fom
Posts:
1,969
Registered:
12/4/12


Re: What does one call vector geometry without a coordinate system?
Posted:
Sep 2, 2013 7:46 PM


On 9/2/2013 3:24 PM, FredJeffries wrote: > On Saturday, August 31, 2013 3:28:21 PM UTC7, fom wrote: >> >> I like what is in the link you posted. It clarifies >> many details for me. It differs from Bowen's presentation >> in that it presents an affine space in terms of a >> faithful group action. >> >> f: ExV > E >> >> where I have used V for the translation space. >> >> In Bowen, one has >> >> f: ExE > V >> >> Although I would have to compare the presentations >> more carefully to appreciate any subtle differences, >> Bowen's view seems to be directed toward the topological >> constructions about to be discussed in his presentation. > > There could be something to that. The other paper seems > to come from Jean Gallier's class notes for a computer > science course "Geometric Methods in Computer Science" > > http://www.cis.upenn.edu/~cis610/home09.html > http://www.cis.upenn.edu/~cis610/cis610notes09.html > http://www.cis.upenn.edu/~cis610/cis610syllabus09.html > > "The purpose of this course is to present geometric > methods used in computer vision, robotics and computer graphics." >
I think that the distinction for computer visualization is to make clear the relationship of affine aspects and projective aspects so that they can be treated properly in algorithms implementing the graphics. Visualization essentially corresponds with projective methods because of perspective drawing.
Look at the responses to me from Seymour. Bowen is going directly to Euclidean spaces. His mention of affine spaces is secondary. The text is using inner product spaces, and, this is what will be required to introduce the general distance function in the next few pages. This is what Seymour's correction to my statements reflect.

