ANALYTICAL GEOMETRY

OF

THREE DIMENSIONS.

EQUATIONS OF A POINT.

We have seen that the position of a point in a plane is determined when we know its distances from two straight lines drawn in that plane; in like manner we shall now proceed to show that the position of a point in space, is determined by its distances from three planes.

Let there be three planes YAZ, XAZ, XAY, which we shall suppose to be perpendicular to each other, and whose intersections are the three straight lines AZ, AY, AX, each of which is perpendicular to the other two according to the principles established in the Geometry of Planes. Let us call the distances of a point in space from these three planes a, b, c, and let us suppose these distances are known, then the position of the

X'.

Z

R

point will be completely determined, provided that we have ascertained in the first instance, that the point is situated within the trihedral angle AXYZ.

For, take on the three straight lines AX, AY, AZ, the distances AN, AO, AQ, respectively, equal to a, b, c; through the points N, O, Q, draw planes parallel to the given planes.

Since the two first parallel planes have all their points situated at the distances a and b, respectively, from the planes YAZ, XAZ, it follows that all the points of the straight line PM, which is the common intersection of these two planes have exclusively the property of being at the same distances from the planes YAZ, XAZ. Hence the point sought must be situated in the straight line PM. Again, the point sought must be situated somewhere in the third plane PnQo which is parallel to XAY, since all the points in this plane have exclusively the property of being at the distance c from the plane XAY. Hence the point sought must be the point P in which the third plane cuts the common intersection of the two first, and thus its position is altogether determined. We may designate by x the distance of a point from the plane YAZ reckoned along AX;

U U

We may designate by y the distance of a point from the plane XAZ reckoned along AY;

We may designate by z the distance of a point from the plane XAY reckoned along AZ.

So that AX, AY, AZ, the intersections of the three planes, two and two, will be the axes of x's, of y's, and of z's. They are called conjointly Axes of Coordinates, the three planes the Co-ordinate Planes, and the three distances the Co-ordinates of a point. These terms are all analogous to those already employed in Analytical Geometry of two dimensions.

The plane YAZ perpendicular to the axis of x's, is called the plane yz ; The plane XAZ perpendicular to the axis of y's, is called the plane æz; The plane XAY perpendicular to the axis of z's, is called the plane xy. This last plane is usually represented in a horizontal position, and the two others in a vertical position.

It follows from what has been said above that the equations

x = a, yb, z = c

(a, b, c being known quantities) are sufficient to determine the position of a point in space, they are for that reason called the Equations of a point in space. We must remark, that, since the three co-ordinate planes when prolonged indefinitely determine eight trihedral angles, viz. four formed above the plane of xy, and four formed below the same plane; it is necessary for us to express analytically in which of these eight angles the point is situated. It is sufficient for this purpose, to extend to planes the principles which have been applied to distances from points and straight lines, that is to say, if we regard as POSITIVE distances reckoned along AX to the right of A, we must regard as NEGAtive distances reckoned along AX to the left of A, that is to say, in the direction AX, the remark applies to the two other co-ordinate axes.

We must therefore consider in the quantities a, b, c, not only the numerical value of these quantities, but also the signs with which they are affected, in order that we may be enabled to determine in which of the eight trihedral angles about the point A the required point is situated.

According to this principle we have, in order to express completely the position of a point in space, the following combinations:

x = + a, y = + b, z = + c,
a, y = + b, z = + c,

x=

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point situated in the angle AXYZ, point situated in the angle AX'YZ, x = +a, y=- b, zc, point situated in the angle AXY'Z, x = + a, y = + b, z = c, point situated in the angle AXYZ', b, zc, point situated in the angle AX'Y'Z, + b, z — — c, point situated in the angle AX'YZ', b, z= c, point situated in the angle AXY'Z, c, point situated in the angle AXY'Z',

x=

a, y = x=1 a, y =

x = + a, y =
x=- a, y ==

b, z= in all, eight combinations, viz. two

systems in which the signs are the same,

three in which one sign is negative and the two others positive, and three in which one sign is positive and the two others negative.

There are also some particular positions of the point which it is notice. For example, in order to express that a point is situated in the plane ay, we must write that its distance from that plane is nothing, and we shall have for the equation of such a point

proper

to

x = a, yb, z = 0.

Similarly, a point situated on the axis of a's, whose distances from the planes xz and xy are nothing at the same time, will have for its equation x = a, y = 0, z = 0

and so for other points situated on the planes or on the co-ordinate axes.

The planes parallel to the three co-ordinate planes, and which have served to determine the position of the point P, constitute, along with these, a rectangular parallelopipedon of which the twelve edges, which are equal, taken four and four, are the three co-ordinates x, y, z, of the point P.

DEFINITION.—If from any point in space, a straight line be drawn perpendicular to a given plane, the foot of the perpendicular is called the projection of the given point upon the given plane.

In like manner, if from every point of any line in space, whether straight or curved, perpendiculars be drawn to any given plane, the line traced out by the feet of the perpendiculars upon the given plane, is called the projection of the given line upon the given plane.

If we suppose that x = a, y = b, z= c are the equations of the point P, the co-ordinates of the point M are ....................

the co-ordinates of the point n are ....... which gives for the co-ordinates of the point o

x = a, y = = b,

x = α, 2 = c,

y = b, z = c.

From which it appears, that if the projection of a point P upon two of the coordinate planes be known, the third projection will also necessarily be known. When the co-ordinates are not at right

angles to each other, in which case the axes AX, AY, AZ, make with each other any angle whatever, and are called oblique axes, the equations of a point P are still

x = α, y = = b, z = c.

But in this case, a, b, c, express distances reckoned parallel to these axes, and the projections of the point P are obtained by the straight lines PM, Pn, Po, respectively, parallel to AX, AY, AZ.

N

P

In other respects, every thing that has been said with regard to rectangular axes, is applicable to oblique axes also.

In what follows we shall always suppose the axes rectangular, unless the contrary is specified.

PROPOSITION. To find an expression for the distance between two points in space, whose co-ordinates are known.

Join NM which determines a trapezium PQMN, in the plane of this trape zium draw Q parallel to MN, and in the plane xy, draw NL parallel to AX. The right angled triangles PQO, MNL, give

And

But

PQ PO2 + QO2 = PO2 + MN2
MN NL + ML' = mu3 + ML3
PQ2 = mn2 + ML' + PO'

the expression required.

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If one of the given points be the origin, then

x = 0 y"= 0 2′′ = 0

and the above expression becomes

d = √ x2 + y2 + z2.

The last formula may be derived directly from the figure at the beginning of the

chapter, as follows:

To find the equation to a straight line in space. The projections of a straight line on two planes is sufficient to determine its position, and hence it follows that a straight line will be determined analyti‐ cally, if we know the equations of its projections upon two of the three co-ordinate planes, We generally consider the projections of the straight line on the planes of xz, and yz; and since these two planes have AZ for their common axis, this line is regarded in each of the planes as the axis of abscissas; AX is, therefore, the axis of ordinates in the plane of xz, and AY is the axis of ordinates in the plane of yz.

Let MN be any straight line in space, an

mn, m'n' its projections on the planes xz, yz then the equations of these two projections will be of the form

x = az + a

expresses not only the relation between the co-or linates of any point in the

straight line mn, but also the relation between the co-ordinates of any point in the plane MNnm drawn through MN perpendicular to the plane of xz. In like manner, the equation

y = bz + 3

belongs not only to the straight line m'n', but likewise to all the points of the projecting plane m'n'NM drawn perpendicular to the plane of yz through the straight line MN.

It appears then, that this system of equations holds for all the points of the straight line MN, which is the intersection of the two planes perpendicular to the planes of xz and yz, and holds good for the points of this straight line alone. These equations therefore are, in this sense, the equations to the straight line itself, although, in the first instance, we established them separately as the equations of the projections.

It follows from this, that the elimination of the variable z between these two equations, gives rise to a third equation between x and y, viz.

b (x — α) = a (y — ß)

which represents the straight line m”n” the projection of MN on the plane of xy; or, more generally, this equation belongs to all the points of the projecting plane MNn"m" drawn through MN perpendicular to the plane of xy.

When the straight line passes through the origin, its projections will also pass through the origin, in this case the distances a, ß, are nothing, and the straight line is represented by the system of equations

x = az

y = bz }

The straight line may be situated in one of the co-ordinate planes, for example in the plane of xz. In this case, for all points in this straight line

y=0

and the system of equations representing the straight line becomes

x = az + α }
y= = 0

"}

that is to say, in this case we shall have b = 0, 60, which is evident from the figure, for the projection of the straight line on the plane of yz will coincide with

AZ.

When the constants a, b, a, ß, are given a priori, the position of the straight line is completely determined. In order to obtain its different points we must give a succession of particular values to one of the variables, z for example, in each of the equations x = az + a, y = bz + 3, by means of which we shall obtain corresponding values for the two other variables x, y. Then let z=z' then x = az + a = p a known quantity,

y = bx + ß = q a known quantity.

Take in AX a distance AM = p;
From P draw Mm parallel to AY and ≈ q;
From m draw mP perpendicular to xy and
=2;

The point P thus determined belongs to the straight line, and in the same manner we may obtain all the other points.

It may be required however, to determine the constants a, b, a, 3, conformably to certain conditions, which gives rise to a