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- "Then the unique great circle perpendicular to point P3 is line L3 in S1."
- How can a circle be perpendicular to a point?
- "L1 is the unique great circle in a plane perpendicular to the line through the pair of points P1"
- Why is P1 referred to as a pair of points when it was originally defined as a single point? Is the author referring to the antipodal equivalent as well here? If so, then a definate article for "the line" doesn't make sense since there is no unique line through antipodal points.
- Update: I think I understand what you were trying to say now. By line, I think you mean a line through euclidian 3D space, although earlier you defined a line to be a great circle. I will submit a re-wording of your example if I am successful in coming up with a more clearly stated revision.
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- The footnote
- >>*ยน When we say a line through pair of points P, or simply a line through P, we refer to the three dimensional euclidian line that passes through the antipodal points represented by P. When we say that a geodesic line, or a great circle, L is perpendicular to the line passing through the pair of points P, we mean that L lies on the plane that is perpendicular to, and intersects at the midpoint of, the straight line segment in euclidian space that connects the antipodal points that is represented by P. In other words, L is the set of points equidistant in euclidian space to the antipodal points represented by P. L is unique for P. <<
- is exactly right: it is precisely what was meant! Apparently there was no conscious awareness of just how the meanings of "line" and "circle" were being overloaded. --AugPi 00:55, 18 Feb 2005 (UTC)
[edit] Points and lines in the plane
It seems to me this section makes it harder to understand than is really necessary, since using (m, b) to represent a line with slope m and intercept b misses some lines, and also loses the symmetrical quality of projective space. I'm inclined to think it needs a massive rewrite, starting from the point of view of the section I added on duality in terms of vector spaces. Gene Ward Smith 14:38, 5 May 2006 (UTC)
- I suggest moving most of that section to Duality (projective geometry)/Proofs, in the same fashion as has been done in Category:Article proofs. Such busy-body expositions may be useful to some, but do make the article much more opaque. linas 14:31, 12 December 2006 (UTC)
