34. In the general case of a singly infinite system of lines, the locus is a ruled surface (orregulus). 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, orscroll; one on which they do is called a developable surface ortorse.Suppose, for instance, that the equations of a line (depending onthe variable parameter θ) are x/a + y/c = θ (1 + y/b), x/a − z/c = (1/θ)(1 − y/b); then, eliminating θ we have x²/a² − z²/c² = 1 − y²/b², or say, x²/a² + y²/b² − z²/c² = 1, the 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 φ) arex+z= φ(1 −y),x−z=1(1 +y).acbacφbIt 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, Oy1, Ozx1, the mutual cosine-inclinations being shown by the diagram—xyzx1αβγy1αβ′γ′z1α″β″γ″that is, α, β, γ are the cosine-inclinations of Ox1to Ox, Oy, Oz; α′, β′, γ′ those of Oy1, &c.And this diagram gives also the linear expressions of the coordinates (x1, y1, z1) or (x, y, z) of either set in terms of those of the other set; we thus havex1= α x + β y + γ z,x = αx1+ α′y1+ α″z1,y1= α′x + β′y + γ′z,y = βx1+ β′y1+ β″z1,z1= α″x + β″y + γ″z,z = γx1+ γ′y1+ γ″z1,which are obtained by projection, as above explained. Each of these equations is, in fact, nothing else than the before-mentioned equation p = α′ξ + β′η + γ′ζ, 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 x² + y² + z² = x1² + y1² + z1², it appears that these satisfy the relations—α²+ β²+ γ²= 1,α² +α′²+ α″²= 1,α′²+ β′²+ γ′²= 1,β²+ β′²+ β″²= 1,α″²+ β″²+ γ″²= 1,γ²+ γ′²+ γ″²= 1,α′a″+ β′β″+ γ′γ″= 0,βγ+β′γ′+ β″γ″= 0,α″α+ β″β+ γ″γ= 0,γα+ γ′α′+ γ″α″= 0,αα′+ ββ′+ γγ′= 0,αβ+α′β′+ α″β″= 0,either set of six equations being implied in the other set.It follows that the square of the determinantα,β,γα′,β′,γ′α″,β″,γ″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 Ox1, Oy1, Oz1are 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, y1, z1in 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 α = β′γ″ − β″γ′, and eight like ones, obtained from this by cyclical interchanges of the letters α, β, γ, 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 writtenxyzx11 + λ² − μ² − ν²2(λμ − ν)2(νλ + μ)y12(λμ + ν)1 − λ² + μ² − ν²2(μν + λ)z12(νλ − μ)2(μν + λ)1 − λ² − μ² + ν²÷ (1 + λ² + μ² + ν²),the meaning being that the coefficients in the transformation are fractions, with numerators expressed as in the table, and the common denominator.37.The Species of Quadric Surfaces.—Surfaces represented by equations of the second degree are calledquadricsurfaces. Quadric surfaces are eitherproperorspecial. 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 theproperquadric 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):(1)z = x²/2a + y²/2b, elliptic paraboloid.(2)z = x²/2a − y²/2b, hyperbolic paraboloid.(3)x²/a² + y²/b² + z²/c² = 1, ellipsoid.(4)x²/a² + y²/b² − z²/c² = 1, hyperboloid of one sheet.(5)x²/a² + y²/b² − z²/c² = −1, hyperboloid of two sheets.Fig. 61.It is at once seen that these are distinct surfaces; and the equations also show very readily the general form and mode of generation of the several surfaces.In the elliptic paraboloid (fig. 61) the sections by the planes of zx and zy are the parabolasz =x², z =y²,2a2bhaving the common axes Oz; and the section by any plane z = γ parallel to that of xy is the ellipseγ =x²+y²;2a2bso that the surface is generated by a variable ellipse moving parallel to itself along the parabolas as directrices.Fig. 62.Fig. 63.Fig. 64.In the hyperbolic paraboloid (figs. 62 and 63) the sections by the planes of zx, zy are the parabolas z = x²/2a, z = − y²/2b, having the opposite axes Oz, Oz′, and the section by a plane z = γ parallel to that of xy is the hyperbola γ = x²/2a − y²/2b, which has its transverse axis parallel to Ox or Oy according as γ is positive or negative. The surface is thus generated by a variable hyperbola moving parallel to itself along the parabolas as directrices. The form is best seen from fig. 63, which represents the sections by planes parallel to the plane of xy, or say the contour lines; the continuous lines are the sections above the plane of xy, and the dotted lines the sections below this plane. The form is, in fact, that of a saddle.In the ellipsoid (fig. 64) the sections by the planes of zx, zy, and xy are each of them an ellipse, and the section by any parallel plane is also an ellipse. The surface may be considered as generated by an ellipse moving parallel to itself along two ellipses as directrices.In the hyperboloid of one sheet (fig. 65), the sections by the planes of zx, zy are the hyperbolasx²−z²= 1,y²−z²= 1,c²c²b²c²having a common conjugate axis zOz′; 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, we get in like manner the generators of the other system; there will be the same general configuration when a ≠ b. The hyperbolic paraboloid is also covered by two systems of rectilinear generators as a method like that used in § 34 establishes without difficulty. The figures should be studied to see how they can lie.Fig. 65.Fig. 66.In the hyperboloid of two sheets (fig. 66) the sections by the planes of zx and zy are the hyperbolasz²−x²= 1,z²−y²= 1,c²a²c²b²having a common transverse axis along z′Oz; the section by any plane z = ±γ parallel to that of xy is the ellipsex²+y²=γ²− 1,a²b²c²provided γ² > 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 θ, which may in particular be one of themselves. Use the notation x′, x″ for dx/dθ, d²x/dθ², and similarly as to y and z. Only a few formulae will be given. Call the current coordinates (ξ, η, ζ).Thetangentat (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(ξ − x)/x′ = (η − y)/y′ = (ζ − z)/z′.Theosculating planeat (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 is(ξ − x) (y′z″ − y″z′) + (η − y) (z′x″ − z″x′) + (ζ − z) (x′y″ − x″y′) = 0.Thenormal planeis the plane through (x, y, z) at right angles to the tangent line,i.e.the planex′(ξ − x) + y′ (η − y) + z′ (ζ − z) = 0.It cuts the osculating plane in a line called theprincipal normal. Every line through (x, y, z) in the normal plane is a normal. The normal perpendicular to the osculating plane is called thebinormal. 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, z) 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 ofspherical curvature. It cuts the osculating plane in a circle, called thecircle of absolute curvature; and the centre and radius of this circle are the centre and radius of absolute curvature. The centre of absolute curvature is the limiting position of the point where the 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 ƒ(x, y, z) = 0. 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∂ƒ(ξ − x) +∂ƒ(η − y) +∂ƒ(ζ − z) = 0.∂x∂y∂zThis plane is called thetangent planeat (x, y, z). One line through (x, y, z) is at right angles to the tangent plane. This is the normal(ξ − x)/∂ƒ= (η − y)/∂ƒ= (ζ − z)/∂ƒ.∂x∂y∂zThe 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 calledchief-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 theprincipal tangentsat 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 calledlines 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 twoprincipal radii of curvatureof 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 z = b respectively), the surface has in the neighbourhood of the point the form of the paraboloidz =x²+y²,2a2band the chief-tangents are determined by the equation 0 = x²/2a + y²/2b. 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 θ to that of zx is given by the equation1=cos² θ+sin² θ.ρabThe section in question is that by a plane through the normal and a line in the tangent plane inclined at an angle θ 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 φ; then it is shown without difficulty (Meunier’s theorem) that the radius of curvature of this inclined section of the surface is = ρ cos φ.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 mentionis to be made of R. Descartes,Géométrie(Leyden, 1637); John Wallis,Tractatus de sectionibus conicis nova methodo expositis(1655,Opera mathematica, i., Oxford, 1695); de l’Hospital,Traité analytique des sections coniques(Paris, 1720); Leonhard Euler,Introductio in analysin infinitorum, ii. (Lausanne, 1748); Gaspard Monge, “Application d’algèbre à la géométrie” (Journ. École Polytech., 1801); Julius Plücker,Analytisch-geometrische Entwickelungen, 3 Bde. (Essen, 1828-1831);System der analytischen Geometrie(Berlin, 1835); G. Salmon,A Treatise on Conic Sections(Dublin, 1848; 6th ed., London, 1879); Ch. Briot and J. Bouquet,Leçons de géométrie analytique(Paris, 1851; 16th ed., 1897); M. Chasles,Traité de géométrie supérieure(Paris, 1852); Wilhelm Fiedler,Analytische Geometrie der Kegelschnittenach G. Salmon frei bearbeitet (Leipzig, 5te Aufl., 1887-1888); N.M. Ferrers,An Elementary Treatise on Trilinear Coordinates(London, 1861); Otto Hesse,Vorlesungen aus der analytischen Geometrie(Leipzig, 1865, 1881); W.A. Whitworth,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. Clebsch-F. 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 theEncyklopädie der mathematischen Wissenschaften, Bd. iii. 1, 2.
34. In the general case of a singly infinite system of lines, the locus is a ruled surface (orregulus). 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, orscroll; one on which they do is called a developable surface ortorse.
Suppose, for instance, that the equations of a line (depending onthe variable parameter θ) are x/a + y/c = θ (1 + y/b), x/a − z/c = (1/θ)(1 − y/b); then, eliminating θ we have x²/a² − z²/c² = 1 − y²/b², or say, x²/a² + y²/b² − z²/c² = 1, the 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 φ) are
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, Oy1, Ozx1, the mutual cosine-inclinations being shown by the diagram—
that is, α, β, γ are the cosine-inclinations of Ox1to Ox, Oy, Oz; α′, β′, γ′ those of Oy1, &c.
And this diagram gives also the linear expressions of the coordinates (x1, y1, z1) or (x, y, z) of either set in terms of those of the other set; we thus have
which are obtained by projection, as above explained. Each of these equations is, in fact, nothing else than the before-mentioned equation p = α′ξ + β′η + γ′ζ, 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 x² + y² + z² = x1² + y1² + z1², it appears that these satisfy the relations—
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 Ox1, Oy1, Oz1are 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, y1, z1in 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 α = β′γ″ − β″γ′, and eight like ones, obtained from this by cyclical interchanges of the letters α, β, γ, 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.
37.The Species of Quadric Surfaces.—Surfaces represented by equations of the second degree are calledquadricsurfaces. Quadric surfaces are eitherproperorspecial. 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 theproperquadric 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):
It is at once seen that these are distinct surfaces; and the equations also show very readily the general form and mode of generation of the several surfaces.
In the elliptic paraboloid (fig. 61) the sections by the planes of zx and zy are the parabolas
having the common axes Oz; and the section by any plane z = γ 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 planes of zx, zy are the parabolas z = x²/2a, z = − y²/2b, having the opposite axes Oz, Oz′, and the section by a plane z = γ parallel to that of xy is the hyperbola γ = x²/2a − y²/2b, which has its transverse axis parallel to Ox or Oy according as γ is positive or negative. The surface is thus generated by a variable hyperbola moving parallel to itself along the parabolas as directrices. The form is best seen from fig. 63, which represents the sections by planes parallel to the plane of xy, or say the contour lines; the continuous lines are the sections above the plane of xy, and the dotted lines the sections below this plane. The form is, in fact, that of a saddle.
In the ellipsoid (fig. 64) the sections by the planes of zx, zy, and xy are each of them an ellipse, and the section by any parallel plane is also an ellipse. The surface may be considered as generated by an ellipse moving parallel to itself along two ellipses as directrices.
In the hyperboloid of one sheet (fig. 65), the sections by the planes of zx, zy are the hyperbolas
having a common conjugate axis zOz′; 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, we get in like manner the generators of the other system; there will be the same general configuration when a ≠ b. The hyperbolic paraboloid is also covered by two systems of rectilinear generators as a method like that used in § 34 establishes without difficulty. The figures should be studied to see how they can lie.
In the hyperboloid of two sheets (fig. 66) the sections by the planes of zx and zy are the hyperbolas
having a common transverse axis along z′Oz; the section by any plane z = ±γ parallel to that of xy is the ellipse
provided γ² > 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 θ, which may in particular be one of themselves. Use the notation x′, x″ for dx/dθ, d²x/dθ², and similarly as to y and z. Only a few formulae will be given. Call the current coordinates (ξ, η, ζ).
Thetangentat (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
(ξ − x)/x′ = (η − y)/y′ = (ζ − z)/z′.
Theosculating planeat (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 is
(ξ − x) (y′z″ − y″z′) + (η − y) (z′x″ − z″x′) + (ζ − z) (x′y″ − x″y′) = 0.
Thenormal planeis the plane through (x, y, z) at right angles to the tangent line,i.e.the plane
x′(ξ − x) + y′ (η − y) + z′ (ζ − z) = 0.
It cuts the osculating plane in a line called theprincipal normal. Every line through (x, y, z) in the normal plane is a normal. The normal perpendicular to the osculating plane is called thebinormal. 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, z) 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 ofspherical curvature. It cuts the osculating plane in a circle, called thecircle of absolute curvature; and the centre and radius of this circle are the centre and radius of absolute curvature. The centre of absolute curvature is the limiting position of the point where the 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 ƒ(x, y, z) = 0. 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
This plane is called thetangent planeat (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 calledchief-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 theprincipal tangentsat 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 calledlines 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 twoprincipal radii of curvatureof 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 z = 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 0 = x²/2a + y²/2b. 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 θ 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 θ 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 φ; then it is shown without difficulty (Meunier’s theorem) that the radius of curvature of this inclined section of the surface is = ρ cos φ.
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 mentionis to be made of R. Descartes,Géométrie(Leyden, 1637); John Wallis,Tractatus de sectionibus conicis nova methodo expositis(1655,Opera mathematica, i., Oxford, 1695); de l’Hospital,Traité analytique des sections coniques(Paris, 1720); Leonhard Euler,Introductio in analysin infinitorum, ii. (Lausanne, 1748); Gaspard Monge, “Application d’algèbre à la géométrie” (Journ. École Polytech., 1801); Julius Plücker,Analytisch-geometrische Entwickelungen, 3 Bde. (Essen, 1828-1831);System der analytischen Geometrie(Berlin, 1835); G. Salmon,A Treatise on Conic Sections(Dublin, 1848; 6th ed., London, 1879); Ch. Briot and J. Bouquet,Leçons de géométrie analytique(Paris, 1851; 16th ed., 1897); M. Chasles,Traité de géométrie supérieure(Paris, 1852); Wilhelm Fiedler,Analytische Geometrie der Kegelschnittenach G. Salmon frei bearbeitet (Leipzig, 5te Aufl., 1887-1888); N.M. Ferrers,An Elementary Treatise on Trilinear Coordinates(London, 1861); Otto Hesse,Vorlesungen aus der analytischen Geometrie(Leipzig, 1865, 1881); W.A. Whitworth,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. Clebsch-F. 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 theEncyklopädie der mathematischen Wissenschaften, Bd. iii. 1, 2.