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.↩︎