In space of three dimensions there are two fundamentally different kinds of loci, of which the simplest examples are the plane and the straight line.

A particle which moves along a straight line has only one degree of freedom. Its direction of motion is fixed; its position can be completely fixed by one measurement of position, by its distance from a fixed point on the line. If we take the line as our fundamental line \(\Lambda\) of Ch. I, the position of any of its points is determined by a single coordinate \(x\). A particle which moves in a plane, on the other hand, has two degrees of freedom; its position can only be fixed by the determination of two coordinates.

A locus represented by a single equation \[z = f(x, y)\] plainly belongs to the second of these two classes of loci, and is called a surface. It may or may not (in the obvious simple cases it will) satisfy our common-sense notion of what a surface should be.

The considerations of § 31 may evidently be generalised so as to give definitions of a function \(f(x, y, z)\) of three variables (or of functions of any number of variables). And as in § 32 we agreed to adopt \(f(x, y) = 0\) as the standard form of the equation of a plane curve, so now we shall agree to adopt \[f(x, y, z) = 0\] as the standard form of equation of a surface.

The locus represented by two equations of the form \(z = f(x, y)\) or \(f(x, y, z) = 0\) belongs to the first class of loci, and is called a curve. Thus a straight line may be represented by two equations of the type \(Ax + By + Cz + D = 0\). A circle in space may be regarded as the intersection of a sphere and a plane; it may therefore be represented by two equations of the forms \[(x – \alpha)^{2} + (y – \beta)^{2} + (z – \gamma)^{2} = \rho^{2},\quad Ax + By + Cz + D = 0.\]

Examples XIX

1. What is represented by three equations of the type \(f(x, y, z) = 0\)?

2. Three linear equations in general represent a single point. What are the exceptional cases?

3. What are the equations of a plane curve \(f(x, y) = 0\) in the plane \(XOY\), when regarded as a curve in space? [\(f(x, y) = 0\), \(z = 0\).]

4. Cylinders. What is the meaning of a single equation \(f(x, y) = 0\), considered as a locus in space of three dimensions?

[All points on the surface satisfy \(f(x, y) = 0\), whatever be the value of \(z\). The curve \(f(x, y) = 0\), \(z = 0\) is the curve in which the locus cuts the plane \(XOY\). The locus is the surface formed by drawing lines parallel to \(OZ\) through all points of this curve. Such a surface is called a cylinder.]

5. Graphical representation of a surface on a plane. Contour Maps. It might seem to be impossible to represent a surface adequately by a drawing on a plane; and so indeed it is: but a very fair notion of the nature of the surface may often be obtained as follows. Let the equation of the surface be \(z = f(x, y)\).

If we give \(z\) a particular value \(a\), we have an equation \(f(x, y) = a\), which we may regard as determining a plane curve on the paper. We trace this curve and mark it \((a)\). Actually the curve \((a)\) is the projection on the plane \(XOY\) of the section of the surface by the plane \(z = a\). We do this for all values of \(a\) (practically, of course, for a selection of values of \(a\)). We obtain some such figure as is shown in Fig. 17. It will at once suggest a contoured Ordnance Survey map: and in fact this is the principle on which such maps are constructed. The contour line \(1000\) is the projection, on the plane of the sea level, of the section of the surface of the land by the plane parallel to the plane of the sea level and \(1000\) ft. above it.1

6. Draw a series of contour lines to illustrate the form of the surface \(2z = 3xy\).

7. Right circular cones. Take the origin of coordinates at the vertex of the cone and the axis of \(z\) along the axis of the cone; and let \(\alpha\) be the semi-vertical angle of the cone. The equation of the cone (which must be regarded as extending both ways from its vertex) is \(x^{2} + y^{2} – z^{2}\tan^{2} \alpha = 0\).

8. Surfaces of revolution in general. The cone of Ex. 7 cuts \(ZOX\) in two lines whose equations may be combined in the equation \(x^{2} = z^{2}\tan^{2}\alpha\). That is to say, the equation of the surface generated by the revolution of the curve \(y = 0\), \(x^{2} = z^{2}\tan^{2}\alpha\) round the axis of \(z\) is derived from the second of these equations by changing \(x^{2}\) into \(x^{2} + y^{2}\). Show generally that the equation of the surface generated by the revolution of the curve \(y = 0\), \(x = f(z)\), round the axis of \(z\), is \[\sqrt{x^{2} + y^{2}} = f(z).\]

9. Cones in general. A surface formed by straight lines passing through a fixed point is called a cone: the point is called the vertex. A particular case is given by the right circular cone of Ex. 7. Show that the equation of a cone whose vertex is \(O\) is of the form \(f(z/x, z/y) = 0\), and that any equation of this form represents a cone. [If \((x, y, z)\) lies on the cone, so must \((\lambda x, \lambda y, \lambda z)\), for any value of \(\lambda\).]

10. Ruled surfaces. Cylinders and cones are special cases of surfaces composed of straight lines. Such surfaces are called ruled surfaces.

The two equations \[\begin{equation*} x = az + b,\quad y = cz + d, \tag {1}\end{equation*}\] represent the intersection of two planes, a straight line. Now suppose that \(a\), \(b\), \(c\), \(d\) instead of being fixed are functions of an auxiliary variable \(t\). For any particular value of \(t\) the equations give a line. As \(t\) varies, this line moves and generates a surface, whose equation may be found by eliminating \(t\) between the two equations . For instance, in Ex. 7 the equations of the line which generates the cone are \[x = z\tan \alpha\cos t,\quad y = z\tan \alpha\sin t,\] where \(t\) is the angle between the plane \(XOZ\) and a plane through the line and the axis of \(z\).

Another simple example of a ruled surface may be constructed as follows. Take two sections of a right circular cylinder perpendicular to the axis and at a distance \(l\) apart (Fig. 18a). We can imagine the surface of the cylinder to be made up of a number of thin parallel rigid rods of length \(l\), such as \(PQ\), the ends of the rods being fastened to two circular rods of radius \(a\).

Now let us take a third circular rod of the same radius and place it round the surface of the cylinder at a distance \(h\) from one of the first two rods (see Fig. 18a, where \(Pq = h\)). Unfasten the end \(Q\) of the rod \(PQ\) and turn \(PQ\) about \(P\) until \(Q\) can be fastened to the third circular rod in the position \(Q’\). The angle \(qOQ’ = \alpha\) in the figure is evidently given by \[l^{2} – h^{2} = qQ’^{2} = \left (2a\sin\tfrac{1}{2} \alpha\right)^{2}.\] Let all the other rods of which the cylinder was composed be treated in the same way. We obtain a ruled surface whose form is indicated in Fig. 18b. It is entirely built up of straight lines; but the surface is curved everywhere, and is in general shape not unlike certain forms of table-napkin rings (Fig. 18c).

  1. We assume that the effects of the earth’s curvature may be neglected.↩︎

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