Class Monday 3/10: Central Projection and Perspective drawing

Light Ray Paths

Light ray paths through an "eye" or a sensor at point E in 3-space model the set of lines through E.  For this version of the theory we do not distinguish between the opposite rays on the same line, but count the whole line as one path.

Light Ray paths and a sphere

Let S be a sphere centered at E, then each light ray path m intersects S in two antipodal points M1 and M2.  So we can set up a correspondence between light ray paths and pairs of antipodal points on the sphere.

Light Ray paths and a plane

If p is a plane not through E, then each line m through E (with some exceptions noted shortly) intersects the plane p in a point P(m).  This sets of a correspondence between points M on the line and light ray paths (lines) EP (m). The exception is that lines m parallel to p do not intersect.  We will "fix" this by adding points at infinity.  This is explained on the next page.

Light Ray paths and perspective drawing

Let p and q be planes not through E.  We think of p as a plane in the "real world" and q as a transparent "easel" on which we draw by sighting through the easel.  This means that the object we are drawing at P, the image P' on the easel, and our eye at E are collinear.

To say this a different way: (1) Start with a point P on p (2) then if m = EP, P = P(m) (3) let P' be the point Q(m), the intersection of m with q.  We call correspondence from points in p to points in q the central projection from p to q with center E.

This perspective drawing has several important properties that we will explore in class.

• If q_E is the plane through E parallel to q, then the points on the intersection line g of q_E and p do not project to points on q.
• Points on a line j on p are projected to points on a line j' on q., with the exception noted above of the points of intersection of j and g.
• If p_E is the plane through E parallel to p, then the points on the intersection line h of p_E and q are not the projections of any point on plane p.
• A set of parallel lines on p is projected either to a set of parallel lines or a set of concurrent lines that meet at a point of line h.
  

Adding Points at Infinity to a Euclidean (or affine) plane

The special cases of points on lines g and h above can be removed as special cases if we add some points at infinity to each plane.  This is one way to do this:  For each line j in a plane, add one point I(j) at infinity with the rule that if two lines are parallel, then they have the same point at infinity – if j is parallel to k, then I(j) = I(k).  This makes it literally true that two parallel lines in the extended Euclidean plane meet at a point at infinity.

If this seems too exotic or arbitrary, there are alternate definitions that are quite concrete:

• As noted before, the points on plane p correspond to a set of light paths through E.  The new points at infinity on p correspond to the light paths through E that are parallel to p and thus do not intersect p at a (finite) point. 
• In the sphere model for light paths, the points at infinity on p correspond to the pairs of antipodal points on the great circle parallel to plane p.

From our study or perspective drawing we see that for two parallel lines m and n,

• For points M on m, the light paths EM are lines through E in the plane containing E and m.  There is one line e through E in this plane that is parallel to m.  This is the line that we consider the line through E and I(m).
• For the parallel line n, the story is the same.  But the line through E parallel to n is also parallel to m, so it is the same line e.  This line e is the intersection of the plane containing E and m and the plane containing E and n. This means that the light ray corresponding to the point at infinity on m and the light ray corresponding to the point at infinity on n are the same when m is parallel to n.
• The set of all the points at infinity is the set of light paths all lie in the plane p_E parallel to p through E.  Thus the points project to a line on another plane q (including one point at infinity on q!).  So this means that the set of points at infinity on plane q should be considered as just another line on the extended plane of p.