Linear Algebra

Topic: Projective Geometry 335to emphasize that they are not two different points, the pair of spots togethermake one projective point.While drawing a point as a pair of antipodal spots is not as natural as the onespot-per-pointdome mode, on the other hand the awkwardness of the domemodel is gone, in that if as a line of view slides from north to south, no suddenchanges happen on the picture. This model of central projection is uniform —the three cases are reduced to one.So far we have described points in projective geometry. What about lines?What a viewer at the origin sees as a line is shown below as a great circle, theintersection of the model sphere with a plane through the origin.(One of the projective points on this line is shown to bring out a subtlety.Because two antipodal spots together make up a single projective point, thegreat circle’s behind-the-paper part is the same set of projective points as itsin-front-of-the-paper part.) Just as we did with each projective point, we willalso describe a projective line with a triple of reals. For instance, the membersof this plane through the origin in R 3⎛⎞{ ⎝ x y⎠ ∣ x + y − z = 0}zproject to a line that we can described with the triple ( 1 1 −1 ) (we use rowvectors to typographically distinguish lines from points). In general, for anynonzero three-wide row vector L ⃗ we define the associated line in the projectiveplane, to be the set L = {kL ⃗ ∣ k ∈ R and k ≠ 0} of nonzero multiples of L. ⃗The reason that this description of a line as a triple is convienent is thatin the projective plane, a point v and a line L are incident — the point lieson the line, the line passes throught the point — if and only if a dot productof their representatives v 1 L 1 + v 2 L 2 + v 3 L 3 is zero (Exercise 4 shows that thisis independent of the choice of representatives ⃗v and L). ⃗ For instance, theprojective point described above by the column vector with components 1, 2,and 3 lies in the projective line described by ( 1 1 −1 ) , simply because any