Linear Algebra

334 Chapter Four. DeterminantsFor a better model, put the projector P at the origin. Imagine that P iscovered by a glass hemispheric dome. As P looks outward, anything in the lineof vision is projected to the same spot on the dome. This includes things onthe line between P and the dome, as in the case of projection by the movieprojector. It includes things on the line further from P than the dome, as inthe case of projection by the painter. It also includes things on the line that liebehind P , as in the case of projection by a pinhole.l = {k ·( 123)∣∣k ∈ R}From this perspective P , all of the spots on the line are seen as the same point.Accordingly, for any nonzero vector ⃗v ∈ R 3 , we define the associated point vin the projective plane to be the set {k⃗v ∣ k ∈ R and k ≠ 0} of nonzero vectorslying on the same line through the origin as ⃗v. To describe a projective pointwe can give any representative member of the line, so that the projective pointshown above can be represented in any of these three ways.⎛⎝ 1 ⎞ ⎛2⎠⎝ 1/3⎞ ⎛2/3⎠⎝ −2⎞−4⎠3 1 −6Each of these is a homogeneous coordinate vector for v.This picture, and the above definition that arises from it, clarifies the descriptionof central projection but there is something awkward about the domemodel: what if the viewer looks down? If we draw P ’s line of sight so thatthe part coming toward us, out of the page, goes down below the dome thenwe can trace the line of sight backward, up past P and toward the part of thehemisphere that is behind the page. So in the dome model, looking down givesa projective point that is behind the viewer. Therefore, if the viewer in thepicture above drops the line of sight toward the bottom of the dome then theprojective point drops also and as the line of sight continues down past theequator, the projective point suddenly shifts from the front of the dome to theback of the dome. This discontinuity in the drawing means that we often haveto treat equatorial points as a separate case. That is, while the railroad trackdiscussion of central projection has three cases, the dome model has two.We can do better than this. Consider a sphere centered at the origin. Anyline through the origin intersects the sphere in two spots, which are said to beantipodal. Because we associate each line through the origin with a point in theprojective plane, we can draw such a point as a pair of antipodal spots on thesphere. Below, the two antipodal spots are shown connected by a dashed line