is AFAIK barely explored and thus poorly understood, I thought it might be fruitful to first consider what geometry might look like in finite-dimensional space, but one in which the usual coefficient field of real numbers is augmented with non-finite quantities like the hyperreals or surreals. For example, what could we say about the properties of geometry in *R

^{3}or No

^{3}?

One thing for sure: the usual geometry still works as before in the finite subspace of this augmented space -- you'd have the old set of finite regular polytopes, etc.. But now there may be new objects that are introduced. For example, one could now construct a sphere that contains a linear apeirotope, for example. It would be a sphere of non-finite radius. One may also construct polyhedra with "points at infinity", i.e., one or more points that contain non-finite coordinates. One may also construct infinitesimal polytopes, or polytopes that contain infinitesimal elements. One may conceivably also embed the Long Line in this space (though that has yet to be confirmed).

One thing that could potentially fundamentally alter geometry in such a space is the possibility of defining certain infinite series to converge to real numbers displaced by an infinitesimal. OT1H this could be used to distinguish between things that in normal geometry are indistinguishible, but OTOH it may also prove problematic when certain computations yield results that don't match up because they differ by an infinitesimal.

What else can be said about such a space? Any ideas? How would one go about computing, say, the angle between a line in finite subspace and a line containing one or more non-finite coefficients? What about vector operations like dot products and linear transformations? What can be said about linear transformations involving non-finite or infinitesimal coefficients?