« PreviousContinue »
In the case of a curve we have between the coordinates (x, y, z) a | twofold relation: two equations f(x, y, z) =0, (x, y, z)=0 give such a relation; i.e. the curve is here considered as the intersection of two surfaces (but the curve is not always the complete intersection of two surfaces, and there are hence difficulties); or, again, the coordinates may be given each of them as a function of a single variable parameter. The form y=(x), z=(x), where two of the coordinates are given in terms of the third, is a particular case of each of these modes of representation.
29. The remarks under plane geometry as to descriptive and metrical propositions, and as to the non-metrical character of the method of coordinates when used for the proof of a descriptive proposition, apply also to solid geometry; and they might be illustrated in like manner by the instance of the theorem of the radical centre of four spheres. The proof is obtained from the consideration that S and S' being each of them a function of the form x+++ ax+by+cz+d, the difference S-S' is a mere linear function of the coordinates, and consequently that S-S'=o is the equation of the plane containing the circle of intersection of the two spheres S=0. and S'=0.
30. Metrical Theory. The foundation in solid geometry of the metrical theory is in fact the before-mentioned theorem that if a finite right line PQ be projected upon any other line 00' by lines perpendicular to 00', then the length of the projection P'Q' is equal to the length of PQ into the cosine of its inclination to P'Q'or (in the form in which it is now convenient to state the theorem) the perpendicular distance P'Q' of two parallel planes is equal to the inclined distance PQ into the cosine of the inclination. The principle of § 16, that the algebraical sum of the projections of the sides of any closed polygon on any line is zero, or that the two sets of sides of the polygon which connect a vertex A and a vertex B have the same sum of projections+y2+s2, viz. an equation on the line, in sign and magnitude, as we pass from A to B, is applicable when the sides do not all lie in one plane.
31. Consider the skew quadrilateral QMNP, the sides QM, MN, NP being respectively parallel to the three rectangular axes Ox, FIG. 60. Oy, Oz; let the lengths of these sides be 7., and that of the side QP be p; and let the cosines of the inclinations (or say the cosine-inclinations) of p to the three axes be a, B, Y; then projecting successively on the three sides and on QP we have
ξ, η, ζ = ρα, ρβ, ργ.
whence p2++, which is the relation between a distance and its projections &, 7, upon three rectangular axes. And from the same equations we obtain a2+2+2=1, which is a relation connecting the cosine-inclinations of a line to three rectangular axes. Suppose we have through Q any other line QT, and let the cosineinclinations of this to the axes be a', ',, and 8 be its cosineinclination to QP; also let p be the length of the projection of QP upon QT; then projecting on QT we have
so that the sine of the inclination can only be expressed as a square
root. These formulae are the foundation of spherical trigonometry. 32. Straight Lines, Planes and Spheres.-The foregoing formulae give at once the equations of these loci.
For first, taking Q to be a fixed point, coordinates (a, b, c), and the cosine-inclinations (a, B, v) to be constant, then P will be a point in the line through Q in the direction thus determined; or, taking (x, y, z) for its coordinates, these will be the current coordinates of a point in the line. The values of , n, then are x-a, y-b, z-c, and we thus have
these are expressions for the current coordinates in terms of a parameter p, which is in fact the distance from the fixed point (a, b, c).
which (omitting the last equation,p) are the equations of the line through the point (a, b, c), the cosine-inclinations to the axes being a. B. Y, and these quantities being connected by the relation a2+8+2=1. This equation may be omitted, and then a, B, Y, instead of being equal, will only be proportional, to the cosineinclinations.
Using the last equation, and writing
x, y, z=a+ap, b+Bp, c+rp,
It is easy to see that, if the coordinates (x, y, z) are connected by any two linear equations, these equations can always be brought into the foregoing form, and hence that the two linear equations represent a line.
Secondly, taking for greater simplicity the point Q to be coincident with the origin, and a', B', y', p to be constant, then p is the perpendicular distance of a plane from the origin, and a', ', ' are the cosineinclinations of this distance to the axes (a+8+2=1). Pis any point in this plane, and taking its coordinates to be (x, y, z) then (..) are = (x, y, z), and the foregoing equation p=a's+sa+7'> becomes a'x+B'y+r'z=p, . which is the equation of the plane in question.
If, more generally, Q is not coincident with the origin, then, taking its coordinates to be (a, b, c), and writing p instead of p, the equation is
and we thence have pip-(aa'+bB+cy), which is an expression for the perpendicular distance of the point (a, b, c) from the plane
It is obvious that any linear equation Ax+By+C+D=o between the coordinates can always be brought into the foregoing form, and hence that such an equation represents a plane.
and the distance QP, p, to be constant, say this is =d, then, as Thirdly, supposing Q to be a fixed point, coordinates (a, b, c), before, the values of E, n, are x-a, y-b, z-c, and the equation ++= p2 becomes
If the lines all pass through a given point, then the surface is a cone; and, in particular, if the lines are all parallel to a given line, then the surface is a cylinder.
Beginning with this last case, suppose the lines are parallel to the line x=ms, y=nz, the equations of a line of the system are x=mz+a, y=nz+b,-where a, b are supposed to be functions of the variable parameter, or, what is the same thing, there is between them a relation f(a, b)=0: we have a=x-mz, b=y-ns, and the result of the elimination of the parameter therefore is f(x-ms, y-nz)=0, which is thus the general equation of the cylinder the generating lines whereof are parallel to the line x=mz, y=n. The equation of the section by the plane z =o is f(x, y) =o, and conversely if the cylinder be determined by means of its curve of intersection And in the last equation substituting for §, 7, ♪ their values pa, with the plane s=0,,then, taking the equation of this curve to be PB, Py we find f(x, y)=o, the equation of the cylinder is f(x-ma, y-nz)=0. Thus, if the curve of intersection be the circle (x-a)2+(y−8)2=y3, we have (x-ms-a)2+(y—ns—ß)2 = as the equation of an oblique cylinder on this base, and thus also (x−a)2+(y−8)= as the equation of the right cylinder.
p=a′E+B'n+y's = pd.
which is an expression for the mutual cosine-inclination of two lines, the cosine-inclinations of which to the axes are a, B, and a', ', respectively. We have of course a++2=1 and a2+8+= 1; and hence also
If the lines all pass through a given point (a, b, c), then the equations of a line are x-a=a(2-c), y—b=B(s—c), where a, 8 are functions of the variable parameter, or, what is the same thing, there exists between them an equation f(a, B)=0; the elimination
of the parameter gives, therefore, (==) =0; and this
which is the equation of the sphere, coordinates of the centre=(a,b,c), and radius=d.
A quadric equation wherein the terms of the second order are
can always, it is clear, be brought into the foregoing form; and it thus appears that this is the equation of a sphere, coordinates of the centre A, B, C, and squared radius=(A+B+C2)-D.
33. Cylinders, Cones, ruled Surfaces. If the two equations of a straight line involve a parameter to which any value may be given, we have a singly infinite system of lines. They cover a surface, and the equation of the surface is obtained by eliminating the parameter between the two equations.
equation, or, what is the same thing, any homogeneous equation f(x-a, y-b. 2-c)=0, or, taking f to be a rational and integral function of the order n, say (*) (x—a, y—b, z—c)" =0, is the general equation of the cone having the point (a, b, c) for its vertex. Taking the vertex to be at the origin, the equation is (*) (x, y, z)"=0; and, in particular, (*) (x, y, z) =o is the equation of a cone of the second order, or quadricone, having the origin for its vertex.
34. In the general case of singly infinite system of lines, the locus is a ruled surface (or regulus). Now, when a line is changing its position in space, it may be looked upon as in a state of turning about some point in itself, while that point is, as a rule, in a state of moving out of the plane in which the turning takes place. If instantaneously it is only in a state of turning, it is usual, though not strictly accurate, to say that it intersects its consecutive position. A regulus such that consecutive lines on it do not intersect, in this sense, is called a skew surface, or scroll; one on which they do is called a developable surface or torse.
Suppose, for instance, that the equations of a line (depending on
the variable parameter ) are then, eliminating 0, we have 23+25-03the equation of a quadric surface, afterwards called the hyperboloid of one sheet; this surface is consequently a scroll. It is to be remarked that we have upon the surface a second singly infinite series of lines; the equations of a line of this second system (depending on the variable parameter ø) are
+ − + (1-3). - - ÷ (1+3).
It is easily shown that any line of the one system intersects every line of the other system.
Considering any curve (of double curvature) whatever, the tangent lines of the curve form a singly infinite system of lines, each line intersecting the consecutive line of the system,-that is, they form a developable, or torse; the curve and torse are thus inseparably connected together, forming a single geometrical figure. An osculating plane of the curve (see 38 below) is a tangent plane of the torse all along a generating line.
35. Transformation of Coordinates.-There is no difficulty in changing the origin, and it is for brevity assumed that the origin remains unaltered. We have, then, two sets of rectangular axes, Ox, Oy, Oz, and Ox1, Oyi, Oz, the mutual cosine-inclinations being shown by the diagram—
that is, a, B, y are the cosine-inclinations of Ox, to Ox, Oy, Oz; a', B', 'those of Oy1, &c.
And this diagram gives also the linear expressions of the coordinates (x1, 1, 1) or (x, y, z) of either set in terms of those of the other set; we thus have
The by equations of the second degree are called quadric surfaces. Quadric surfaces are either proper or special. The special ones arise when the coefficients in the general equation are limited to satisfy certain special equations; they comprise (1) plane-pairs, including in particular one plane twice repeated, and (2) cones, including in particular cylinders; there is but one form of cone, but cylinders may be elliptic, parabolic or hyperbolic.
A discussion of the general equation of the second degree shows that the proper quadric surfaces are of five kinds, represented respectively, when referred to the most convenient axes of reference, by equations of the five types (a and b.positive):
which are obtained by projection, as above explained. Each of these equations is, in fact, nothing else than the before-mentioned equation pa'+'n+y's, adapted to the problem in hand.
But we have to consider the relations between the nine coefficients. By what precedes, or by the consideration that we must have identically++82=x2+2+22, it appears that these satisfy
a2 +B2 +y2 =1,
either set of six equations being implied in the other set. It follows that the square of the determinant
is-1; and hence that the determinant itself is 1. The distinction of the two cases is an important one: if the determinant is +1, then the axes Oxi, Oy, Os are such that they can by a rotation about O be brought to coincide with Ox, Oy, Oz respectively; if it is-1, then they cannot. But in the latter case, by measuring x1, y, z in the opposite directions we change the signs of all the coefficients and so make the determinant to be +1; hence the former case need alone be considered, and it is accordingly assumed that the determinant is = +1. This being so, it is found that we have the equality a='"-B'y', and eight like ones, obtained from this by cyclical interchanges of the letters a, 8, Y, and of unaccented, singly and doubly accented letters.
36. The nine cosine-inclinations above are, as has been seen, connected by six equations. It ought then to be possible to express them all in terms of three parameters. An elegant means of doing this has been given by Rodrigues, who has shown that the tabular expression of the formulae of transformation may be written
the meaning being that the coefficients in the transformation are fractions, with numerators expressed as in the table, and the common denominator.
having the common axes Oz; and the section by any plane z=Y parallel to that of xy is the ellipse
so that the surface is generated by
a variable ellipse moving parallel to itself along the parabolas as directrices.
In the hyperbolic paraboloid (figs. 62 and 63) the sections by the - having the opposite planes of zx, sy are the parabolas z= axes Oz, Oz', and the section by a plane = parallel to that of xy is the hyperbola y=-2, which has its transverse axis parallel to Ox or Oy according as y is positive or negative. The surface is thus
In the hyperboloid of one sheet (fig. 65), the sections by the planes | absolute curvature is the limiting position of the point where the of zx, zy are the hyperbolas principal normal at (x, y, z) is cut by the normal plane at a neighbouring point, as that point moves up to (x, y, z).
39. Differential Geometry of Surfaces. Let (x, y, z) be any chosen point on a surface f(x, y, z) =o. As a second point of the surface moves up to (x, y, z), its connector with (x, y, z) tends to a limiting position, a tangent line to the surface at (x, y, z). All these tangent lines at (x, y, z), obtained by approaching (x, y, z) from different directions on a surface, lie in one plane 3/4 (E− x) + 21 (n − y) + 3/2 (5-2) =0. af
having a common conjugate axis O'; the section by the plane of x, y, and that by any parallel plane, is an ellipse; and the surface may be considered as generated by a variable ellipse moving parallel to itself along the two hyperbolas as directrices. If we imagine two equal and parallel circular disks, their points connected by strings of equal lengths, so that these are the generators of a right circular cylinder, and if we turn one of the disks about its centre through an angle in its plane, the strings in their new positions will be one system of generators of a hyperboloid of one sheet, for which a=b; and if we turn it through the same angle in the opposite direction,
provided >c2, and the surface, consisting of two distinct portions or sheets, may be considered as generated by a variable ellipse moving parallel to itself along the hyperbolas as directrices.
38. Differential Geometry of Curves.-For convenience consider the coordinates (x, y, z) of a point on a curve in space to be given as functions of a variable parameter 0, which may in particular be one of themselves. Use the notation x', x" for dx/do, d2x/de, and similarly as to y and z. Only a few formulae will be given. Call the current coordinates (§, n, ).
The tangent at (x, y, z) is the line tended to as a limit by the connector of (x, y, z) and a neighbouring point of the curve when the latter moves up to the former: its equations are
(E-x)/x' (7-y)/y' = (5 − z)/2′.
The osculating plane at (x, y, z) is the plane tended to as a limit by that through (x, y, z) and two neighbouring points of the curve as these, remaining distinct, both move up to (x, y, z): its one equation
(§−x) (y′z′′ −y′′z')+(n−y)(z'x′′−z′′x′)+(5−z)(x′y′′ −x′′y′) = o. The normal plane is the plane through (x, y, z) at right angles to the tangent line, i.e. the plane
It cuts the osculating plane in a line called the principal normal. Every line through (x, y, z) in the normal plane is a normal. The normal perpendicular to the osculating plane is called the binormal. A tangent, principal normal, and binormal are a convenient set of rectangular axes to use as those of reference, when the nature of a curve near a point on it is to be discussed.
Through (x, y, s) and three neighbouring points, all on the curve, passes a single sphere; and as the three points all move up to (x, y, z) continuing distinct, the sphere tends to a limiting size and position. The limit tended to is the sphere of closest contact with the curve at x, y, z); its centre and radius are called the centre and radius of spherical curvature. It cuts the osculating plane in a circle, called the circle of absolute curvature; and the centre and radius of this circle are the centre and radius of absolute curvature. The centre of
This plane is called the tangent plane at (x, y, z). One line through (x, y, z) is at right angles to the tangent plane. This is the normal
The tangent plane is cut by the surface in a curve, real or imaginary, with a node or double point at (x, y, z). Two of the tangent lines touch this curve at the node. They are called the "chief tangents" (Haupt-tangenten) at (x, y, z); they have closer contact with the surface than any other tangents.
In the case of a quadric surface the curve of intersection of a tangent and the surface is of the second order and has a node, it must therefore consist of two straight lines. Consequently a quadric surface is covered by two sets of straight lines, a pair through every point on it; these are imaginary for the ellipsoid, hyperboloid of two sheets, and elliptic paraboloid.
A surface of any order is covered by two singly infinite systems of curves, a pair through every point, the tangents to which are all chief tangents at their respective points of contact. These are called chief-tangent curves; on a quadric surface they are the above straight lines.
40. The tangents at a point of a surface which bisect the angles between the chief tangents are called the principal tangents at the point. They are at right angles, and together with the normal constitute a convenient set of rectangular axes to which to refer the surface when its properties near the point are under discussion. At a special point which is such that the chief tangents there run to the circular points at infinity in the tangent plane, the principal tangents are indeterminate; such a special point is called an umbilic of the surface.
There are two singly infinite systems of curves on a surface, a pair cutting one another at right angles through every point upon it, all tangents to which are principal tangents of the surface at their respective points of contact. These are called lines of curvature, because of a property next to be mentioned.
As a point Q moves in an arbitrary direction on a surface from coincidence with a chosen point P, the normal at it, as a rule, at once fails to meet the normal at P; but, if it takes the direction of a line of curvature through P, this is instantaneously not the case. We have thus on the normal two centres of curvature, and the distances of these from the point on the surface are the two principal radii of curvature of the surface at that point; these are also the radii of curvature of the sections of the surface by planes through the normal and the two principal tangents respectively; or say they are the radii of curvature of the normal sections through the two principal tangents respectively. Take at the point the axis of z in the direction of the normal, and those of x and y in the directions of the principal tangents respectively, then, if the radii of curvature be a, b (the signs being such that the coordinates of the two centres of curvature are z=a and b respectively), the surface has in the neighbourhood of the point the form of the paraboloid
and the chief-tangents are determined by the equation o= + 3/2 The two centres of curvature may be on the same side of the point or on opposite sides; in the former case a and b have the same sign, the paraboloid is elliptic, and the chief-tangents are imaginary; in the latter case a and b have opposite signs, the paraboloid is hyperbolic, and the chief-tangents are real.
The normal sections of the surface and the paraboloid by the same plane have the same radius of curvature; and it thence readily follows that the radius of curvature of a normal section of the surface
by a plane inclined at an angle to that of zx is given by the equation
The section in question is that by a plane through the normal and a line in the tangent plane inclined at an angle to the principal tangent along the axis of x. To complete the theory, consider the section by a plane having the same trace upon the tangent plane, but inclined to the normal at an angle ; then it is shown without difficulty (Meunier's theorem) that the radius of curvature of this inclined section of the surface is = cos .
AUTHORITIES.-The above article is largely based on that by Arthur Cayley in the 9th edition of this work. Of early and important recent publications on analytical geometry, special mention
is to be made of R. Descartes, Géométrie (Leyden, 1637); John | are called Complexes; Congruences, and Ruled Surfaces or Skews Wallis, Tractatus de sectionibus conicis nova methodo expositis (1655, respectively. A Complex is thus a system of lines satisfying one Opera mathematica, i., Oxford, 1695); de l'Hospital, Traité analytique condition-that is, the coordinates are connected by a single relation; des sections coniques (Paris, 1720); Leonhard Euler, Introductio in and the degree of the complex is the degree of this equation supposing analysin infinitorum, ii. (Lausanne, 1748); Gaspard Monge," Appli- it to be algebraic. The lines of a complex of the nth degree which cation d'algèbre à la géométrie" (Journ. Ecole Polytech., 1801); pass through any point lie on a cone of the nth degree, those which Julius Plücker, Analytisch-geometrische Entwickelungen, 3 Bde. lie in any plane envelop a curve of the nth class and there are n lines (Essen, 1828-1831); System der analytischen Geometrie (Berlin, of the complex in any plane pencil; the last statement combines 1835); G. Salmon, A Treatise on Conic Sections (Dublin, 1848; the former two, for it shows that the cone is of the nth degree and 6th ed., London, 1879); Ch. Briot and J. Bouquet, Leçons de géo- the curve is of the nth class. To find the lines common to four métrie analytique (Paris, 1851; 16th ed., 1897); M. Chasles, Traité complexes of degrees n1, na, na, na, we have to solve five equations, viz. de géométrie supérieure (Paris, 1852); Wilhelm Fiedler, Analytische the four complex equations together with the quadratic equation Geometrie der Kegelschnitte nach G. Salmon frei bearbeitet (Leipzig, connecting the line coordinates, therefore the number of common 5te Aufl., 1887-1888); N. M. Ferrers, An Elementary Treatise on lines is 2N2N 2N 4- As an example of complexes we have the lines Trilinear Coordinates (London, 1861); Otto Hesse, Vorlesungen meeting a twisted curve of the nth degree, which form a complex aus der analytischen Geometrie (Leipzig, 1865, 1881); W. A. Whit- of the nth degree. worth, Trilinear Coordinates and other Methods of Modern Analytical Geometry (Cambridge, 1866): J Booth, A Treatise on Some New Geometrical Methods (London, i., 1873; ii.. 1877); A. ClebschF. Lindemann, Vorlesungen über Geometrie, Bd. i. (Leipzig, 1876, 2te Aufl., 1891); R. Baltser, Analytische Geometrie (Leipzig, 1882); Charlotte A. Scott, Modern Methods of Analytical Geometry (London, 1894); G. Salmon, A Treatise on the Analytical Geometry of three Dimensions (Dublin, 1862; 4th ed., 1882); Salmon-Fiedler, Analytische Geometrie des Raumes (Leipzig, 1863; 4te Aufl., 1898); P. Frost, Solid Geometry (London, 3rd ed., 1886; 1st ed., Frost and J. Wolstenholme). See also E. Pascal, Repertorio di matematiche superiori, II. Geometria (Milan, 1900), and articles now appearing in the Encyklopädie der mathematischen Wissenschaften, Bd. iii. 1, 2. (E. B. EL.)
A Congruence is the set of lines satisfying two conditions: thus a finite number m of the lines pass through any point, and a finite number # lie in any plane; these numbers are called the degree and class respectively, and the congruence is symbolically written (m, n).
V. LINE GEOMETRY
Line geometry is the name applied to those geometrical investigations in which the straight line replaces the point as element. Just as ordinary geometry deals primarily with points and systems of points, this theory deals in the first instance with straight lines and systems of straight lines. In two dimensions there is no necessity for a special line geometry, inasmuch as the straight line and the point are interchangeable by the principle of duality; but in three dimensions the straight line is its own reciprocal, and for the better discussion of systems of lines we require some new apparatus, e.g., a system of coordinates applicable to straight lines rather than to points. The essential features of the subject are most easily elucidated y analytical methods: we shall therefore begin with the notion of line coordinates, and in order to emphasize the merits of the system of coordinates ultimately adopted, we first notice a system without these advantages, but often useful in special investigations.
In ordinary Cartesian coordinates the two equations of a straight line may be reduced to the form y=rx+s, 2=x+u, and r, s, t, may be regarded as the four coordinates of the line. These coordinates lack symmetry: moreover, in changing from one base of reference to another the transformation is not linear, so that the degree of an equation is deprived of real significance. For purposes of the general theory we employ homogeneous coordinates; if 111 and yw are two points on the line, it is easily verified that the six determinants of the array
are in the same ratios for all point-pairs on the line, and further, that when the point coordinates undergo a linear transformation so also do these six determinants. We therefore adopt these six determinants for the coordinates of the line, and express them by the symbols 1. A, m. u, n, where xw-x2W1, X = y12- Y11, &c. There is the further advantage that if abicid, and azbacada be two planes through the line, the six determinants
| abc3d: |
are in the same ratios as the foregoing, so that except as regards a factor of proportionality we have λ=bc-b2c11 = c1d2-cod1, &c. The identical relation Ix+mu+nv=o reduces the number of independent constants in the six coordinates to four, for we are only concerned with their mutual ratios; and the quadratic character of this relation marks an essential difference between point geometry, and line geometry. The condition of intersection of two lines is IX'+1'λ+mp'+m'u+nv'+n'v=o
where the accented letters refer to the second line. If the coordinates are Cartesian and I, m, n are direction cosines, the quantity on the left is the mutual moment of the two lines.
Since a line depends on four constants, there are three distinct types of configurations arising in line geometry--those containing a triplyinfinite, a doubly-infinite and a singly-infinite number of lines; they
The simplest example of a congruence is the system of lines constituted by all those that pass through m points and those that lie in n planes; through any other point there pass m of these lines, and in any other plane there lie n, therefore the congruence is of degree m and class n. It has been shown by G. H. Halphen that the number of lines common to two congruences is mm'+nn', which may be verified by taking one of them to be of this simple type. The lines meeting two fixed lines form the general (1, 1) congruence; and the chords of a twisted cubic form the general type of a (1, 3) congruence; Halphen's result shows that two twisted cubics have in general ten common chords. As regards the analytical treatment, the difficulty is of the same nature as that arising in the theory of curves in space, for a congruence is not in general the complete intersection of two complexes.
A Ruled Surface, Regulus or Skew is a configuration of lines which satisfy three conditions, and therefore depend on only one parameter. Such lines all lie on a surface, for we cannot draw one the surface; the simplest example, that of a quadric surface, is through an arbitrary point; only one line passes through a point of really two skews on the same surface.
The degree of a ruled surface qua line geometry is the number of which meets a given line is the degree of the surface qua point geoits generating lines contained in a linear complex. Now the number metry, and as the lines meeting a given line form a particular case of linear complex, it follows that the degree is the same from whichever point of view we regard it. The lines common to three complexes of degrees, nnn, form a ruled surface of degree 2n111; but not every ruled surface is the complete intersection of three complexes.
In the case of a complex of the first degree (or linear complex) the lines through a fixed point lie in a plane called the polar plane or nul-plane of that point, and those lying in a fixed plane pass through a point called the nul-point or pole of the plane. If the nul-plane of A pass through B, then the nul-plane of B will pass through A; the nul-planes of all points on one line, pass through another line 4. The relation between I, and is reciprocal; any line of the complex that meets one will also meet the other, and every line meeting both belongs to the complex. They are called conjugate or polar lines with respect to the complex. On these principles can be founded a theory of reciprocation with respect to a linear complex.
This may be aptly illustrated by an elegant example due to A. Voss. Since a twisted cubic can be made to satisfy twelve conditions, it might be supposed that a finite number could be drawn to touch four given lines, but this is not the case. For, suppose one such can be drawn, then its reciprocal with respect to any linear complex containing the four lines is a curve of the third class, i.e. another twisted cubic, touching the same four lines, which are unaltered in the process of reciprocation; as there is an infinite number of complexes containing the four lines, there is an infinite number of cubics touching the four lines, and the problem is poristic.
The following are some geometrical constructions relating to the unique linear complex that can be drawn to contain five arbitrary lines:
To construct the nul-plane of any point O, we observe that the two lines which meet any four of the given five are conjugate lines of the complex, and the line drawn through O to meet them is therefore a ray of the complex; similarly, by choosing another four we can find another ray through O: these rays lie in the nulplane, and there is clearly a result involved that the five lines so obtained all lie in one plane. A reciprocal construction will enable us to find the nul-point of any plane. Proceeding now to the metrical properties and the statical and dynamical applications, we remark that there is just one line such that the nul-plane of any point on it is perpendicular to it. This is called the central axis; if d be the shortest distance, 0 the angle between it and a ray of the complex, then d tan @p, where p is a constant called the pitch or parameter. Any system of forces can be reduced to a force R along a certain line, and a couple G perpendicular to that line; the lines of nul-moment
for the system form a linear complex of which the given line is the |
The lines common to the two-complexes also belong to an infinite number of linear complexes, of which two reduce to single straight lines. These two lines are conjugate lines with respect to each of the complexes, but they may coincide, and then some simple modifications are required. The locus of the central axis of this system of complexes is a surface of the third degree called the cylindroid, which plays a leading part in the theory of screws as developed synthetically by Ball. Since a linear complex has an invariant of the second degree in its coefficients, it follows that two linear complexes have a lineo-linear invariant. This invariant is fundamental: if the complexes be both straight lines, its vanishing is the condition of their intersection as given above; if only one of them be a straight line, its vanishing is the condition that this line should belong to the other complex. When it vanishes for any two complexes they are said to be in involution or apolar; the nul-points P, Q of any plane then divide harmonically the points in which the plane meets the common conjugate lines, and each complex is its own reciprocal with respect to the other. As regards a configuration of these linear complexes, the common lines from one system of generators of a quadric, and the doubly infinite system of complexes containing the common lines, include an infinite number of straight lines which form the other system of generators of the same quadric.
General line co
If the equation of a linear complex is Al+Bm+Cn+Dλ+Eμ+ | Fo, then for a line not belonging to the complex we may regard the expression on the left-hand side as a multiple of the moment of the line with respect to the complex, the word moment being used in the statical sense; and we infer ordinates. that when the coordinates are replaced by linear functions of themselves the new coordinates are multiples of the moments of the line with respect to six fixed complexes. The essential features of this coordinate system are the same as those of the original one, viz. there are six coordinates connected by a quadratic equation, but this relation has in general a different form. By suitable choice of the six fundamental complexes, as they may be called, this connecting relation may be brought into other simple forms of which we mention two: (i.) When the six are mutually in involution it can be reduced to x2+x2+x2+x2+x2+x2=0; (ii.) When the first four are in involution and the other two are the lines common to the first four it is x+x+x2+x2-2x=0. These generalized coordinates might be explained without reference to actual magnitude, just as homogeneous point coordinates can be; the essential remark is that the equation of any coordinate to zero represents a linear complex, a point of view which includes our original system, for the equation of a coordinate to zero represents all the lines meeting an edge of the fundamental tetrahedron.
The system of coordinates referred to six complexes mutually in involution was introduced by Felix Klein, and in many cases is more useful than that derived directly from point coordinates; e.g. in the discussion of quadratic complexes: by means of it Klein has developed an analogy between line geometry and the geometry of spheres as treated by G. Darboux and others. In fact, in that geometry a point is represented by five coordinates, connected by a relation of the same type as the one just mentioned when the five fundamental spheres are mutually at right angles and the equation of a sphere is of the first degree. Extending this to four dimensions of space, we obtain an exact analogue of line geometry, in which (i.) a point corresponds to a line; (ii) a linear complex to a hypersphere; (iii) two linear complexes in involution to two orthogonal hyperspheres; (iv) a linear complex and two conjugate lines to a hypersphere and two inverse points. Many results may be obtained by this principle, and more still are suggested by trying to extend the properties of circles to spheres in three and four dimensions. Thus the elementary theorem, that, given four lines, the circles circumscribed to the four triangles formed by them are concurrent, may be extended to six hyperplanes in four dimensions; and then we can derive a result in line geometry by translating the inverse of this theorem. Again, just as there is an infinite number of spheres touching a surface at a given point, two of them having contact of a closer nature, so there is an infinite number of linear complexes touching a non-linear complex at a given line, and three of these have contact of a closer nature (cf. Klein, Math. Ann. v.).
Sophus Lie has pointed out a different analogy with sphere geometry. Suppose, in fact, that the equation of a sphere of radius
so that a+b+c-d; then introducing the quantity e to make this equation homogeneous, we may regard the sphere as given by the six coordinates a, b, c, d, e, r connected by the equation a+ b2+c22-de=o, and it is easy to see that two spheres touch if
the polar form 2aa,+2bb,+2cc,-2rr, -de, -a,e vanishes. Comparing this with the equation x2+x+x2+x-2x=0 given above, it appears that this sphere geometry and line geometry are identical, for we may write a=x, b=x2, C=xs, 7=x+5=1, d=3s. =x; but it is to be noticed that a sphere is really replaced by two lines whose coordinates only differ in the sign of x,, so that they are polar lines with respect to the complex x,o. Two spheres which touch correspond to two lines which intersect, or more accurately to two pairs of lines (p, p') and(g, q'), of which the pairs (p, g) and (p, q') both intersect. By this means the problem of describing a sphere to touch four given spheres is reduced to that of drawing a pair of lines (, ) (of which t intersects one line of the four pairs (pp'), (qq′), (rr'), (ss'), and ' intersects the remaining four). We may, however, ignore the accented letters in translating theorems, for a configuration of lines and its polar with respect to a linear complex have the same projective properties. In Lie's transformation a linear complex corresponds to the totality of spheres cutting a given sphere at a given angle. A most remarkable result is that lines of curvature in the sphere geometry become asymptotic lines in the line geometry.
Some of the principles of line geometry may be brought into clearer light by admitting the ideas of space of four and five dimensions.
Thus, regarding the coordinates of a line as homogeneous coordinates in five dimensions, we may say that line geometry, is equivalent to geometry on a quadric surface in five dimensions. A linear complex is represented by a hyperplane section; and if two such complexes are in involution, the corresponding hyperplanes are conjugate with respect to the fundamental quadric. By projecting this quadric stereographically into space of four dimensions we obtain Klein's analogy. In the same way geometry in a linear complex is equivalent to geometry on a quadric in four dimensions; when two lines intersect the representative points are on the same generator of this quadric. Stereographic projection, therefore, converts a curve in a linear complex, i.e. one whose tangents all belong to the complex, into one whose tangents intersect a fixed conic: when this conic is the imaginary circle at infinity the curve is what Lie calls a minimal curve. Curves in a linear complex have been extensively studied. The osculating plane at any point of such a curve is the nul-plane of the point with respect to the complex, and points of superosculation always coincide in pairs at the points of contact of stationary tangents. When a point of such a curve is given, the osculating plane is determined, hence all the curves through a given point with the same tangent have the same torsion.
The lines through a given point that belong to a complex of the nth degree lie on a cone of the nth degree! if this cone has a double line the point is said to be a singular point. Similarly, Non-linear a plane is said to be singular when the envelope of the lines in it has a double tangent. It is very remarkable that the same surface is the locus of the singular points plexes. and the envelope of the singular planes: this surface is called the singular surface, and both its degree and class are in general 2n (n-1), which is equal to four for the quadratic complex.
The singular lines of a complex F =o are the lines common to F and the complex
8F 8F, 8F 8F, &F F
As already mentioned, at each line I of a complex there is an infinite number of tangent linear complexes, and they all contain the lines adjacent to. If now / be a singular line, these complexes all reduce to straight lines which form a plane pencil containing the line I. Suppose the vertex of the pencil is A, its plane a, and one of its lines & then being a complex line near 1, meets &, or more accurately the mutual moment of ', and is of the second order of small quantities. If P be a point on /, a line through P quite near / in the plane a will meet and is therefore a line of the complex; hence the complex-cones of all points on / touch a and the complex-curves of all planes through touch / at A. It follows that is a double line of the complex-cone of A, and a double tangent of the complexcurve of a. Conversely, a double line of a cone or curve is a singular line, and a singular line clearly touches the curves of all planes through it in the same point. Suppose now that the consecutive line is also a singular line, A' being the allied singular point, a the singular plane and any line of the pencil (A', a') so that § is a tangent line at ' to the complex: the mutual moments of the pairs, and I, are each of the second order; hence the plane a meets the lines / and ' in two points very near A. This being true for all singular planes, near a the point of contact of a with its envelope is in A, i.e. the locus of singular points is the same as the envelope of singular planes. Further, when a line touches a complex it touches the singular surface, for it belongs to a plane pencil like (Aa), and thus in Klein's analogy the analogue of a locus of a hypersurface being a bitangent line of the complex is also a bitangent line of the singular surface. The theory of cosingular complexes is thus brought into line with that of confocal surfaces in four dimensions, and guided by these principles the existence of cosingular quadratic complexes can easily be established, the analysis required being almost the same as that invented for confocal cyclides by Darboux