CHAPTER XVI

STRAIGHT LINE AND PLANE

172. Lines and Points in a Plane.-THEOREM I. A straight line contained by a given plane has its traces on the traces of the plane. The traces of the line are points in the planes of projection, but they are also points in the plane containing the line, therefore the traces of the line must lie on the intersection of the plane with the planes of projection, that is, on the traces of the plane.

Referring to the pictorial projection, Fig. 414, LA and LB are the traces of a plane and AB is a straight line contained by the plane ALB. The points A and B are the traces of AB.

THEOREM II. A straight line which is parallel to one of the planes of projection and is contained by a given plane is parallel to the trace of that plane on the plane of projection to which the line is parallel. Referring to the pictorial projection, Fig. 414, CD is a horizontal line lying in the plane LAB. CD is parallel to LA the horizontal trace of the plane. Hence cd the plan of CD is parallel to LA. c'd', the elevation of CD, is parallel to XY. This line has only one trace C which is a vertical trace. EF is a line parallel to the vertical plane of projection and lying in the plane ALB. EF is parallel to LB the vertical trace of the plane. Hence e'f' the elevation of EF is parallel to LB. ef, the plan of EF, is parallel to XY. This line has only one trace E which is a horizontal trace.

Fig. 415 shows the various lines referred to above by ordinary plan and elevation.

It is obvious that the distance of CD from the horizontal plane of projection is equal to the distance of its elevation from XY, also the distance of EF from the vertical plane of projection is equal to the distance of its plan from XY.

A knowledge of the foregoing principles will enable the student to solve a number of problems.

For example. Given the traces of a plane (Fig. 415) and the plan p of a point contained by the plane, to find the elevation of the point. Through p draw ab to cut the horizontal trace of the plane at a and XY at b. Consider ab as the plan of a line lying in the plane. The elevation a'b' of this line is found as shown, and a'b' must contain p' the elevation required. Instead of taking the line ab, a'b' lying in the

plane and containing the point P, the horizontal line cd, c'd' lying in the plane and containing the point I may be taken, or the line ef, e'f"

parallel to the vertical plane of projection may be used to find p'. If p is given p may be found by working the foregoing construction backwards.

Again, given the traces of a plane and the distances of a point in the plane from the planes of projection to find its plan and elevation. Place in the plane (Fig. 415) a horizontal line cd, c'd' whose distance from the horizontal plane of projection is equal to the given distance of the point from the latter plane. Next place in the plane a line ef, ef whose distance from the vertical plane of projection is equal to the given distance of the point from the latter plane. p the intersection of cd and ef is the plan, and p' the intersection of c'd' and ef is the elevation required.

In another set of problems it is required to find the traces of a plane which contains given points or lines. Three points which are not in the same straight line will fix a plane, also two parallel straight lines or two intersecting straight lines, or one straight line and one point not in that line. These problems are solved by making use of the facts that if two points are in a plane the straight line joining them is in that plane and the traces of this line are on the traces of the plane.

173. To draw the Traces of a Plane which shall contain a given Line and have a given Inclination to one of the Planes of Projection.-Let ab, a'b' (Fig. 417) be the given line, and let the given inclination of the plane be to the horizontal plane.

Draw the projections of a cone having its vertex at the vertical

trace of the given line, its base on the horizontal plane of projection, and its slant side inclined to its base at the given angle (6).

A line through the horizontal trace of the given line, to touch the base of the cone, will be the horizontal trace of the plane required, and a line through the point where this horizontal trace meets the ground line and through the vertical trace of the given line will be the vertical trace of the plane required. The foregoing construction will be readily understood by referring to the pictorial projection shown in Fig. 416.

If the horizontal trace of the given line falls outside the base of the cone two tangents from this point to the base of the cone can be drawn and there will therefore be two planes which will satisfy the given conditions. In Figs. 416 and 417 only one of the two possible planes is shown. If the horizontal trace of the given line falls on the circumference of the base of the cone only one tangent from this point to the base of the cone can be drawn and there will then be only one plane which will satisfy the given conditions, and this plane will have the same inclination as the given line. If the horizontal trace of the given line falls inside the base of the cone no tangent from this point to the base of the cone can be drawn, which shows that the problem is impossible when the inclination of the plane is less than the inclination of the line.

In the construction just explained it has been assumed that the horizontal and vertical traces of the given line have been accessible, it now remains to be shown how the problem is to be dealt with when these points are inaccessible.

Referring to Fig. 419, the horizontal and vertical traces of the given line ab, a'b' are supposed to be inaccessible.

Draw the projections of two cones having their vertices at points ce' and dd' in the given line, their bases on the horizontal plane,

and their slant sides inclined to their bases at the given angle (#). The horizontal trace of the required plane is a common tangent to the bases of the two cones. The vertical trace of the plane may be found by taking a point ee' in the horizontal trace of the plane and joining this point with two points (say cc' and dd') in the given line. The line

joining the vertical traces of these two lines is the vertical trace of the plane required. A reference to the pictorial projection (Fig. 418) will make the construction clear.

It should be noticed that although four tangents may be drawn to the bases of the cones shown in Figs. 418 and 419 only the outside tangents, and not the cross tangents, can be the horizontal traces of planes which are tangential to the two cones.

If the inclination of the required plane be given to the vertical plane of projection instead of to the horizontal plane the bases of the cones must be placed on the former plane instead of on the latter.

174. In a given Plane to place a Line having a given Inclination to one of the Planes of Projection.-The inclination of the line must not exceed the inclination of the plane.

Let the given inclination be to the horizontal plane.

Take a point C (Fig. 421) on the ground line and draw Cb' making the given angle with XY and meeting the vertical trace of the given plane at b'. Draw b'b perpendicular to XY meeting the latter at b. With centre b and radius bC describe an arc of a circle to cut the horizontal trace of the given plane at a. Draw aa' perpendicular to XY meeting it at a'. ab is the plan and a'b' the elevation of the line required.

The correctness of the construction is obvious, for AB (Fig. 420) is evidently on the given plane and it is also on the surface of a cone whose base is on the horizontal plane and whose base angle is 0.

The construction is applicable whatever be the positions of the traces of the given plane.

If the given inclination be to the vertical plane of projection instead of to the horizontal plane, the construction will present no difficulty to the student who has followed that just given.

175. To draw the Projections of a Line which shall pass through a given Point, have a given Inclination, and be parallel to a given Plane. By the construction of the preceding Art. place in the given plane a line having the given inclination. Through the plan and elevation of the given point draw lines parallel to the plan and elevation respectively of the line which has been placed in the given plane. These will be the projections required.

That the line whose projections have thus been found fulfils the given conditions is not difficult to see; for it is parallel to a line which has the given inclination, and must therefore have itself the given inclination. Also it can never meet the given plane, for if it did it could not be parallel to any line in that plane, therefore it is parallel to the given plane.

176. To draw the Traces of a Plane which shall contain a given Point and be parallel to a given Plane.-The horizontal and vertical traces of the plane required must be parallel to the horizontal and vertical traces respectively of the given plane.

Pp' (Figs. 422 and 423) is the given point and H.T. and V.T. are

the traces of the given plane.