33. Loci in space

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 $Λ$ 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 $A x + B y + C z + D = 0$ . 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 $(x - α)^{2} + (y - β)^{2} + (z - γ)^{2} = ρ^{2}, A x + B y + C z + D = 0.$

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 $X O Y$ , 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

X O Y

. The locus is the surface formed by drawing lines parallel to

O Z

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 $X O Y$ 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 $2 z = 3 x y$ .

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 $α$ 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 $x^{2} + y^{2} - z^{2} \tan^{2} α = 0$ .

8. Surfaces of revolution in general. The cone of Ex. 7 cuts $Z O X$ in two lines whose equations may be combined in the equation $x^{2} = z^{2} \tan^{2} α$ . That is to say, the equation of the surface generated by the revolution of the curve $y = 0$ , $x^{2} = z^{2} \tan^{2} α$ 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 $(λ x, λ y, λ z)$ , for any value of $λ$ .]

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

The two equations $\begin{matrix} (1) & x = a z + b, y = c z + d, \end{matrix}$ 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 $x = z \tan α \cos t, y = z \tan α \sin t,$ where $t$ is the angle between the plane $X O Z$ 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 $P Q$ , 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 $P q = h$ ). Unfasten the end $Q$ of the rod $P Q$ and turn $P Q$ about $P$ until $Q$ can be fastened to the third circular rod in the position $Q^{'}$ . The angle $q O Q^{'} = α$ in the figure is evidently given by $l^{2} - h^{2} = q Q^{' 2} = {(2 a \sin \frac{1}{2} α)}^{2} .$ 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).

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

\leftarrow

32. Curves in a plane

Main Page

MISCELLANEOUS EXAMPLES ON CHAPTER II

\to