Linear Algebra

Topic: Projective Geometry 345in fact the dome is not quite an entire hemisphere, or else when the viewer islooking exactly along the equator then there are two points in the line on thedome. Instead we define it so that the points on the equator with a positive ycoordinate, as well as the point where y = 0 and x is positive, are on the domebut the other equatorial points are not.) This discontinuity means that we oftenhave to treat equatorial points as a separate case. That is, while the railroadtrack discussion of central projection has three cases, the dome model has two.We can do better, we can reduce to having no separate cases. Consider asphere centered at the origin. Any line through the origin intersects the spherein two spots, which are antipodal. Because we associate each line through theorigin with a point in the projective plane, we can draw such a point as a pairof antipodal spots on the sphere. Below, we show the two antipodal spotsconnected by a dashed line to emphasize that they are not two different points,the pair of spots together make one projective point.While drawing a point as a pair of antipodal spots is not as natural as theone-spot-per-point dome 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. This model of central projection is uniform.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.(We’ve included one of the projective points on this line 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 describe with the row vector (1 1 −1) (we use arow vector to typographically set lines apart from points). In general, for any