Lehmer, Derrick Norman - An Elementary Course in Synthetic Projective Geometry

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[Figure 37]

FIG. 37

[Figure 38]

FIG. 38

*131. Involution of points on a point-row of the second order.* It is
important to note also, in Steiner's construction, that we have obtained
two point-rows of the second order superposed on the same conic, and have
paired the points of one with the points of the other in such a way that
the correspondence is double. We may then extend the notion of involution
to point-rows of the second order and say that _the points of a conic are
paired in involution when they are corresponding __ points of two
projective point-rows superposed on the conic, and when they correspond to
each other doubly._ With this definition we may prove the theorem: _The
lines joining corresponding points of a point-row of the second order in
involution all pass through a fixed point __U__, and the line joining any
two points __A__, __B__ meets the line joining the two corresponding
points __A'__, __B'__ in the points of a line __u__, which is the polar of
__U__ with respect to the conic._ For take _A_ and _A'_ as the centers of
two pencils, the first perspective to the point-row _A'_, _B'_, _C'_ and
the second perspective to the point-row _A_, _B_, _C_. Then, since the
common ray of the two pencils corresponds to itself, they are in
perspective position, and their axis of perspectivity _u_ (Fig. 38) is the
line which joins the point _(AB', A'B)_ to the point _(AC', A'C)_. It is
then immediately clear, from the theory of poles and polars, that _BB'_
and _CC'_ pass through the pole _U_ of the line _u_.

*132. Involution of rays.* The whole theory thus far developed may be
dualized, and a theory of lines in involution may be built up, starting
with the complete quadrilateral. Thus,

_The three pairs of rays which may be drawn from a point through the three
pairs of opposite vertices of a complete quadrilateral are said to be in
involution. If the pairs __aa'__ and __bb'__ are fixed, and the line __c__
describes a pencil, the corresponding line __c'__ also describes a pencil,
and the rays of the pencil are said to be paired in the involution
determined by __aa'__ and __bb'__._

*133. Double rays.* The self-corresponding rays, of which there are two
or none, are called _double rays_ of the involution. Corresponding rays of
the involution are harmonic conjugates with respect to the double rays. To
the theorem of Desargues (§ 125) which has to do with the system of conics
through four points we have the dual:

_The tangents from a fixed point to a system of conics tangent to four
fixed lines form a pencil of rays in involution._

*134.* If a conic of the system should go through the fixed point, it is
clear that the two tangents would coincide and indicate a double ray of
the involution. The theorem, therefore, follows:

_Two conics or none may be drawn through a fixed point to be tangent to
four fixed lines._

*135. Double correspondence.* It further appears that two projective
pencils of rays which have the same center are in involution if two pairs
of rays correspond to each other doubly. From this it is clear that we
might have deemed six rays in involution as six rays which pass through a
point and also through six points in involution. While this would have
been entirely in accord with the treatment which was given the
corresponding problem in the theory of harmonic points and lines, it is
more satisfactory, from an aesthetic point of view, to build the theory of
lines in involution on its own base. The student can show, by methods
entirely analogous to those used in the second chapter, that involution is
a projective property; that is, six rays in involution are cut by any
transversal in six points in involution.

*136. Pencils of rays of the second order in involution.* We may also
extend the notion of involution to pencils of rays of the second order.
Thus, _the tangents to a conic are in involution when they are
corresponding rays of two protective pencils of the second order
superposed upon the same conic, and when they correspond to each other
doubly._ We have then the theorem:

*137.* _The intersections of corresponding rays of a pencil of the second
order in involution are all on a straight line __u__, and the intersection
of any two tangents __ab__, when joined to the intersection of the
corresponding tangents __a'b'__, gives a line which passes through a fixed
point __U__, the pole of the line __u__ with respect to the conic._

*138. Involution of rays determined by a conic.* We have seen in the
theory of poles and polars (§ 103) that if a point _P_ moves along a line
_m_, then the polar of _P_ revolves about a point. This pencil cuts out on
_m_ another point-row _P'_, projective also to _P_. Since the polar of _P_
passes through _P'_, the polar of _P'_ also passes through _P_, so that
the correspondence between _P_ and _P'_ is double. The two point-rows are
therefore in involution, and the double points, if any exist, are the
points where the line _m_ meets the conic. A similar involution of rays
may be found at any point in the plane, corresponding rays passing each
through the pole of the other. We have called such points and rays
_conjugate_ with respect to the conic (§ 100). We may then state the
following important theorem:

*139.* _A conic determines on every line in its plane an involution of
points, corresponding points in the involution __ being conjugate with
respect to the conic. The double points, if any exist, are the points
where the line meets the conic._

*140.* The dual theorem reads: _A conic determines at every point in the
plane an involution of rays, corresponding rays being conjugate with
respect to the conic. The double rays, if any exist, are the tangents from
the point to the conic._

PROBLEMS

1. Two lines are drawn through a point on a conic so as always to make
right angles with each other. Show that the lines joining the points where
they meet the conic again all pass through a fixed point.

2. Two lines are drawn through a fixed point on a conic so as always to
make equal angles with the tangent at that point. Show that the lines
joining the two points where the lines meet the conic again all pass
through a fixed point.

3. Four lines divide the plane into a certain number of regions.
Determine for each region whether two conics or none may be drawn to pass
through points of it and also to be tangent to the four lines.

4. If a variable quadrangle move in such a way as always to remain
inscribed in a fixed conic, while three of its sides turn each around one
of three fixed collinear points, then the fourth will also turn around a
fourth fixed point collinear with the other three.

5. State and prove the dual of problem 4.

6. Extend problem 4 as follows: If a variable polygon of an even number
of sides move in such a way as always to remain inscribed in a fixed
conic, while all its sides but one pass through as many fixed collinear
points, then the last side will also pass through a fixed point collinear
with the others.

7. If a triangle _QRS_ be inscribed in a conic, and if a transversal _s_
meet two of its sides in _A_ and _A'_, the third side and the tangent at
the opposite vertex in _B_ and _B'_, and the conic itself in _C_ and _C'_,
then _AA'_, _BB'_, _CC'_ are three pairs of points in an involution.

8. Use the last exercise to solve the problem: Given five points, _Q_,
_R_, _S_, _C_, _C'_, on a conic, to draw the tangent at any one of them.

9. State and prove the dual of problem 7 and use it to prove the dual of
problem 8.

10. If a transversal cut two tangents to a conic in _B_ and _B'_, their
chord of contact in _A_, and the conic itself in _P_ and _P'_, then the
point _A_ is a double point of the involution determined by _BB'_ and
_PP'_.

11. State and prove the dual of problem 10.

12. If a variable conic pass through two given points, _P_ and _P'_, and
if it be tangent to two given lines, the chord of contact of these two
tangents will always pass through a fixed point on _PP'_.

13. Use the last theorem to solve the problem: Given four points, _P_,
_P'_, _Q_, _S_, on a conic, and the tangent at one of them, _Q_, to draw
the tangent at any one of the other points, _S_.

14. Apply the theorem of problem 9 to the case of a hyperbola where the
two tangents are the asymptotes. Show in this way that if a hyperbola and
its asymptotes be cut by a transversal, the segments intercepted by the
curve and by the asymptotes respectively have the same middle point.

15. In a triangle circumscribed about a conic, any side is divided
harmonically by its point of contact and the point where it meets the
chord joining the points of contact of the other two sides.

CHAPTER IX - METRICAL PROPERTIES OF INVOLUTIONS

[Figure 39]

FIG. 39

*141. Introduction of infinite point; center of involution.* We connect
the projective theory of involution with the metrical, as usual, by the
introduction of the elements at infinity. In an involution of points on a
line the point which corresponds to the infinitely distant point is called
the _center_ of the involution. Since corresponding points in the
involution have been shown to be harmonic conjugates with respect to the
double points, the center is midway between the double points when they
exist. To construct the center (Fig. 39) we draw as usual through _A_ and
_A'_ any two rays and cut them by a line parallel to _AA'_ in the points
_K_ and _M_. Join these points to _B_ and _B'_, thus determining on _AK_
and _AN_ the points _L_ and _N_. _LN_ meets _AA'_ in the center _O_ of the
involution.

*142. Fundamental metrical theorem.* From the figure we see that the
triangles _OLB'_ and _PLM_ are similar, _P_ being the intersection of KM
and LN. Also the triangles _KPN_ and _BON_ are similar. We thus have

_OB : PK = ON : PN_

and

_OB' : PM = OL : PL;_

whence

_OB . OB' : PK . PM = ON . OL : PN . PL._

In the same way, from the similar triangles _OAL_ and _PKL_, and also
_OA'N_ and _PMN_, we obtain

_OA . OA' : PK . PM = ON . OL : PN . PL,_

and this, with the preceding, gives at once the fundamental theorem, which
is sometimes taken also as the definition of involution:

_OA . OA' = OB . OB' = __constant__,_

or, in words,

_The product of the distances from the center to two corresponding points
in an involution of points is constant._

*143. Existence of double points.* Clearly, according as the constant is
positive or negative the involution will or will not have double points.
The constant is the square root of the distance from the center to the
double points. If _A_ and _A'_ lie both on the same side of the center,
the product _OA . OA'_ is positive; and if they lie on opposite sides, it
is negative. Take the case where they both lie on the same side of the
center, and take also the pair of corresponding points _BB'_. Then, since
_OA . OA' = OB . OB'_, it cannot happen that _B_ and _B'_ are separated
from each other by _A_ and _A'_. This is evident enough if the points are
on opposite sides of the center. If the pairs are on the same side of the
center, and _B_ lies between _A_ and _A'_, so that _OB_ is greater, say,
than _OA_, but less than _OA'_, then, by the equation _OA . OA' = OB .
OB'_, we must have _OB'_ also less than _OA'_ and greater than _OA_. A
similar discussion may be made for the case where _A_ and _A'_ lie on
opposite sides of _O_. The results may be stated as follows, without any
reference to the center:

_Given two pairs of points in an involution of points, if the points of
one pair are separated from each other by the points of the other pair,
then the involution has no double points. If the points of one pair are
not separated from each other by the points of the other pair, then the
involution has two double points._

*144.* An entirely similar criterion decides whether an involution of
rays has or has not double rays, or whether an involution of planes has or
has not double planes.

[Figure 40]

FIG. 40

*145. Construction of an involution by means of circles.* The equation
just derived, _OA . OA' = OB . OB'_, indicates another simple way in which
points of an involution of points may be constructed. Through _A_ and _A'_
draw any circle, and draw also any circle through _B_ and _B'_ to cut the
first in the two points _G_ and _G'_ (Fig. 40). Then any circle through
_G_ and _G'_ will meet the line in pairs of points in the involution
determined by _AA'_ and _BB'_. For if such a circle meets the line in the
points _CC'_, then, by the theorem in the geometry of the circle which
says that _if any chord is __ drawn through a fixed point within a circle,
the product of its segments is constant in whatever direction the chord is
drawn, and if a secant line be drawn from a fixed point without a circle,
the product of the secant and its external segment is constant in whatever
direction the secant line is drawn_, we have _OC . OC' = OG . OG' =_
constant. So that for all such points _OA . OA' = OB . OB' = OC . OC'_.
Further, the line _GG'_ meets _AA'_ in the center of the involution. To
find the double points, if they exist, we draw a tangent from _O_ to any
of the circles through _GG'_. Let _T_ be the point of contact. Then lay
off on the line _OA_ a line _OF_ equal to _OT_. Then, since by the above
theorem of elementary geometry _OA . OA' = OT__2__ = OF__2_, we have one
double point _F_. The other is at an equal distance on the other side of
_O_. This simple and effective method of constructing an involution of
points is often taken as the basis for the theory of involution. In
projective geometry, however, the circle, which is not a figure that
remains unaltered by projection, and is essentially a metrical notion,
ought not to be used to build up the purely projective part of the theory.

*146.* It ought to be mentioned that the theory of analytic geometry
indicates that the circle is a special conic section that happens to pass
through two particular imaginary points on the line at infinity, called
the _circular points_ and usually denoted by _I_ and _J_. The above method
of obtaining a point-row in involution is, then, nothing but a special
case of the general theorem of the last chapter (§ 125), which asserted
that a system of conics through four points will cut any line in the plane
in a point-row in involution.

[Figure 41]

FIG. 41

*147. Pairs in an involution of rays which are at right angles. Circular
involution.* In an involution of rays there is no one ray which may be
distinguished from all the others as the point at infinity is
distinguished from all other points on a line. There is one pair of rays,
however, which does differ from all the others in that for this particular
pair the angle is a right angle. This is most easily shown by using the
construction that employs circles, as indicated above. The centers of all
the circles through _G_ and _G'_ lie on the perpendicular bisector of the
line _GG'_. Let this line meet the line _AA'_ in the point _C_ (Fig. 41),
and draw the circle with center _C_ which goes through _G_ and _G'_. This
circle cuts out two points _M_ and _M'_ in the involution. The rays _GM_
and _GM'_ are clearly at right angles, being inscribed in a semicircle.
If, therefore, the involution of points is projected to _G_, we have found
two corresponding rays which are at right angles to each other. Given now
any involution of rays with center _G_, we may cut across it by a straight
line and proceed to find the two points _M_ and _M'_. Clearly there will
be only one such pair unless the perpendicular bisector of _GG'_ coincides
with the line _AA'_. In this case every ray is at right angles to its
corresponding ray, and the involution is called _circular_.

*148. Axes of conics.* At the close of the last chapter (§ 140) we gave
the theorem: _A conic determines at every point in its plane an involution
of rays, corresponding rays __ being conjugate with respect to the conic.
The double rays, if any exist, are the tangents from the point to the
conic._ In particular, taking the point as the center of the conic, we
find that conjugate diameters form a system of rays in involution, of
which the asymptotes, if there are any, are the double rays. Also,
conjugate diameters are harmonic conjugates with respect to the
asymptotes. By the theorem of the last paragraph, there are two conjugate
diameters which are at right angles to each other. These are called axes.
In the case of the parabola, where the center is at infinity, and on the
curve, there are, properly speaking, no conjugate diameters. While the
line at infinity might be considered as conjugate to all the other
diameters, it is not possible to assign to it any particular direction,
and so it cannot be used for the purpose of defining an axis of a
parabola. There is one diameter, however, which is at right angles to its
conjugate system of chords, and this one is called the _axis_ of the
parabola. The circle also furnishes an exception in that every diameter is
an axis. The involution in this case is circular, every ray being at right
angles to its conjugate ray at the center.

*149. Points at which the involution determined by a conic is circular.*
It is an important problem to discover whether for any conic other than
the circle it is possible to find any point in the plane where the
involution determined as above by the conic is circular. We shall proceed
to the curious problem of proving the existence of such points and of
determining their number and situation. We shall then develop the
important properties of such points.

*150.* It is clear, in the first place, that such a point cannot be on
the outside of the conic, else the involution would have double rays and
such rays would have to be at right angles to themselves. In the second
place, if two such points exist, the line joining them must be a diameter
and, indeed, an axis. For if _F_ and _F'_ were two such points, then,
since the conjugate ray at _F_ to the line _FF'_ must be at right angles
to it, and also since the conjugate ray at _F'_ to the line _FF'_ must be
at right angles to it, the pole of _FF'_ must be at infinity in a
direction at right angles to _FF'_. The line _FF'_ is then a diameter, and
since it is at right angles to its conjugate diameter, it must be an axis.
From this it follows also that the points we are seeking must all lie on
one of the two axes, else we should have a diameter which does not go
through the intersection of all axes--the center of the conic. At least one
axis, therefore, must be free from any such points.

[Figure 42]

FIG. 42

*151.* Let now _P_ be a point on one of the axes (Fig. 42), and draw any
ray through it, such as _q_. As _q_ revolves about _P_, its pole _Q_ moves
along a line at right angles to the axis on which _P_ lies, describing a
point-row _p_ projective to the pencil of rays _q_. The point at infinity
in a direction at right angles to _q_ also describes a point-row
projective to _q_. The line joining corresponding points of these two
point-rows is always a conjugate line to _q_ and at right angles to _q_,
or, as we may call it, a _conjugate normal_ to _q_. These conjugate
normals to _q_, joining as they do corresponding points in two projective
point-rows, form a pencil of rays of the second order. But since the point
at infinity on the point-row _Q_ corresponds to the point at infinity in a
direction at right angles to _q_, these point-rows are in perspective
position and the normal conjugates of all the lines through _P_ meet in a
point. This point lies on the same axis with _P_, as is seen by taking _q_
at right angles to the axis on which _P_ lies. The center of this pencil
may be called _P'_, and thus we have paired the point _P_ with the point
_P'_. By moving the point _P_ along the axis, and by keeping the ray _q_
parallel to a fixed direction, we may see that the point-row _P_ and the
point-row _P'_ are projective. Also the correspondence is double, and by
starting from the point _P'_ we arrive at the point _P_. Therefore the
point-rows _P_ and _P'_ are in involution, and if only the involution has
double points, we shall have found in them the points we are seeking. For
it is clear that the rays through _P_ and the corresponding rays through
_P'_ are conjugate normals; and if _P_ and _P'_ coincide, we shall have a
point where all rays are at right angles to their conjugates. We shall now
show that the involution thus obtained on one of the two axes must have
double points.

[Figure 43]

FIG. 43

*152. Discovery of the foci of the conic.* We know that on one axis no
such points as we are seeking can lie (§ 150). The involution of points
_PP'_ on this axis can therefore have no double points. Nevertheless, let
_PP'_ and _RR'_ be two pairs of corresponding points on this axis (Fig.
43). Then we know that _P_ and _P'_ are separated from each other by _R_
and _R'_ (§ 143). Draw a circle on _PP'_ as a diameter, and one on _RR'_
as a diameter. These must intersect in two points, _F_ and _F'_, and since
the center of the conic is the center of the involution _PP'_, _RR'_, as
is easily seen, it follows that _F_ and _F'_ are on the other axis of the
conic. Moreover, _FR_ and _FR'_ are conjugate normal rays, since _RFR'_ is
inscribed in a semicircle, and the two rays go one through _R_ and the
other through _R'_. The involution of points _PP'_, _RR'_ therefore
projects to the two points _F_ and _F'_ in two pencils of rays in
involution which have for corresponding rays conjugate normals to the
conic. We may, then, say:

_There are two and only two points of the plane where the involution
determined by the conic is circular. These two points lie on one of the
axes, at equal distances from the center, on the inside of the conic.
These points are called the foci of the conic._

*153. The circle and the parabola.* The above discussion applies only to
the central conics, apart from the circle. In the circle the two foci fall
together at the center. In the case of the parabola, that part of the
investigation which proves the existence of two foci on one of the axes
will not hold, as we have but one axis. It is seen, however, that as _P_
moves to infinity, carrying the line _q_ with it, _q_ becomes the line at
infinity, which for the parabola is a tangent line. Its pole _Q_ is thus
at infinity and also the point _P'_, so that _P_ and _P'_ fall together at
infinity, and therefore one focus of the parabola is at infinity. There
must therefore be another, so that

_A parabola has one and only one focus in the finite part of the plane._

[Figure 44]

FIG. 44

*154. Focal properties of conics.* We proceed to develop some theorems
which will exhibit the importance of these points in the theory of the
conic section. Draw a tangent to the conic, and also the normal at the
point of contact _P_. These two lines are clearly conjugate normals. The
two points _T_ and _N_, therefore, where they meet the axis which contains
the foci, are corresponding points in the involution considered above, and
are therefore harmonic conjugates with respect to the foci (Fig. 44); and
if we join them to the point _P_, we shall obtain four harmonic lines. But
two of them are at right angles to each other, and so the others make
equal angles with them (Problem 4, Chapter II). Therefore

_The lines joining a point on the conic to the foci make equal angles with
the tangent._

It follows that rays from a source of light at one focus are reflected by
an ellipse to the other.

*155.* In the case of the parabola, where one of the foci must be
considered to be at infinity in the direction of the diameter, we have

[Figure 45]

FIG. 45

_A diameter makes the same angle with the tangent at its extremity as that
tangent does with the line from its point of contact to the focus (Fig.
45)._

*156.* This last theorem is the basis for the construction of the
parabolic reflector. A ray of light from the focus is reflected from such
a reflector in a direction parallel to the axis of the reflector.

*157. Directrix. Principal axis. Vertex.* The polar of the focus with
respect to the conic is called the _directrix_. The axis which contains
the foci is called the _principal axis_, and the intersection of the axis
with the curve is called the _vertex_ of the curve. The directrix is at
right angles to the principal axis. In a parabola the vertex is equally
distant from the focus and the directrix, these three points and the point
at infinity on the axis being four harmonic points. In the ellipse the
vertex is nearer to the focus than it is to the directrix, for the same
reason, and in the hyperbola it is farther from the focus than it is from
the directrix.

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