It also follows immediately that—
If a plane α 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 which 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.
§ 4. Representation of a Line.—A line is determined either by two points in it or by two planes through it. We get accordingly two representations of it either by projections or by traces.
First.—A line a is represented by its projections a1 and a2 on the two planes π1 and π2. These may be any two lines, for, bringing the planes π1, π2 into their original position, the planes through these lines perpendicular to π1 and π2 respectively will intersect in some line a which has a1, a2 as its projections.
Secondly.—A line a is represented by its traces—that is, by the points in which it cuts the two planes π1, π2. 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.
The fact that there are different methods of representing points and planes, and hence two methods of representing lines, suggests the principle of duality (section ii., Projective Geometry, § 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 represent. Projections do not as a rule do this excepting when the point or line projected lies in one of the planes of projection.
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 α, β, γ; lines in space by small letters a, b, c; horizontal projections by suffixes 1, like A1, a1; vertical projections by suffixes 2, like A2, a2; traces by single and double dashes α′ α″, a′, a″. Hence P1 will be the horizontal projection of a point P in space; a line a will have the projections a1, a2 and the traces a′ and a″; a plane α has the traces α′ and α″.
§ 5. 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 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.
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| 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.
§ 8. Parallel planes have parallel traces, because parallel planes are cut by any plane, hence also by π1 and by π2, 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 the axis, because these lines are the traces of a plane parallel to the given plane.
§ 9. 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 point 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 which 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.
§ 10. 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 defined. 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.
The draughtsman must remember that the lines which he draws are not mathematical lines without thickness, and therefore every drawing is affected by some errors. 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 when, as in the above case, some lines must meet on the axis, or if three points must lie in a line. A careful draughtsman will always avail himself of these checks.
§ 11. To draw a plane through a given point parallel to a given plane α, we draw through the point two lines which are parallel to the plane α, and determine the plane through them; or, as we know that the traces of the required plane are parallel to those of the given one (§ 8), we need only draw one line l through the point parallel to the plane and find one of its traces, say the vertical trace l″; a line through this parallel to the vertical trace of α will be the vertical trace β″ of the required plane β, and a line parallel to the horizontal trace of α meeting β″ on the axis will be the horizontal trace β′.
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| Fig. 42. |
Let A1 A2 (fig. 42) be the given point, α′ α″ the given plane, a line l1 through A1, parallel to α′ and a horizontal line l2 through A2 will be the projections of a line l through A parallel to the plane, because the horizontal plane through this line will cut the plane α in a line c which has its horizontal projection c1 parallel to α′.
§ 12. We now come to the metrical properties of figures.
A line is perpendicular to a plane if the projections of the line are perpendicular to the traces of the plane. We prove it for the horizontal projection. If a line p is perpendicular to a plane α, every plane through p is perpendicular to α; hence also the vertical plane which projects the line p to p1. As this plane is perpendicular both to the horizontal plane and to the plane α, it is also perpendicular to their intersection—that is, to the horizontal trace of α. It follows that every line in this projecting plane, therefore also p1, the plan of p, is perpendicular to the horizontal trace of α.
To draw a plane through a given point A perpendicular to a given line p, we first draw through some point O in the axis lines γ′, γ″ perpendicular respectively to the projections p1 and p2 of the given line. These will be the traces of a plane γ which is perpendicular to the given line. We next draw through the given point A a plane parallel to the plane γ; this will be the plane required.
