38 THEORY OF PROJECTION. 
a pencil with its vertex at P. Let this pencil be cut by any 
other line, as l’, in points A’, B’, C’, D’... The operation of 
constructing the pencil through the point P and the points of 
the range on / is called projecting the range from P. The 
operation of cutting the pencil by another line, as 1’, is called 
taking a section of the pencil. The new range A’, B’, C’, 
D’... on the line 1’ is called a perspective projection of the 
former range, P being the center or vertex of the projection. 
45. One-to-one Correspondence. The points A and A’, B 
and B’, ete., are called corresponding points of the two ranges 
onlandl’. It is evident that to a point such as A on the 
line / there corresponds one and only one point, A’, on l’, the 
corresponding point lying on the same ray through P. This 
is true of every point on / except the point J, where the par- 
allel to /’ through P cuts /, and infinitely distant points on J. 
Therefore, with the exception of these points there is a one- 
to-one correspondence between the points of these two ranges. 
46. The Point at Infinity. In order to make this one-to- 
one correspondence hold without any exceptions we adopt the 
following convention. We say that two parallel lines meet 
in one infinitely distant point. According to Euclid’s hypothe- 
sis PJ is the only ray through P parallel to /’; J is therefore 
the only point on / which corresponds to points at infinity on 
l’. Inthe same way I’ is the only point on l’ which corre- 
sponds to points at infinity on /. One-to-one correspondence 
of points on the two ranges is therefore general with no ex- 
ceptions, if we assume but a single point at infinity on each 
of the lines / and /’. 
47. Self-corresponding Point. Wealso see that the point 
O on | corresponds to the point O onl’; in other words, the 
point of intersection of the lines / and 1’ is a self-correspond- 
ing point on the two ranges. Two ranges connected by a 
perspective projection are characterized by the facts that they 
have a self-corresponding point and that the rays joining cor- 
responding points meet in a point. They are sometimes 
