CHAPTER V - PENCILS OF RAYS OF THE SECOND ORDER
79. Pencil of rays of the second order defined. If the corresponding points of two projective point-rows be joined by straight lines, a system of lines is obtained which is called a pencil of rays of the second order. This name arises from the fact, easily shown (§ 57), that at most two lines of the system may pass through any arbitrary point in the plane. For if through any point there should pass three lines of the system, then this point might be taken as the center of two projective pencils, one projecting one point-row and the other projecting the other. Since, now, these pencils have three rays of one coincident with the corresponding rays of the other, the two are identical and the two point-rows are in perspective position, which was not supposed.
80. Tangents to a circle. To get a clear notion of this system of lines, we may first show that the tangents to a circle form a system of this kind. For take any two tangents, u and u', to a circle, and let A and B be the points of contact (Fig. 19). Let now t be any third tangent with point of contact at C and meeting u and u' in P and P' respectively. Join A, B, P, P', and C to O, the center of the circle. Tangents from any point to a circle are equal, and therefore the triangles POA and POC are equal, as also are the triangles P'OB [pg 49] and P'OC. Therefore the angle POP' is constant, being equal to half the constant angle AOC + COB. This being true, if we take any four harmonic points, P1, P2, P3, P4, on the line u, they will project to O in four harmonic lines, and the tangents to the circle from these four points will meet u' in four harmonic points, P'1, P'2, P'3, P'4, because the lines from these points to O inclose the same angles as the lines from the points P1, P2, P3, P4 on u. The point-row on u is therefore projective to the point-row on u'. Thus the tangents to a circle are seen to join corresponding points on two projective point-rows, and so, according to the definition, form a pencil of rays of the second order.
81. Tangents to a conic. If now this figure be projected to a point outside the plane of the circle, and any section of the resulting cone be made by a plane, we can easily see that the system of rays tangent to any conic section is a pencil of rays of the second order. The converse is also true, as we shall see later, and a pencil of rays of the second order is also a set of lines tangent to a conic section.
82. The point-rows u and u' are, themselves, lines of the system, for to the common point of the two point-rows, considered as a point of u, must correspond some point of u', and the line joining these two corresponding points is clearly u' itself. Similarly for the line u.
83. Determination of the pencil. We now show that it is possible to assign arbitrarily three lines, a, b, and c, of [pg 50] the system (besides the lines u and u'); but if these three lines are chosen, the system is completely determined.
This statement is equivalent to the following:
Given three pairs of corresponding points in two projective point-rows, it is possible to find a point in one which corresponds to any point of the other.
We proceed, then, to the solution of the fundamental
Problem. Given three pairs of points, AA', BB', and CC', of two projective point-rows u and u', to find the point D' of u' which corresponds to any given point D of u.
On the line a, joining A and A', take two points, S and S', as centers of pencils perspective to u and u' respectively (Fig. 20). The figure will be much simplified if we take S on BB' and S' on CC'. SA and S'A' are corresponding rays of S and S', and the two pencils are therefore in perspective position. It is not difficult to see that the axis of perspectivity m is the line joining B' and C. Given any point D on u, to find the corresponding point D' on u' we proceed as follows: Join D to S and note where the joining line meets m. Join this point to S'. This last line meets u' in the desired point D'.
We have now in this figure six lines of the system, a, b, c, d, u, and u'. Fix now the position of u, u', b, c, and d, and take four lines of the system, a1, a2, a3, a4, which meet b in four harmonic points. These points project to [pg 51] D, giving four harmonic points on m. These again project to D', giving four harmonic points on c. It is thus clear that the rays a1, a2, a3, a4 cut out two projective point-rows on any two lines of the system. Thus u and u' are not special rays, and any two rays of the system will serve as the point-rows to generate the system of lines.
84. Brianchon's theorem. From the figure also appears a fundamental theorem due to Brianchon:
If 1, 2, 3, 4, 5, 6 are any six rays of a pencil of the second order, then the lines l = (12, 45), m = (23, 56), n = (34, 61) all pass through a point.
85. To make the notation fit the figure (Fig. 21), make a=1, b = 2, u' = 3, d = 4, u = 5, c = 6; or, interchanging two of the lines, a = 1, c = 2, u = 3, d = 4, u' = 5, b = 6. Thus, by different namings of the lines, it appears that not more than 60 different Brianchon points are possible. If we call 12 and 45 opposite vertices of a circumscribed hexagon, then Brianchon's theorem may be stated as follows:
The three lines joining the three pairs of opposite vertices of a hexagon circumscribed about a conic meet in a point.
86. Construction of the pencil by Brianchon's theorem. Brianchon's theorem furnishes a ready method of determining a sixth line of the pencil of rays of the second [pg 52] order when five are given. Thus, select a point in line 1 and suppose that line 6 is to pass through it. Then l = (12, 45), n = (34, 61), and the line m = (23, 56) must pass through (l, n). Then (23, ln) meets 5 in a point of the required sixth line.
87. Point of contact of a tangent to a conic. If the line 2 approach as a limiting position the line 1, then the intersection (1, 2) approaches as a limiting position the point of contact of 1 with the conic. This suggests an easy way to construct the point of contact of any tangent with the conic. Thus (Fig. 22), given the lines 1, 2, 3, 4, 5 to construct the point of contact of 1=6. Draw l = (12,45), m =(23,56); then (34, lm) meets 1 in the required point of contact T.
88. Circumscribed quadrilateral. If two pairs of lines in Brianchon's hexagon coalesce, we have a theorem concerning a quadrilateral circumscribed about a conic. It is easily found to be (Fig. 23)
The four lines joining the two opposite pairs of vertices and the two opposite points of contact of a quadrilateral circumscribed about a conic all meet in a point. The consequences of this theorem will be deduced later.
[pg 53]
89. Circumscribed triangle. The hexagon may further degenerate into a triangle, giving the theorem (Fig. 24) The lines joining the vertices to the points of contact of the opposite sides of a triangle circumscribed about a conic all meet in a point.
90. Brianchon's theorem may also be used to solve the following problems:
Given four tangents and the point of contact on any one of them, to construct other tangents to a conic. Given three tangents and the points of contact of any two of them, to construct other tangents to a conic.
91. Harmonic tangents. We have seen that a variable tangent cuts out on any two fixed tangents projective point-rows. It follows that if four tangents cut a fifth in four harmonic points, they must cut every tangent in four harmonic points. It is possible, therefore, to make the following definition:
Four tangents to a conic are said to be harmonic when they meet every other tangent in four harmonic points.
92. Projectivity and perspectivity. This definition suggests the possibility of defining a projective correspondence between the elements of a pencil of rays of the second order and the elements of any form heretofore discussed. In particular, the points on a tangent are said to be perspectively related to the tangents of a conic when each point lies on the tangent which corresponds to it. These notions are of importance in the higher developments of the subject.
[pg 54]
93. Brianchon's theorem may also be applied to a degenerate conic made up of two points and the lines through them. Thus(Fig. 25),
If a, b, c are three lines through a point S, and a', b', c' are three lines through another point S', then the lines l = (ab', a'b), m = (bc', b'c), and n = (ca', c'a) all meet in a point.
94. Law of duality. The observant student will not have failed to note the remarkable similarity between the theorems of this chapter and those of the preceding. He will have noted that points have replaced lines and lines have replaced points; that points on a curve have been replaced by tangents to a curve; that pencils have been replaced by point-rows, and that a conic considered as made up of a succession of points has been replaced by a conic considered as generated by a moving tangent line. The theory upon which this wonderful law of duality is based will be developed in the next chapter.
PROBLEMS
1. Given four lines in the plane, to construct another which shall meet them in four harmonic points.
2. Where are all such lines found?
3. Given any five lines in the plane, construct on each the point of contact with the conic tangent to them all.
[pg 55]4. Given four lines and the point of contact on one, to construct the conic. ("To construct the conic" means here to draw as many other tangents as may be desired.)
5. Given three lines and the point of contact on two of them, to construct the conic.
6. Given four lines and the line at infinity, to construct the conic.
7. Given three lines and the line at infinity, together with the point of contact at infinity, to construct the conic.
8. Given three lines, two of which are asymptotes, to construct the conic.
9. Given five tangents to a conic, to draw a tangent which shall be parallel to any one of them.
10. The lines a, b, c are drawn parallel to each other. The lines a', b', c' are also drawn parallel to each other. Show why the lines (ab', a'b), (bc', b'c), (ca', c'a) meet in a point. (In problems 6 to 10 inclusive, parallel lines are to be drawn.)
CHAPTER VI - POLES AND POLARS
95. Inscribed and circumscribed quadrilaterals. The following theorems have been noted as special cases of Pascal's and Brianchon's theorems:
If a quadrilateral be inscribed in a conic, two pairs of opposite sides and the tangents at opposite vertices intersect in four points, all of which lie on a straight line.
If a quadrilateral be circumscribed about a conic, the lines joining two pairs of opposite vertices and the lines joining two opposite points of contact are four lines which meet in a point.
96. Definition of the polar line of a point. Consider the quadrilateral K, L, M, N inscribed in the conic (Fig. 26). It determines the four harmonic points A, B, C, D which project from N in to the four harmonic points M, B, K, O. Now the tangents at K and M meet in P, a point on the line AB. The line AB is thus determined entirely by [pg 57] the point O. For if we draw any line through it, meeting the conic in K and M, and construct the harmonic conjugate B of O with respect to K and M, and also the two tangents at K and M which meet in the point P, then BP is the line in question. It thus appears that the line LON may be any line whatever through O; and since D, L, O, N are four harmonic points, we may describe the line AB as the locus of points which are harmonic conjugates of O with respect to the two points where any line through O meets the curve.
97. Furthermore, since the tangents at L and N meet on this same line, it appears as the locus of intersections of pairs of tangents drawn at the extremities of chords through O.
98. This important line, which is completely determined by the point O, is called the polar of O with respect to the conic; and the point O is called the pole of the line with respect to the conic.
99. If a point B is on the polar of O, then it is harmonically conjugate to O with respect to the two intersections K and M of the line BC with the conic. But for the same reason O is on the polar of B. We have, then, the fundamental theorem
If one point lies on the polar of a second, then the second lies on the polar of the first.
100. Conjugate points and lines. Such a pair of points are said to be conjugate with respect to the conic. Similarly, lines are said to be conjugate to each other with respect to the conic if one, and consequently each, passes through the pole of the other.
[pg 58]
101. Construction of the polar line of a given point. Given a point P, if it is within the conic (that is, if no tangents may be drawn from P to the conic), we may construct its polar line by drawing through it any two chords and joining the two points of intersection of the two pairs of tangents at their extremities. If the point P is outside the conic, we may draw the two tangents and construct the chord of contact (Fig. 27).
102. Self-polar triangle. In Fig. 26 it is not difficult to see that AOC is a self-polar triangle, that is, each vertex is the pole of the opposite side. For B, M, O, K are four harmonic points, and they project to C in four harmonic rays. The line CO, therefore, meets the line AMN in a point on the polar of A, being separated from A harmonically by the points M and N. Similarly, the line CO meets KL in a point on the polar of A, and therefore CO is the polar of A. Similarly, OA is the polar of C, and therefore O is the pole of AC.
103. Pole and polar projectively related. Another very important theorem comes directly from Fig. 26.
As a point A moves along a straight line its polar with respect to a conic revolves about a fixed point and describes a pencil projective to the point-row described by A.
For, fix the points L and N and let the point A move along the line AQ; then the point-row A is projective to the pencil LK, and since K moves along the conic, the pencil LK is projective to the pencil NK, which in turn is projective to the point-row C, which, finally, is projective to the pencil OC, which is the polar of A.
[pg 59]104. Duality. We have, then, in the pole and polar relation a device for setting up a one-to-one correspondence between the points and lines of the plane—a correspondence which may be called projective, because to four harmonic points or lines correspond always four harmonic lines or points. To every figure made up of points and lines will correspond a figure made up of lines and points. To a point-row of the second order, which is a conic considered as a point-locus, corresponds a pencil of rays of the second order, which is a conic considered as a line-locus. The name 'duality' is used to describe this sort of correspondence. It is important to note that the dual relation is subject to the same exceptions as the one-to-one correspondence is, and must not be appealed to in cases where the one-to-one correspondence breaks down. We have seen that there is in Euclidean geometry one and only one ray in a pencil which has no point in a point-row perspective to it for a corresponding point; namely, the line parallel to the line of the point-row. Any theorem, therefore, that involves explicitly the point at infinity is not to be translated into a theorem concerning lines. Further, in the pencil the angle between two lines has nothing to correspond to it in a point-row perspective to the pencil. Any theorem, therefore, that mentions angles is not translatable into another theorem by means of the law of duality. Now we have seen that the notion of the infinitely distant point on a line involves the notion of dividing a segment into any number of equal parts—in other words, of measuring. If, therefore, we call any theorem that has to do with the line at infinity or with [pg 60] the measurement of angles a metrical theorem, and any other kind a projective theorem, we may put the case as follows:
Any projective theorem involves another theorem, dual to it, obtainable by interchanging everywhere the words 'point' and 'line.'
105. Self-dual theorems. The theorems of this chapter will be found, upon examination, to be self-dual; that is, no new theorem results from applying the process indicated in the preceding paragraph. It is therefore useless to look for new results from the theorem on the circumscribed quadrilateral derived from Brianchon's, which is itself clearly the dual of Pascal's theorem, and in fact was first discovered by dualization of Pascal's.
106. It should not be inferred from the above discussion that one-to-one correspondences may not be devised that will control certain of the so-called metrical relations. A very important one may be easily found that leaves angles unaltered. The relation called similarity leaves ratios between corresponding segments unaltered. The above statements apply only to the particular one-to-one correspondence considered.
PROBLEMS
1. Given a quadrilateral, construct the quadrangle polar to it with respect to a given conic.
2. A point moves along a straight line. Show that its polar lines with respect to two given conics generate a point-row of the second order.
[pg 61]3. Given five points, draw the polar of a point with respect to the conic passing through them, without drawing the conic itself.
4. Given five lines, draw the polar of a point with respect to the conic tangent to them, without drawing the conic itself.
5. Dualize problems 3 and 4.
6. Given four points on the conic, and the tangent at one of them, draw the polar of a given point without drawing the conic. Dualize.
7. A point moves on a conic. Show that its polar line with respect to another conic describes a pencil of rays of the second order.
Suggestion. Replace the given conic by a pair of protective pencils.
8. Show that the poles of the tangents of one conic with respect to another lie on a conic.
9. The polar of a point A with respect to one conic is a, and the pole of a with respect to another conic is A'. Show that as A travels along a line, A' also travels along another line. In general, if A describes a curve of degree n, show that A' describes another curve of the same degree n. (The degree of a curve is the greatest number of points that it may have in common with any line in the plane.)