An Elementary Course in Synthetic Projective Geometry Part 5

*55. Point-row of the second order.* The question naturally arises, What is the locus of points of intersection of corresponding rays of two projective pencils which are not in perspective position? This locus, which will be discussed in detail in subsequent chapters, is easily seen to have at most two points in common with any line in the plane, and on account of this fundamental property will be called a _point-row of the second order_. For any line _u_ in the plane of the two pencils will be cut by them in two projective point-rows which have at most two self-corresponding points. Such a self-corresponding point is clearly a point of intersection of corresponding rays of the two pencils.

*56.* This locus degenerates in the case of two perspective pencils to a pair of straight lines, one of which is the axis of perspectivity and the other the common ray, any point of which may be considered as the point of intersection of corresponding rays of the two pencils.

*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 s.p.a.ce may easily be obtained by joining the points and lines considered in the plane theorems to a point _S_ in s.p.a.ce. Two projective pencils give rise to two projective axial pencils with axes intersecting. Corresponding planes meet in lines which all pa.s.s 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 s.p.a.ce not pa.s.sing 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 s.p.a.ce, 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 pa.s.s through any ray through _S_. It is called _a pencil of planes of the second order_.

PROBLEMS

*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.

4. Two boys walk along two paths _u_ and _u'_ each holding a string which they keep stretched tightly between them. They both move at constant but different rates of speed, letting out the string or drawing it in as they walk. How many times will the line of the string pa.s.s over any given point in the plane of the paths?

5. Trace the lines of the string when the two boys move at the same rate of speed in the two paths but do not start at the same time from the point where the two paths intersect.

6. A s.h.i.+p is sailing on a straight course and keeps a gun trained on a point on the sh.o.r.e. Show that a line at right angles to the direction of the gun at its muzzle will pa.s.s through any point in the plane twice or not at all. (Consider the point-row at infinity cut out by a line through the point on the sh.o.r.e at right angles to the direction of the gun.)

7. Two lines _u_ and _u'_ revolve about two points _U_ and _U'_ respectively in the same plane. They go in the same direction and at the same rate of speed, but one has an angle a the start of the other. Show that they generate a point-row of the second order.

8. Discuss the question given in the last problem when the two lines revolve in opposite directions. Can you recognize the locus?

CHAPTER IV - POINT-ROWS OF THE SECOND ORDER

*60. Point-row of the second order defined.* We have seen that two fundamental forms in one-to-one correspondence may sometimes generate a form of higher order. Thus, two point-rows (-- 55) generate a system of rays of the second order, and two pencils of rays (-- 57), a system of points of the second order. As a system of points is more familiar to most students of geometry than a system of lines, we study first the point-row of the second order.

*61. Tangent line.* We have shown in the last chapter (-- 55) that the locus of intersection of corresponding rays of two projective pencils is a point-row of the second order; that is, it has at most two points in common with any line in the plane. It is clear, first of all, that the centers of the pencils are points of the locus; for to the line _SS'_, considered as a ray of _S_, must correspond some ray of _S'_ which meets it in _S'_. _S'_, and by the same argument _S_, is then a point where corresponding rays meet. Any ray through _S_ will meet it in one point besides _S_, namely, the point _P_ where it meets its corresponding ray.

Now, by choosing the ray through _S_ sufficiently close to the ray _SS'_, the point _P_ may be made to approach arbitrarily close to _S'_, and the ray _S'P_ may be made to differ in position from the tangent line at _S'_ by as little as we please. We have, then, the important theorem

_The ray at __S'__ which corresponds to the common ray __SS'__ is tangent to the locus at __S'__._

In the same manner the tangent at _S_ may be constructed.

*62. Determination of the locus.* We now show that _it is possible to a.s.sign arbitrarily the position of three points, __A__, __B__, and __C__, on the locus (besides the points __S__ and __S'__); but, these three points being chosen, the locus is completely determined._

*63.* This statement is equivalent to the following:

_Given three pairs of corresponding rays in two projective pencils, it is possible to find a ray of one which corresponds to any ray of the other._

*64.* We proceed, then, to the solution of the fundamental

PROBLEM: _Given three pairs of rays, __aa'__, __bb'__, and __cc'__, of two protective pencils, __S__ and __S'__, to find the ray __d'__ of __S'__ which corresponds to any ray __d__ of __S__._

[Figure 12]

FIG. 12

Call _A_ the intersection of _aa'_, _B_ the intersection of _bb'_, and _C_ the intersection of _cc'_ (Fig. 12). Join _AB_ by the line _u_, and _AC_ by the line _u'_. Consider _u_ as a point-row perspective to _S_, and _u'_ as a point-row perspective to _S'_. _u_ and _u'_ are projectively related to each other, since _S_ and _S'_ are, by hypothesis, so related. But their point of intersection _A_ is a self-corresponding point, since _a_ and _a'_ were supposed to be corresponding rays. It follows (-- 52) that _u_ and _u'_ are in perspective position, and that lines through corresponding points all pa.s.s through a point _M_, the center of perspectivity, the position of which will be determined by any two such lines. But the intersection of _a_ with _u_ and the intersection of _c'_ with _u'_ are corresponding points on _u_ and _u'_, and the line joining them is clearly _c_ itself. Similarly, _b'_ joins two corresponding points on _u_ and _u'_, and so the center _M_ of perspectivity of _u_ and _u'_ is the intersection of _c_ and _b'_. To find _d'_ in _S'_ corresponding to a given line _d_ of _S_ we note the point _L_ where _d_ meets _u_. Join _L_ to _M_ and get the point _N_ where this line meets _u'_. _L_ and _N_ are corresponding points on _u_ and _u'_, and _d'_ must therefore pa.s.s through _N_. The intersection _P_ of _d_ and _d'_ is thus another point on the locus. In the same manner any number of other points may be obtained.

*65.* The lines _u_ and _u'_ might have been drawn in any direction through _A_ (avoiding, of course, the line _a_ for _u_ and the line _a'_ for _u'_), and the center of perspectivity _M_ would be easily obtainable; but the above construction furnishes a simple and instructive figure. An equally simple one is obtained by taking _a'_ for _u_ and _a_ for _u'_.