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This eBook from the Gutenberg Project consists of approximately 96 pages of information about An Elementary Course in Synthetic Projective Geometry.

57.  Pencils of rays of the second order. Similar investigations may be made concerning the system of lines joining corresponding points of two projective point-rows.  If we project the point-rows to any point in the plane, we obtain two projective pencils having the same center.  At most two pairs of self-corresponding rays may present themselves.  Such a ray is clearly a line joining two corresponding points in the two point-rows.  The result may be stated as follows:  The system of rays joining corresponding points in two protective point-rows has at most two rays in common with any pencil in the plane. For that reason the system of rays is called a pencil of rays of the second order.

58. In the case of two perspective point-rows this system of rays degenerates into two pencils of rays of the first order, one of which has its center at the center of perspectivity of the two point-rows, and the other at the intersection of the two point-rows, any ray through which may be considered as joining two corresponding points of the two point-rows.

59.  Cone of the second order. The corresponding theorems in space may easily be obtained by joining the points and lines considered in the plane theorems to a point S in space.  Two projective pencils give rise to two projective axial pencils with axes intersecting.  Corresponding planes meet in lines which all pass through S and through the points on a point-row of the second order generated by the two pencils of rays.  They are thus generating lines of a cone of the second order, or quadric cone, so called because every plane in space not passing through S cuts it in a point-row of the second order, and every line also cuts it in at most two points.  If, again, we project two point-rows to a point S in space, we obtain two pencils of rays with a common center but lying in different planes.  Corresponding lines of these pencils determine planes which are the projections to S of the lines which join the corresponding points of the two point-rows.  At most two such planes may pass through any ray through S.  It is called a pencil of planes of the second order.


1. A man A moves along a straight road u, and another man B moves along the same road and walks so as always to keep sight of A in a small mirror M at the side of the road.  How many times will they come together, A moving always in the same direction along the road?

2.  How many times would the two men in the first problem see each other in two mirrors M and N as they walk along the road as before? (The planes of the two mirrors are not necessarily parallel to u.)

3.  As A moves along u, trace the path of B so that the two men may always see each other in the two mirrors.

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