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 projections of both. Hence
If two lines, given by their projections, intersect, the intersection of their planes 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.
§ 6. To find the projections of a line which joins two points A, B given by their projections A1, A2 and B1, B2, we join A1, B1 and A2, B2; 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 themselves and the feet of the perpendiculars from them to the axis.
| Fig. 41. |
Hence if a′ a″ (fig. 41) are the traces of a line a, and if the perpendiculars 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 projections a1, a2 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 a1 and a2 in the traces a′ and a″.
To find the line of intersection 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.
§ 7. To decide whether a point A, given by its projections, lies in a plane α, given by its traces, we draw a line p by joining A to some point in the plane α 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 A1 to some point p′ in the trace α′; this gives p1; and the point where the perpendicular from p′ to the axis cuts the latter we join to A2; this gives p2. 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 α.