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.