PKOJECTION 801 Pi If the plane is parallel to the axis its traces are parallel to the axis. Of these one may be at infinity ; then the plane will cut one of the planes of projection at infinity and will be parallel to it. Thus 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. AVe call it simply the third nlane, and denote it by 7r 3 . As it is perpendicular to ir 1; it may ue taken as the plane of elevation, its line of intersection y with ir l being the axis, and be turned down to coincide with n^. 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 rabattecl about y to the horizontal plane. A plane a through the axis x will then show in it a trace a 3 . In fig. 13 the lines OC and OP will thus be the traces of a plane through the axis x which makes an angle POQ with the horizontal plane. We can also find the trace which any other plane makes with 7r 3 . In rabatting the plane 7r 3 its trace OB with the plane ir. 2 will come to the position OD. Hence a plane having the traces CA and CB will have with the third plane the trace /8 3 , or AD if OD = OB. It also follows immediately that If a plane a is perpendicular 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 ivhich is perpendicular to the horizontal plane has its vertical trace perpendicular to the axis. Any plane which is perpendicular to the vertical plane has its horizontal trace perpendicular to the axis and the vertical projections of all points in the plane lie in this trace. 31. REPRESENTATION OF A LINE. A line is determined either by two points in it or by two planes through it. We get accord ingly two representations of it either by projections or by traces. First. A line a is represented by its projections a 1 and a on the two planes ir l and ir. 2 . These may be any two lines, for, bringing the planes 7r 1; ir 2 ^ their original position, the planes through these lines perpendicular to n^ and ir 2 respectively will intersect in some line a which has a 1} a 2 as its projections. Secondly. A line a is represented by its traces that is, by the points in which it cuts the two planes TTJ, ir 2 . Any two points may be taken as the traces of a line in space, for it is determined when the planes are in their original position as the line joining the two traces. This representation becomes undetermined if the two traces coincide in the axis. In this case we again use a third plane, or else the projections of the line. 32. The fact that there are different methods of representing points and planes, and hence two methods of representing lines, suggests the principle of duality (G. 41). It is worth while to keep this in mind. It is also worth remembering that traces of planes or lines always lie in the planes or lines which they repre sent. Projections do not as a rule do this excepting when the point or line projected lies in one of the planes of projection. 33. Having now shown how to represent points, planes, and lines, we have to state the conditions which must hold in order that these elements may lie one in the other, or else that the figure formed by them may possess certain metrical properties. It will be found that the former are very much simpler than the latter. Before we do this, however, we shall explain the notation used ; for it is of great importance to have a systematic notation. We shall denote points in space by capitals A, B, C ; planes in space by Greek letters o, /3, 7 ; lines in space by small letters a, b, c ; horizontal projections by suffixes 1, like A 1( a 1 ; vertical pro jections by suffixes 2, like A 2 , a 2 ; traces by single and double dashes a a", a a". Hence Pj will be the horizontal projection of a point P in space ; a line a will have the projections a 1( a 2 and the traces a and a" ; a plane a has the traces o and a". 34. If a point lies in a line, the projections of the point lie in the projections of the line. If a line lies in a plane, the traces of the line lie in the traces of the plane. These propositions follow at once from the definitions of the projections and of the traces. If a point lies in two lines its projections must lie in the pro jections of both. Hence Fig. 14. If two lines, given by their projections, intersect, the intersection of their plans and the intersection of their elevations must lie in a line perpendicular to the axis, because they must be the projections of the point common to the two lines. Similarly If two lines given by their traces lie in the same plane or intersect, then the lines joining their horizontal and vertical traces respectively must meet on the axis, because they must be the traces of the plane through them. 35. To find the projections of a line which joins two points A, B given by their projections A 1} A 2 and Bj, B 2 , we join A lt B x and A 2 , B 2 ; these will be the projections required. For example, the traces of a line are two points in the line whose projections are known or at all events easily found. They are the traces them selves and the feet of the perpendiculars from them to the axis. Hence if a , a" (fig. 14) are the. traces of a line a, and if the per pendiculars from them cut the axis in P and Q respectively^ then the line a Q will be the horizontal and a"P the vertical projection of the line. Conversely, if the projec tions i, 2 of a line are given, and if these cut the axis in Q and P respectively, then the perpendiculars Pa and Qa" to the axis drawn through these points cut the projections a and a 2 in the traces a and a". To find the line of intersec tion of two planes, we observe that this line lies in both planes ; its traces must therefore lie in the traces of both. Hence the points where the horizontal traces of the given planes meet will be the horizontal, and the point where the vertical traces meet the vertical trace of the line required. 36. To decide whether a point A, given by its projections, lies in a plane a, given by its traces, we draw a line^ by joining A to some point in the plane o and determine its traces. If these lie in the traces of the plane, then the line, and therefore the point A, lies in the plane ; otherwise not. This is conveniently done by joining Aj to some point p in the trace a ; this gives p 1 ; and the point where the perpendicular from p to the axis cuts the latter we join to A 2 ; this gives p 2 . If the vertical trace of this line lies in the vertical trace of the plane, then, and then only, does the line p, and with it the point A, lie in the plane o. 37. Parallel planes have parallel traces, because parallel planes are cut by any plane, hence also by TT I and by ir 2 , in parallel lines. Parallel lines have parallel projections, because points at infinity are projected to infinity. If a line is parallel to a plane, then lines through the traces of the line and parallel to the traces of the plane must meet on tJie axis, because these lines are the traces of a plane parallel to the given plane. 38. To draw a plane through two intersecting lines or through two parallel lines, we determine the traces of the lines; the lines joining their horizontal and vertical traces respectively will be the horizontal and vertical traces of the plane. They will meet, at a finite point or at infinity, on the axis if the lines do intersect. To draw a plane through a line and a poitti without the line, we join the given point to any point in the line and determine the plane through this and the given line. To draw a plane through three points ivhich are not in a line, we draw two of the lines which each join two of the given points and draw the plane through them. If the traces of all three lines AB, BC, CA be found, these must lie in two lines which meet on the axis. 39. We have in the last example got more points, or can easily get more points, than are necessary for the determination of the figure required in this case the traces of the plane. This will happen in a great many constructions and is of considerable importance. It may happen that some of the points or lines obtained are not convenient in the actual construction. The horizontal traces of the lines AB and AC may, for instance, fall very near together, in which case the line joining them is not well de fined. Or, one or both of them may fall beyond the drawing paper, so that they are practically non-existent for the construction. _ In this case the traces of the line BC may be used. Or, if the vertical traces of AB and AC are both in convenient position, so that the vertical trace of the required plane is found and one of the horizontal traces is got, then we may join the latter to the point where the vertical trace cuts the axis. Furthermore the draughtsman will never forget that the lines which he draws are not mathematical lines without thickness. For this reason alone every drawing is affected by some errors. And inaccuracies also come in in drawing the lines required in the con struction. It is therefore very desirable to be able constantly to check the latter. Such checks always present themselves when the same result can be obtained by different constructions, or
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