Page images
PDF
EPUB

If, on the other hand, we take a point P in the plane of a conic, I conjugates with regard to the foci. If therefore the two foci F, and we get to each line a through P one conjugate line which joins P F be joined to P, these lines will be harmonic with regard to the to the pole of a. These pairs of conjugate lines through P form an involution in the pencil at P. The focal rays of this involution are the tangents drawn from P to the conic. This gives the theorem reciprocal to the last, viz:

A conic determines in every pencil in its plane an involution, corresponding lines being conjugate lines with regard to the conic.

If the point is without the conic the involution is hyperbolic, the tangents from the points being the focal rays.

If the point lies on the conic the involution is parabolic, the tangens at the point counting for coincident focal rays.

If the point is within the conic the involution is elliptic, having no focal rays.

It will further be seen that the involution determined by a conic on any line is a section of the involution, which is determined by the conic at the pole P of p.

$83. Foci.-The centre of a pencil in which the conic determines a circular involution is called a "focus" of the conic.

In other words, a focus is such a point that every line through it is perpendicular to its conjugate line. The polar to a focus is called a directrix of the conic.

From the definition it follows that every focus lies on an axis, for the line joining a focus to the centre of the conic is a diameter to which the conjugate lines are perpendicular; and every line joining two foci is an axis, for the perpendiculars to this line through the foci are conjugate to it. These conjugate lines pass through the pole of the line, the pole lies therefore at infinity, and the line is a diameter, hence by the last property an axis.

It follows that all foci lie on one axis, for no line joining a point in one axis to a point in the other can be an axis.

As the conic determines in the pencil which has its centre at a focus a circular involution, no tangents can be drawn from the focus to the conic. Hence each focus lies within a conic; and a directrix does not cut the conic.

Further properties are found by the following considerations: § 84. Through a point P one line p can be drawn, which is with regard to a given conic conjugate to a given line q, viz. that line which joins the point P to the pole of the line q. If the line q is made to describe a pencil about a point Q, then the line p will describe a pencil about P. These two pencils will be projective, for the line P passes through the pole of q, and whilst q describes the pencil Q, its pole describes a projective row, and this row is perspective to the pencil P.

We now take the point P on an axis of the conic, draw any line through it, and from the pole of p draw a perpendicular g to p. which have their centres at P and Q, the lines p and q are conjugate lines at right angles to one another. Besides, to the axis as a ray in either pencil will correspond in the other the perpendicular to the axis ($72). The conic generated by the intersection of corresponding lines in the two pencils is therefore the circle on PQ as diameter, so that every line in P is perpendicular to its corresponding line in Q.

To every point P on an axis of a conic corresponds thus a point Q, such that conjugate lines through P and Q are perpendicular. We shall show that these point-pairs P, Q form an involution. To do this let us move P along the axis, and with it the line p, keeping the latter parallel to itself. Then P describes a row, pa perspective pencil (of parallels), and the pole of p a projective row. At the same time the line q describes a pencil of parallels perpendicular top, and perspective to the row formed by the pole of p. The point Q, therefore, where a cuts the axis, describes a row projective to the row of points P. The two points P and Q describe thus two projective rows on the axis; and not only does P as a point in the first row correspond to Q, but also Q as a point in the first corresponds to P. The two rows therefore form an involution. The centre of this involution, it is easily seen, is the centre of the conic.

A focus of this involution has the property that any two conjugate lines through it are perpendicular; hence, it is a focus to the conic.

Such involution exists on each axis. But only one of these can have foci, because all foci lie on the same axis. The involution on one of the axes is elliptic, and appears (§ 80) therefore as the section of two circular involutions in two pencils whose centres lie in the other axis. These centres are foci, hence the one axis contains two foci, the other axis.none; or every central conic has two foci which lie on one axis equidistant from the centre.

The axis which contains the foci is called the principal axis; in case of an hyperbola it is the axis which cuts the curve, because the foci lie within the conic.

In case of the parabola there is but one axis. The involution on this axis has its centre at infinity. One focus is therefore at infinity, the one focus only is finite. A parabola has only one focus.

$85. If through any point P (fig. 34) on a conic the tangent PT and the normal PN (.e. the perpendicular to the tangent through the point of contact) be drawn, these will be conjugate lines with regard to the conic, and at right angles to each other. They will therefore cut the principal axis in two points, which are conjugate in the involution considered in § 84; hence they are harmonic

B

FIG. 34

tangent and normal. As the latter are perpendicular, they will bisect the angles between the other pair. Hence

The lines joining any point on a conic to the two foci are equally inclined to the tangent and normal at that point. In case of the parabola this becomes

The line joining any point on a parabola to the focus and the diameter through the point, are equally inclined to the tangent and normal at that point.

A

From the definition of a focus it follows that

The segment of a tangent between the directrix and the point of contact is seen from the focus belonging to the directrix under a right angle, because the lines joining the focus to the ends of this segment are conjugate with regard to the conic, and therefore perpendicular.

With equal ease the following theorem is proved:

The two lines which join the points of contact of two tangents each to one focus, but not both to the same, are seen from the intersection of the tangents under equal angles.

$86. Other focal properties of a conic are obtained by the following considerations:

Let F (fig. 35) be a focus to a conic, ƒ the corresponding directrix, A and B the points of contact of two tangents meeting at T, and P the point where the line AB cuts the direc

trix. Then TF will be the polar of P (because A polars of F and T meet TF

B

PF are conjugate lines
through a focus, and
therefore perpendicular.
They are further har-
monic conjugates with
regard to FA and FB
($8 64 and 13), so that A
they bisect the angles
formed by these lines.
This by the way
proves-

The segments between B
the point of intersection
of two tangents to a conic
and their points of con-
tact are seen from a focus
under equal angles.

If we next draw through A and B lines parallel to TF, then the points A1. B1 where these cut the directrix will be harmonic conjugates with regard to P and the point where FT cuts the directrix. The lines FT and FP bisect therefore also the angles between FA, and FB1. From this it follows easily that the triangles FAA, and FBB, are equiangular, and therefore similar, so that FA : AA1 =FB : BB1.

FIG. 35.

The triangles AAA, and BBB, formed by drawing perpendiculars from A and B to the directrix are also similar, so that AA, AA BB: BB2. This, combined with the above proportion, gives FA: AA, FB: BB. Hence the theorem:

The ratio of the distances of any point on a conic from a focus and the corresponding directrix is constant.

To determine this ratio we consider its value for a vertex on the principal axis. In an ellipse the focus lies between the two vertices on this axis, hence the focus is nearer to a vertex than to the corresponding directrix. Similarly, in an hyperbola a vertex is nearer

[ocr errors]

to the directrix than to the focus. In a parabola the vertex lies halfway between directrix and focus.

It follows in an ellipse the ratio between the distance of a point from the focus to that from the directrix is less than unity, in the parabola it equals unity, and in the hyperbola it is greater than unity.

It is here the same which focus we take, because the two foci lie symmetrical to the axis of the conic. If now P is any point on the conic having the distances 7, and 7, from the foci and the distances d and d from the corresponding directrices, then r1/d1 =r2/d2=e, where e is constant. Hence also T12

e.

In the ellipse, which lies between the directrices, d+dh is constant, therefore also +2. In the hyperbola on the other hand d-d is constant, equal to the distance between the directrices, therefore in this case - is constant.

If we call the distances of a point on a conic from the focus its focal distances we have the theorem:

In an ellipse the sum of the focal distances is constant; and in an hyperbola the difference of the focal distances is constant.

This constant sum or difference equals in both cases the length of the principal axis.

PENCIL OF CONICS

$87. Through four points A, B, C, D in a plane, of which no three lie in a line, an infinite number of conics may be drawn, viz. through these four points and any fifth one single conic. This system of conics is called a pencil of conics. Similarly, all conics touching four fixed lines form a system such that any fifth tangent determines one and only one conic. We have here the theorems: The pairs of points in which any line is cut by a system of conics through four fixed points

The pairs of tangents which can be drawn from a point to system of conics touching four

a

are in involution.

fixed lines are in involution.

We prove the first theorem only. Let ABCD (fig. 36) be the four-point, then any line will cut two opposite sides AC, BD in

[blocks in formation]

or

But this is, according to § 77 (7), the condition that M, N are corresponding points in the involution determined by the point pairs E, E, F, Fin which the line t cuts pairs of opposite sides of the four-point ABCD. This involution is independent of the particular,

chosen.

§88. There follow several important theorems: Through four points two, one, or no conics may be drawn which touch any given line, according as the involution determined by the given four-point on the line has real, coincident or imaginary foci.

Two, one, or no conics may be drawn which touch four given lines and pass through a given point, according as the involution determined by the given four-side at the point has real, coincident or imaginary focal rays.

For the conic through four points which touches a given line has its point of contact at a focus of the involution determined by the four-point on the line.

As a special case we get, by taking the line at infinity: Through four points of which none is at infinity either two or no parabolas may be drawn.

The problem of drawing a conic through four points and touching a given line is solved by determining the points of contact on the line, that is, by determining the foci of the involution in which the line cuts the sides of the four-point. The corresponding remark holds for the problem of drawing the conics which touch four lines and pass through a given point.

RULED QUADRIC SURFACES

the same plane, in which case lines joining corresponding points 889. We have considered hitherto projective rows which he in envelop a conic. We shall now consider projective rows whose bases do not meet. In this case, corresponding points will be joined by lines which do not lie in a plane, but on some surface, which like every surface generated by lines is called a ruled surface. This surface clearly contains the bases of the two rows.

|

If the points in either row be joined to the base of the other, we obtain two axial pencils which are also projective, those planes being corresponding which pass through corresponding points in the given rows. If A', A be two corresponding points, a, a' the planes in the axial pencils passing through them, then AA' will be the line of intersection of the corresponding planes a, a' and also the line joining corresponding points in the rows.

If we cut the whole figure by a plane this will cut the axial pencils in two projective flat pencils, and the curve of the second order generated by these will be the curve in which the plane cuts the surface. Hence

The locus of lines joining corresponding points in two projective rows which do not lie in the same plane is a surface which contains the bases of the rows, and which can also be generated by the lines of intersection of corresponding planes in two projective axial pencils. This surface is cut by every plane in a curve of the second order, hence either in a conic or in a line-pair. No line which does not lie altogether on the surface can have more than two points in common with the surface, which is therefore said to be of the second order or is called a ruled quadric surface.

That no line which does not lie on the surface can cut the surface in more than two points is seen at once if a plane be drawn through the line, for this will cut the surface in a conic. It follows also that a line which contains more than two points of the surface lies altogether on the surface.

§ 90. Through any point in space one line can always be drawn cutting two given lines which do not themselves meet. If therefore three lines in space be given of which no two meet, then through every point in either one line may be drawn cutting

the other two.

If a line moves so that it always cuts three given lines of which no two meet, then it generates a ruled quadric surface.

Let a, b, c be the given lines, and p, q,r. lines cutting them in the points A, A', A'...; B, B', B' ...; C, C', C... respectively; then the planes through a containing p, q, r, and the planes through & containing the same lines, may be taken as corresponding planes in two axial pencils which are projective, because both pencils cut the line c in the same row, C, C, C...; the surface can therefore be gener ated by projective axial pencils.

Of the lines p, q, r... no two can meet, for otherwise the lines a, b, c which cut them would also lie in their plane. There is a single infinite number of them, for one passes through each point of s These lines are said to form a set of lines on the surface.

[blocks in formation]

892. We establish a correspondence between the lines and planes in pencils in space, or reciprocally between the points and lines in two or more planes, but consider principally pencils.

In two pencils we may either make planes correspond to planes and lines to lines, or else planes to lines and lines to planes. li hereby the condition be satisfied that to a flat, or axial, pencil corresponds in the first case a projective flat, or axial, pencil, and in the second a projective axial, or flat, pencil, the pencils are said to be projective in the first case and reciprocal in the second.

For instance, two pencils which join two points S, and S to the different points and lines in a given planer are projective (and in perspective position), if those lines and planes be taken as

corresponding which meet the plane in the same point or in the
same line.
In this case every plane through both centres S, and S.
of the two pencils will correspond to itself. If these pencils are
brought into any other position they will be projective (but not
perspective).

The correspondence between two projective pencils is uniquely determined, if to four rays (or planes) in the one the corresponding rays (or planes) in the other are given, provided that no three rays of either set lie in a plane.

Let a, b, c, d be four rays in the one, a', b', c', d' the corresponding rays in the other pencil. We shall show that we can find for every ray e in the first a single corresponding ray e' in the second. Το the axial pencil a (b, c, d ...) formed by the planes which join a to b, c, d..., respectively corresponds the axial pencil a' (b', c', d'...), and this correspondence is determined. Hence, the plane a'e' which corresponds to the plane ae is determined. Similarly, the plane b'e' may be found and both together determine the ray e'.

Similarly the correspondence between two reciprocal pencils is determined if for four rays in the one the corresponding planes in the other are given.

In the pencil S we draw some plane which passes through T, but not through S, or S. It will cut the two conics first at T, and therefore each at some other point which we call A and B respectively. These we join to S by lines a and b, and now establish the required correspondence between the pencils S, and S as follows:To ST shall correspond the plane o, to the plane a the line a, and to B the line b, hence to the flat pencil in a the axial pencil a. These pencils are made projective by aid of the conic in a.

§ 93. We may now combine

1. Two reciprocal pencils.

In the same manner the flat pencil in B, is made projective to the axial pencil b by aid of the conic in B, corresponding elements being

Each ray cuts its corresponding plane in a point, the locus those which meet on the conic. This determines the correspondence, of these points is a quadric surface. for we know for more than four rays in S, the corresponding planes 2. Two projective pencils. in S. The two pencils S and S, thus made reciprocal generate a quadric surface ', which passes through the point S and through the two conics a1 and B1.

Each plane cuts its corresponding plane in a line, but a
ray as a rule does not cut its corresponding ray. The
locus of points where a ray cuts its corresponding ray
is a twisted cubic. The lines where a plane cuts its
corresponding plane are secants

The two surfaces and have therefore the points S and S, and the conics a, and 8, in common. To show that they are identical, we draw a plane through S and S2, cutting each of the conics a, and B in two points, which will always be possible. This plane cuts and ' in two conics which have the point S and the points where it cuts a, and B in common, that is five points in all. The conics therefore coincide.

3. Three projective pencils.

The locus of intersection of corresponding planes is a cubic surface. Of these we consider only the first two cases. $94. If two pencils are reciprocal, then to a plane in either corresponds a line in the other, to a flat pencil an axial pencil, and so on. Every line cuts its corresponding plane in a point. If S, and S, be the centres of the two pencils, and P be a point where a line a, in the first cuts its corresponding plane a, then the line by in the pencil S, which passes through P will meet its corresponding plane B, in P. For b is a line in the plane ag. The corresponding plane ẞ, must therefore pass through the line ai, hence through P.

The points in which the lines in Si cut the planes corresponding to them in S, are therefore the same as the points in which the lines in S cut the planes corresponding to them in S1.

The locus of these points is a surface which is cut by a plane in a conic or in a line-pair and by a line in not more than two points unless it lies altogether on the surface. The surface itself is therefore called a quadric surface, or a surface of the second order.

To prove this we consider any line p in space.

The flat pencil in S, which lies in the plane drawn through pin and the corresponding axial pencil in S, determine on p two projective rows, and those points in these which coincide with their corresponding points lie on the surface. But there exist only two, or one, or no such points, unless every point coincides with its corresponding point. In the latter case the line lies altogether on the surface.

This proves also that a plane cuts the surface in a curve of the second order, as no line can have more than two points in common with it. To show that this is a curve of the same kind as those considered before, we have to show that it can be generated by projective flat pencils. We prove first that this is true for any plane through the centre of one of the pencils, and afterwards that every point on the surface may be taken as the centre of such pencil. Let then a be a plane through S. To the flat pencil in S, which it contains corresponds in S a projective axial pencil with axis and this cuts a, in a second flat pencil. These two flat pencils in are projective, and, in general, neither concentric nor perspective. They generate therefore a conic. But if the line 42 passes through S the pencils will have S, as common centre, and may therefore have two, or one, or no lines united with their corresponding lines. The section of the surface by the plane a, will be accordingly a line-pair or a single line, or else the plane a will have only the point S, in common with the surface.

In the first case the point of contact is said to be hyperbolic, in the second parabolic, in the third elliptic.

§ 95. It remains to be proved that every point S on the surface may be taken as centre of one of the pencils which generate the surface. Let S be any point on the surface generated by the reciprocal pencils S, and S. We have to establish a reciprocal correspondence between the pencils S and Si, so that the surface generated by them is identical with . To do this we draw two planes a and B1 through S1, cutting the surface in two conics which we also denote by a, and B. These conics meet at S1, and at some other point T where the line of intersection of a and B1 cuts the surface.

Every line through S, cuts the surface in two points, viz. first in S, and then at the point where it cuts its corresponding plane. If now the corresponding plane passes through Si, as in the case just considered, then the two points where l cuts the surface coincide at S, and the line is called a tangent to the surface with S, as point of contact. Hence if I be a tangent, it lies in that plane 7 which corresponds to the line SS, as a line in the pencil S2. The section of this plane has just been considered. It follows that

All tangents to quadric surface at the centre of one of the reciprocal pencils lie in a plane which is called the tangent plane to the surface at that point as point of contact.

To the line joining the centres of the two pencils as a line in one corresponds in the other the tangent plane at its centre.

The tangent plane to a quadric surface either cuts the surface in two lines, or it has only a single line, or else only a single point in common with the surface.

This proves that all those points P on ' lie on which have the property that the plane SSP cuts the conics a, Bi in two points cach. If the plane SSP has not this property, then we draw a plane SS,P. This cuts each surface in a conic, and these conics have in common the points S, S1, one point on each of the conics a1, B, and one point on one of the conics through S and S, which lie on both surfaces, hence five points. They are therefore coincident, and our theorem is proved.

$96. The following propositions follow:

A quadric surface has at every point a tangent plane.

Every plane section of a quadric surface is a conic or a line-pair Every line which has three points in common with a quadric surface lies on the surface.

Every conic which has five points in common with a quadric surface lies on the surface.

Through two conics which lie in different planes, but have two points common, and through one external point always one quadric surface may be drawn.

97. Every plane which cuts a quadric surface in a line-pair is a tangent plane. For every line in this plane through the centre of the line-pair (the point of intersection of the two lines) cuts the surface in two coincident points and is therefore a tangent to the surface, the centre of the line-pair being the point of contact.

If a quadric surface contains a line, then every plane through this line cuts the surface in a line-pair (or in two coincident lines). For this plane cannot cut the surface in a conic. Hence

If a quadric surface contains one line p then it contains an infinite number of lines, and through every point Q on the surface, one line q can be drawn which cuts p. For the plane through the point Q and the line p cuts the surface in a line-pair which must pass through Q and of which p is one line.

No two such lines q on the surface can meet. For as both meet p their plane would contain p and therefore cut the surface in a triangle.

Every line which cuts three lines q will be on the surface; for it has three points in common with it.

Hence the quadric surfaces which contain lines are the same as the ruled quadric surfaces considered in §§ 89-93, but with one important exception. In the last investigation we have left out of consideration the possibility of a plane having only one line (two coincident lines) in common with a quadric surface.

§ 98. To investigate this case we suppose first that there is one point A on the surface through which two different lines a, b can be drawn, which lie altogether on the surface.

If P is any other point on the surface which lies neither on a nor b, then the plane through P and a will cut the surface in a second line a' which passes through P and which cuts a. Similarly there is a line b' through P which cuts b. These two lines a' and b' may coincide, but then they must coincide with PA.

If this happens for one point P, it happens for every other point Q. For if two different lines could be drawn through Q, then by the same reasoning the line PQ would be altogether on the surface, hence two lines would be drawn through P against the assumption. From this follows:

If there is one point on a quadric surface through which one, but only one, line can be drawn on the surface, then through every point one line

can be drawn, and all these lines meet in a point. The surface is a cone of the second order.

If through one point on a quadric surface, two, and only two, lines can be drawn on the surface, then through every point two lines may be drawn, and the surface is a ruled quadric surface.

If through one point on a quadric surface no line on the surface can be drawn, then the surface contains no lines.

Using the definitions at the end of § 95, we may also say:-. On a quadric surface the points are all hyperbolic, or all parabolic, or all elliptic.

As an example of a quadric surface with elliptical points, we mention the sphere which may be generated by two reciprocal pencils, where to each line in one corresponds the plane perpendicular to it in the other.

$99. Poles and Polar Planes.-The theory of poles and polars with regard to a conic is easily extended to quadric surfaces.

Let P be a point in space not on the surface, which we suppose not to be a cone. On every line through P which cuts the surface in two points we determine the harmonic conjugate Q of P with regard to the points of intersection. Through one of these lines we draw two planes a and B. The locus of the points Q in a is a line a, the polar of P with regard to the conic in which a cuts the surface. Similarly the locus of points Q in B is a line b. This cuts a, because the line of intersection of a and B contains but one point Q. The locus of all points Q therefore is a plane. This plane is called the polar plane of the point P, with regard to the quadric surface. If P lies on the surface we take the tangent plane of P as its polar.

The following propositions hold:

1. Every point has a polar plane, which is constructed by drawing the polars of the point with regard to the conics in which two planes through the point cut the surface.

2. If Q is a point in the polar of P, then P is a point in the polar of Q, because this is true with regard to the conic in which a plane through PQ cuts the surface.

3. Every plane is the polar plane of one point, which is called the Pole of the plane.

The pole to a plane is found by constructing the polar planes of three points in the plane. Their intersection will be the pole.

4. The points in which the polar plane of P cuts the surface are points of contact of tangents drawn from P to the surface, as is easily seen. Hence:

5. The tangents drawn from a point P to a quadric surface form a cone of the second order, for the polar plane of P cuts it in a conic.

6. If the pole describes a line a, its polar plane will turn about another line a', as follows from 2. These lines a and a' are said to be conjugate with regard to the surface.

100. The pole of the line at infinity is called the centre of the surface. If it lies at the infinity, the plane at infinity is a tangent plane, and the surface is called a paraboloid.

The polar plane to any point al infinity passes through the centre, and is called a diametrical plane.

A line through the centre is called a diameter. It is bisected at the centre. The line conjugate to it lies at infinity.

If a point moves along a diameter its polar plane turns about the conjugate line at infinity; that is, it moves parallel to itself, its centre moving on the first line."

The middle points of parallel chords lie in a plane, viz. in the polar plane of the point at infinity through which the chords are drawn. The centres of parallel sections lie in a diameter which is a line conjugate to the line at infinity in which the planes meet.

TWISTED CUBICS

§ 101. If two pencils with centres S, and S2 are made projective, then to a ray in one corresponds a ray in the other, to a plane a plane, to a flat or axial pencil a projective flat or axial pencil, and

| projective pencils meet form a congruence. We shall see this congruence consists of all lines which cut a twisted cubic twice, or of all secants to a twisted cubic.

102. Let be the line SS2 as a line in the pencil S. To it corresponds a line 4 in S. At each of the centres two corresponding lines meet. The two axial pencils with 4 and 4 as axes are projective, and, as their axes meet at S, the intersections of corre sponding planes form a cone of the second order (§ 58), with S, as centre. If #1 and #2 be corresponding planes, then their intersection will be a line pa which passes through S. Corresponding to it in S will be a line p which lies in the plane, and which therefore meets p at some point P. Conversely, if a be any line in S, which meets its corresponding line P at a point P, then to the plane ↳ p will correspond the plane hip, that is, the plane SSP. These planes intersect in pa, so that p is a line on the quadric cone generated by the axial pencils 4 and 4. Hence:

All lines in one pencil which meet their corresponding lines in the other form a cone of the second order which has its centre at the centre of the first pencil, and passes through the centre of the second.

From this follows that the points in which corresponding rays meet lie on two cones of the second order which have the ray joining their centres in common, and form therefore, together with the line SS or h, the intersection of these cones. Any plane cuts each of the cones in a conic. These two conics have necessarily that point in common in which it cuts the line h, and therefore besides either one or three other points. It follows that the curve is of the third order as a plane may cut it in three, but not in more than three, points. Hence:

The locus of points in which corresponding lines on two projective pencils meet is a curve of the third order or a twisted cubic "k, which passes through the centres of the pencils, and which appears as the intersection of two cones of the second order, which have one line in common.

A line belonging to the congruence determined by the pencils is a secant of the cubic; it has two, or one, or no points in common with this cubic, and is called accordingly a secant proper, a tangent, or a secant improper of the cubic. A secant improper may be considered, to use the language of coordinate geometry, as a secant with imaginary points of intersection.

§ 103. If a and a be any two corresponding lines in the two pencils, then corresponding planes in the axial pencils having a, and da as axes generate a ruled quadric surface. If P be any point on the cubic k, and if Pi, pa be the corresponding rays in S, and S, which meet at P, then to the plane ap in S, corresponds a p2 in S. These therefore meet in a line through P.

This may be stated thus:

Those secants of the cubic which cut a ray a1, drawn through the centre S of one pencil, form a ruled quadric surface which passes through both centres, and which contains the twisted cubic k. Of such surfaces an infinite number exists. Every ray through S, or S2 which is not a secant determines one of them.

If, however, the rays a, and a, are secants meeting at A, then the ruled quadric surface becomes a cone of the second order, having A as centre. Or all lines of the congruence which pass through a point on the twisted cubic k form a cone of the second order. In other words, the projection of a twisted cubic from any point in the curve on to any plane is a conic.

If a is not a secant, but made to pass through any point Q in space, the ruled quadric surface determined by a will pass through Q. There will therefore be one line of the congruence passing through Q, and only one. For if two such lines pass through O, then the lines SQ and SQ will be corresponding lines; hence Q will be a point on the cubic k, and an infinite number of secants will pass through it.

Hence:

Through every point in space not on the twisted cubic one and only one secant to the cubic can be drawn.

so on.

There is a double infinite number of lines in a pencil. We shall see that a single infinite number of lines in one pencil meets its corresponding ray, and that the points of intersection form a curve in space.

Of the double infinite number of planes in the pencils each will meet its corresponding plane. This gives a system of a double infinite number of lines in space. We know (85) that there is a quadruple infinite number of lines in space. From among these we may select those which satisfy one or more given conditions. The systems of lines thus obtained were first systematically investigated and classified by Plücker, in his Geometrie des Raumes. He uses the following names:

A treble infinite number of lines, that is, all lines which satisfy one condition, are said to form a complex of lines; e.g. all lines cutting a given line, or all lines touching a surface.

A double infinite number of lines, that is, all lines which satisfy two conditions, or which are common to two complexes, are said to form a congruence of lines; e.g. all lines in a plane, or all lines cutting two curves, or all lines cutting a given curve twice.

Of the two projective pencils at S and S' we may keep the first A single infinite number of lines, that is, all lines which satisfy fixed, and move the centre of the other along the curve. The pencils three conditions, or which belong to three complexes, form a ruled will hereby remain projective, and a plane a in S will be cut by its surface; e.g. one set of lines on a ruled quadric surface, or develop-corresponding plane a' always in the same secant a Whilst S' able surfaces which are formed by the tangents to a curve. moves along the curve the plane a' will turn about o, describing an axial pencil,

It follows that all lines in which corresponding planes in two

§ 104. The fact that all the secants through a point on the cubic form a quadric cone shows that the centres of the projective pencils generating the cubic are not distinguished from any other points on the cubic. If we take any two points S, S' on the cubic, and draw the secants through each of them, we obtain two quadric cones, which have the line SS' in common, and which intersect besides along the cubic. If we make these two pencils having S and S' as centres projective by taking four rays on the one cone as corresponding to the four rays on the other which meet the first on the cubic, the correspondence is determined. These two pencils will generate a cubic, and the two cones of secants having S and S' as centres will be identical with the above cones, for each has five rays in common with one of the first, viz. the line SS' and the four lines determined for the correspondence; therefore these two cones intersect in the original cubic. This gives the theorem

On a twisted cubic any two points may be taken as centres of projective pencils which generate the cubic, corresponding planes being those which meet on the same secant.

AUTHORITIES.-In this article we have given a purely geometrical theory of conics, cones of the second order, quadric surfaces, &c. In doing so we have followed, to a great extent, Reye's Geometrie der Lage, and to this excellent work those readers are referred who wish for a more exhaustive treatment of the subject. Other works especially valuable as showing the development of the subject are: Monge, Géométrie descriptive: Carnot, Géométrie de position (1803), containing a theory of transversals; Poncelet's great work Traité des propriétés projectives des figures (1822); Möbins, Barycentrischer Calcul (1826); Steiner, Abhängigkeit geometrischer Gestalten (1832), containing the first full discussion of the projective relations between rows, pencils, &c.; Von Staudt, Geometrie der Lage (1847) and Beiträge zur Geometrie der Lage (1856-1860), in which a system of geometry is built up from the beginning without any reference to number, so that ultimately a number itself gets a geometrical definition, and in which imaginary elements are systematically introduced into pure geometry; Chasles, Aperçu historique (1837), in which the author gives a brilliant account of the progress of modern geometrical methods, pointing out the advantages of the different purely geometrical methods as compared with the analytical ones, but without taking as much account of the German as of the French authors; Id., Rapport sur les progrès de la géométrie (1870), a continuation of the Aperçu; Id., Traité de géométrie supérieure (1852); Cremona, Introduzione ad una teoria geometrica delle curve piane (1862) and its continuation Preliminari di una teoria geometrica delle superficie (German translations by Curtze). As more elementary books, we mention: Cremona, Elements of Projective Geometry, translated from the Italian by C. Leudesdorf (2nd ed., 1894); J. W. Russell, Pure Geometry (2nd ed., 1905). (O. H.)

III. DESCRIPTIVE GEOMETRY

This branch of geometry is concerned with the methods for representing solids and other figures in three dimensions by drawings in one plane. The most important method is that which was invented by Monge towards the end of the 18th century. It is based on parallel projections to a plane by rays perpendicular to the plane. Such a projection is called orthographic (see PROJECTION, § 18). If the plane is horizontal the projection is called the plan of the figure, and if the plane is vertical the elevation. In Monge's method a figure is represented by its plan and elevation. It is therefore often called drawing in plan and elevation, and sometimes simply orthographic projection.

1. We suppose then that we have two planes, one horizontal, the other vertical, and these we call the planes of plan and of eleva: tion respectively, or the horizontal and the vertical plane, and denote them by the letters and 2. Their line of intersection is called the axis, and will be denoted by xy.

If the surface of the drawing paper is taken as the plane of the plan, then the vertical plane will be the plane perpendicular to it through the axis xy. To bring this also into the plane of the drawing paper we turn it about the axis till it coincides with the horizontal plane. This process of turning one plane down till it coincides with another is called rabatting one to the other. Of course there is no necessity to have one of the two planes horizontal, but even when this is not the case it is convenient to retain the above names.

The whole arrangement will be better understood by referring to

fig. 37. A point A in space is there projected by the perpendicular

FIG. 37.

FIG. 38.

AA and AA, to the planes and so that A, and A, are the horizontal and vertical projections of A.

If we remember that a line is perpendicular to a plane that is perpendicular to every line in the plane if only it is perpendicular to any two intersecting lines in the plane, we see that the axis which is perpendicular both to AA, and to AA, is also perpendicular to AA and to AA because these four lines are all in the same plane. Hence, if the plane ; be turned about the axis till it coincides with the plane, then AA, will be the continuation of AA. This position of the planes is represented in fig. 38, in which the line AA, is perpendicular to the axis x.

Conversely any two points A, A, in a line perpendicular to the axis will be the projections of some point in space when the plane is turned about the axis till it is perpendicular to the plane 1, because in this position the two perpendiculars to the planes and through the points A, and A will be in a plane and therefore meet at some point A.

Representation of Points.-We have thus the following method of representing in a single plane the position of points in space:we take in the plane a line xy as the axis, and then any pair of points A, A, in the plane on a line perpendicular to the axis represent a point A in space. If the line AA, cuts the axis at Ao, and if at A a perpendicular be erected to the plane, then the point A will be in it at a height A1A=AA; above the plane. This gives the position of the point A relative to the plane. In the same way, if in a perpendicular to through A, a point A be taken such that A2A = AA1, then this will give the point A relative to the plane #2.

§ 2. The two planes 1, 2 in their original position divide space into four parts. These are called the four quadrants. We suppose that the plane is turned as indicated in fig. 37, so that the point P comes to Q and R to S, then the quadrant in which the point A lies is called the first, and we say that in the first quadrant a point lies above the horizontal and in front of the vertical plane. Now we go round the axis in the sense in which the plane is turned and come in succession to the second, third and fourth quadrant. In the second a point lies above the plane of the plan and behind the plane of elevation, and so on. In fig. 39, which represents a side view of the planes in fig. 37 the quadrants are marked, and in each a point with its projection is taken. Fig. 38 shows how these are represented when the plane is turned down. We see that

RID2

FIG. 39.

A point lies in the first quadrant if the plan lies below, the elevation the third if the plan lies above, the elevation below; in the fourth if plan above the axis; in the second if plan and elevation both lie above; in and elevation both lie below the axis.

If a point lies in the horizontal plane, its elevation lies in the axis and the plan coincides with the point itself. If a point lies in the with the point itself. If a point lies in the axis, both its plan and vertical plane, its plan lies in the axis and the elevation coincides elevation lie in the axis and coincide with it.

Of each of these propositions, which will easily be seen to be true, the converse holds also.

83. Representation of a Plane.-As we are thus enabled to represent points in a plane, we can represent any finite figure by representing in this way, for the projections of its points completely cover the its separate points. It is, however, not possible to represent a plane planes and , and no plane would appear different from any other. But any plane a cuts each of the planes, in a line. These are point where the latter cuts the plane a. called the traces of the plane. They cut each other in the axis at the

on the axis, and, conversely, any two lines which meet on the axis A plane is determined by its two traces, which are two lines that meet determine a plane.

If the plane is parallel to the axis its traces are parallel to the axis. planes of projection at infinity and will be parallel to it. Thus a Of these one may be at infinity; then the plane will cut one, of the plane parallel to the horizontal plane of the plan has only one finite trace, viz. that with the plane of elevation.

If the plane passes through the axis both its traces coincide with the axis. This is the only case in which the representation of the plane by its two traces fails. A third plane of projection is therefore introduced, which is best taken perpendicular to the other two. We call it simply the third plane and denote it by As it is perpendicular to, it may be taken as the plane of elevation, its line of intersection y with being the axis, and be turned down to coincide with . This is represented in fig. 40. OC is the axis 'xy whilst OA and OB are the traces of the third plane. They lie in one line y. The plane is rabatted about y to the horizontal plane. A plane a through the axis xy will then show in it a trace as. In fig. 40 the lines OC and OP will thus be the traces of a plane through the axis xy, which makes an angle POQ with the horizontal plane.

FIG. 40.

We can also find the trace which any other plane makes with . In rabatting the plane its trace OB with the plane will come to the position OD. Hence a plane & having the traces CA and CB will have with the third plane the trace Ba, or AD if OD=OB

P

[ocr errors]

B... B

II

C, B, 111

[ocr errors]
[ocr errors]

IV

« PreviousContinue »