REPREsENTATION OF POINTS. - We have thus the following method of representing in a single plane the position of points in space : - qee take in the plane a line x as the axis, and then any pair of points Al, A, in the p/ane on a line perpendicular to the axis represent a point A in space. If the line A,A, cuts the axis at Ao, and if at A, a perpendicular be erected to the plane, then the point A will be in it at a height A,A A0A, above the plane. This gives the position of the point A relative to the plane 7ri. in the same way, if in a perpendicular to 72 through A2 a point A be taken such that A,A --.A0A„ then tliis will give the point A relative to the plane 7r, side view of the planes in fig. 10 the quadrants are inarked, and in each a point with its projection is taken. Fig. 11 shows how these are repre-sented when the plane iS down. We see that .4 point /ies in, the first quadrant if the plan lies below, the elevation, above the axis ; in the second if plan and elevation both lie above ; in the third if the plan lies alneve, the eleva-tion below; in the fourth if plan and elevation both lie below the axis.
and the plan coincides with the point itself. If a point lies in the vertical plane, its plan lies in the axis and the elevation coincides with the point itself. If a point lies in the axis, both its plan and elevation lie in the axis and coincide with it.
Of each of these propositions, which will easily be seen to be true, the converse holds also.
..4 plane is determined by its two traces, which are two lines that meet on the axis, and, conversely, any two lines which meet on the axis determine a plane.
I It is very convenient here to make use of the modern extension of the mean• lng of an angle according to which we take as the aught between two non-inter. seeilng lines the angle between two intersecting lines parallel respectively to lbe given ones. If this angle is a right angle, the lines are called perpendkulars. Euclid's definition (Xi. def. 3), anti theorem (XI. 4) may then be stated as in the text. Compare also ticle GEOMETRY (Eceunrtst), § 75, vol. x. p. 386.
If the plane is parallel to the axis its traces are parallel to the axis. Of these one niay be at infinity; then the plane will cut one of the planes of projection at infinity and will be parallel to it. Thns a plane parallel to the horizontal plane of the plan has only one finite trace, viz., that with the plane of elevation.
If the plane passes through, the axis both its traces coincide with the axis. This is the only case in which the representation of the plane by its two traces fails. A third plane of projection is there-fore introduced, which is best taken perpendicular to the other two. We call it simply the third nlane, and denote it by 7r,. As it is perpendicular to T1, it may ue taken as the plane of elevation, its line of' intersection y with 7r, being the axis, and be turned down to coincide with 7ri. This is represented in fig, 13, OC is the axis x whilst OA and OB are the tracesof the third plane. They lie in one line y. The plane is rabatted about y to the horizontal plane. A plane a through the axis x will then show in it a trace a,. In fig. 13 the lines OC and OP will thus - be the traces of a plane throngh the axis x which makes all angle POQ with the horizontal plane.
kVe can also find the trace which any other plane makes with 7r3. rabatting the plane its trace OB with plane s having the traces CA and CB will have with the third plane the trace 0,, or AD if OD= OB.
If a plane a is perpetulicular to the horizontal plane, then every point in it has its horizontal projection in, the horizontal trace of the plane, as all the rays projecting these points lie in the plane itself.
Any plane which, is perpendicular to the horizontal plane has its vertical trace perpendicular to the axis.
Any plane which, is perimndicular to th,c vertical plane has its horizontal trace perpendicular to the axis and the verticcd projections of all points in the plane lie in. this trace.