Analytical Geometry of Three Dimensions

CHAPTER I

COORDINATE SYSTEM: DIRECTIONS

1. Introductory

WE are going to study by algebraic methods the geometry of three dimensional real euclidean space, usually regarded as ordinary space. We adopt the elementary view of analytical geometry, according to which it is merely a matter of convenience to introduce the algebraic method as a tool for the solution of problems having a well-defined meaning apart from the algebra. However, we shall observe that our tool guides us to those problems with which it is best fitted to deal.

This elementary treatment is useful in applications to other parts of mathematics and to mathematical physics. But also, in geometry itself, the student can scarcely hope to appreciate modern abstract treatments without some such introduction. Moreover, the algebraic manipulation in abstract geometry is not essentially different. The form in which it is cast in this book is chosen partly to meet the requirements of the student wishing to pursue the subject further, and for his benefit a note on abstract geometry is given at the end.

The reader is assumed to be acquainted with elementary pure solid geometry, and with simple analytical geometry of two dimensions. The only other special mathematical equipment required is some knowledge of determinants and matrices; for this, reference will be made to Dr Aitken’s Determinants and Matrices in these Texts (quoted as Aitken).

The examples consist almost entirely of auxiliary results needed in the general development. In some cases proofs are indicated and should be completed by the reader; in others proofs should be supplied by the reader as he proceeds. In each section, examples are numbered consecutively, and subsequent reference made by giving the number of the section followed by that of the example. Formulae are similarly treated, except that their numbers are put in brackets. The reader is urged to construct for himself numerical exercises; for riders he must consult larger textbooks and examination papers.

Metrical geometry. The present geometry is metrical, which means that results are expressed, directly or indirectly, in terms of distance and angle. Distance expresses a relationship between a pair of points; angle a relationship between a pair of directions. Both magnitudes have to be measured by comparison with selected standards. It may be remarked that, whereas there is in euclidean space a natural standard angle (the right angle being a convenient unit), the standard of length is quite arbitrary. However, we are here assuming the fundamental properties of these magnitudes, and our initial consideration in applying algebraic methods to the geometry is to find means first of labelling points and directions by algebraic symbols and then of expressing distances and angles in terms of these symbols. Such is the object of this chapter.

Nomenclature. We call three-dimensional euclidean space &. Line will always mean straight line; any other sort of line will be called a curve. If A, B are any two distinct points, then we use the following notation:

"The line AB," or simply AB, means the whole line containing A, B, as distinct from "the segment AB";

AB denotes the same line when sense is relevant and is taken from A to B;

| AB | denotes the length of the segment AB;

(AB) denotes the distance from A to B, sense being relevant;

denotes the vector associated with the segment AB, in the sense from A to B.

If A, B, C are any three non-collinear points, then:

"The plane ABC," or simply ABC, means the whole plane containing A, B, C.

We use the abbreviations: w.r.t. ≡ with respect to; r.h.s. ≡ right-hand side; l.h.s. ≡ left-hand side; and a few others introduced subsequently.

2. Cartesian Coordinates

Consider any fixed point O and any three distinct planes through O. These planes meet in pairs in three non-coplanar lines through O; let X, Y, Z be fixed points, other than O, one on each line. Let P be any point. The lines through P parallel respectively to ΟΧ, OY, OZ meet the planes OΥΖ, OZX, OXY in points L, M, N, say (fig. 1).

FIG. 1.

Write x = | LP | if P, X are on the same side of OYZ, x = — | LP | if P, X are on opposite sides of OYZ, lengths being measured in terms of some selected unit; let y, z be analogously related to | MP |, | NP |. Then, when P is given, the numbers x, y, z are uniquely determined. Conversely, it is seen that, given any three positive or negative numbers x, y, z, there is a unique point P with which these numbers can be associated in the manner described. So We may speak of P as "the point (x, y, z)."

When the points of & are labelled in this fashion, we say that they are referred to origin O and coordinate planes OYZ, OZX, OXY, or coordinate axes OX, OY, OZ. We call x, y, z the (cartesian) coordinates of P in this frame of reference.

Two features should be noted: (i) It must be realised that we can describe any point P only by its relationship to some particular set of points arbitrarily chosen as a system of reference. (ii) When we say that & is real we mean merely that every point of & can be labelled with three real numbers serving as coordinates. Consequently we must ensure that any algebraic theorems to which we attempt to give geometrical interpretations do in fact hold good in the field of real numbers.

If OX, OY, OZ are mutually perpendicular, we call them rectangular or orthogonal axes. L, M, N are then the orthogonal projections of P on the coordinate planes, and x, y, z the perpendicular distances of P from these planes, with appropriate signs attached. Unless otherwise stated, we shall use only such rectangular cartesian coordinates. Also, for definiteness, we shall use right-handed systems, i.e. viewing from O towards X, a rotation from Y towards Z would be that of a right-handed screw, and so on in cyclic order.

1. x, y, z .

Length of a segment. If P1, P2 are the points (x1, y1, z1), (x2, y2, z2), then

This follows from an elementary application of Pythagoras’s theorem to the rectangular parallelepiped having P1, P2 as opposite vertices and edges parallel to OX, OY, OZ.

2. If the axes are oblique and the angles YOZ, ZOX, XQY are Θ, Φ, Ψ, then

3. A necessary and sufficient condition for a cartesian coordinate system to be rectangular is that | OP |² = x² +y² +z² for all positions of P(x, y, z).

Coordinates in general. We shall use the term coordinates with the following general meaning: Let Σ be a collection of geometrical objects such that every member of Σ is labelled by a unique ordered set of n numbers (ξ, η, …, τ) and such that every such set of numbers, in a specified range, is the label of a unique member of Σ. Then ξ, η, …, τ are called the coordinates of the corresponding member of Σ, in this system of labelling.

If the objects are points, we can when desirable distinguish their coordinates as point-coordinates, if planes, as plane-coordinates, and so on.

However, we sometimes find it convenient to replace the n coordinates by the ratios of n + 1 other numbers, or of more than n + 1 numbers connected by certain specified relations. We shall also call these new numbers coordinates, and shall find that we may do so without causing confusion.

3. Projections

Projection will be restricted to mean orthogonal projection. The projection of a point P on a plane Π is the foot of the normal from P to Π. The projection of any other figure on Π is the aggregate of the projections of its points, e.g. the projection of a line s is the intersection of Π with the plane through s perpendicular to Π. The projection of a point P on a line s is the foot of the perpendicular from P on s, i.e. the meet of s with the plane through P normal to s.

The angle θ between two planes Π, Λ is the angle * between the normals from any point to Π, Λ. Let a closed boundary in Π enclose area a, and let its projection in Λ enclose area β. Since lengths parallel to the intersection of Π, Λ are unchanged by the projection, while lengths perpendicular to it are multiplied by cos θ, we find β = cos θ.

The angle φ between a line s and a plane Π is the angle * between s and its projection on Π. Consequently the projection on Π of a segment PQ of s has length | PQ | cos φ.

The angle ψ between two skew lines s, t is the angle* between lines through any point parallel to s, t. The projection on t of a segment PQ of s has length | PQ | cos ψ. But this should be carefully compared with the following paragraph.

Sensed lines. There are two opposed senses of displacement along a line s; we arbitrarily call one positive and the other negative. The positive one is sometimes called merely the sense of s. When we wish to emphasise that s has an assigned sense we denote it by s. P1 P2 being points of s, we reckon the distance (P1, P2) positive if the displacement from P1 to P2 is in the positive sense, and otherwise negative. Thus, if (P1 P2) is positive, (Ρ1 P2) = | P1 P2| = −(P2 P1). If Q is another point of s we say that Q divides the segment P1 P2 in the ratio (P1Q) : (QP2).

FIG. 2

Let t be any other sensed line. We now define the angle χ between s, t χ π) between lines s′, t′ through any point parallel to, and in the same sense as, s, t, respectively (fig. 2, where arrows indicate the senses).

1. The ratio in which Q divides the segment Pl, P2 is positive if Q lies between P1, P2; negative and numerically less than unity if P1 lies between Q, P2; negative and numerically greater than unity if P2 lies between P1, Q.

2. P, Q being points of s, P*, Q* their projections on t, (PQ) has a sign depending on the sense ascribed to s, (P*Q*) one depending on the sense ascribed to t. We call (P*Q*) the projection of (PQ); in all cases (P*Q*) = (PQ) cos χ. [The problem is merely to see that, coupled with our definitions of (PQ), (P*Q*), χ, this formula is equivalent to the standard definition of cos χ.]

3. P1P2 … Pn Pon any (sensed) line is zero.

4. The projection of a given vector on any (sensed) line is the sum of the projections of its components.

4. Direction-cosines and Direction-ratios

Let v be any sensed line through O. We can conveniently describe its orientation by its relation to S, the sphere with centre O and unit radius. For v meets S at the ends of a diameter and one end, say V, is such that (OV) is positive (fig. 3). If v is given, V is a unique point of S; conversely, if V is any given point of S, then v is uniquely determined as the line OV. Let V have coordinates l, m, n; it lies on S if and only if | OV | = 1, i.e. from 1 (1),

Therefore the preceding statement is equivalent to: If v is given, then l, m, n satisfying (1) are uniquely determined; if l, m, n satisfying (1) are given, then