CHAPTER XII PROJECTIONS OF POINTS AND LINES 137. Rules relating to the Projections of a Point.-Fig. 291 is a pictorial projection of a model showing the horizontal and vertical planes of projection in their natural positions together with four points A, B, C, and D in space, A being in the first quadrant or first dihedral angle, B in the second, C in the third, and D in the fourth. The plans and elevations of these points as obtained by dropping perpendiculars from them on to the horizontal and vertical planes of projection respectively are also shown. The positions of the elevations of the points when the vertical plane of projection is turned about XY until it coincides with the horizontal plane are indicated, and in Fig. 292 the various plans and elevations are shown as they appear when the planes of projection are made to coincide and then placed flat on this paper. A careful study of Figs. 291 and 292 should convince the student of the truth of the following rules relating to the projections of a point. (1) The plan of a point is below or above XY according as the point is in front or behind the vertical plane of projection. (2) The elevation of a point is above or below XY according as the point is above or below the horizontal plane of projection. (3) The distance of the plan of a point from XY is equal to the distance of the point from the vertical plane of projection. (4) The distance of the elevation of a point from XY is equal to the distance of the point from the horizontal plane of projection. (5) The plan and elevation of a point are in a straight line perpendicular to XY. The term projector has already been defined and is the line joining a point and a projection of it, but the line joining the plan and elevation of a point is also called a projector. A plan and elevation are also said to be projected, the one from the other. 138. True Length, Inclinations, and Traces of a Line. The projection of a line on a plane will be shorter than the line itself except when the line is parallel to the plane; in the latter case the line and its projection have the same length. A line, its projection on one of the co-ordinate planes, and the projectors from its ends to that plane, form a quadrilateral concerning which everything required for constructing it is known if the plan and elevation of the line are given. One method of finding the true length of a line is therefore to construct this quadrilateral. Let the plan ab and elevation a'b' of a line AB be given as in Fig. 294. Referring to the pictorial projection in Fig. 293, it will be seen that the line AB, its plan ab, and the projectors Aa and Bb form a quadrilateral, the base ab of which is given. Also Aa is equal to a'a,, Bb is equal to bb, and the angles Aab and Bba are right angles. Hence to find the true length of AB, draw (Fig. 294) aA, at right angles to ab and equal to aa'. Next draw bB, at right angles to ab and equal to bb. A,B, is the true length of AB. Note that if the extremities of the line AB are on opposite sides of the horizontal plane, the perpendiculars aA, and bB, must be drawn on opposite sides of the plan ab. The inclination of a line to a plane is the angle between the line and its projection on that plane. Referring to Figs. 293 and 294, it is evident that the inclination of AB to the horizontal plane is the angle between ab and A,B,, so that the construction just given for finding the true length of AB also serves for finding its inclination to the horizontal plane. The inclination of AB to the vertical plane of projection is found by constructing the quadrilateral a'b'BA, in which a'A, and b'B, are at right angles to a'b' and equal to a,a, and bb respectively. AB2 is the true length of AB and the angle between a'b' and A,B, is the inclination of AB to the vertical plane of projection. If aB, be drawn parallel to A,B, (Fig. 294) to meet bB, at B, then aB, will also be the true length of AB, and the angle baB, will be the inclination of AB to the horizontal plane. Also the length of bB is equal to the difference between the distances of B and A from the horizontal plane. Hence the true length of AB and its inclination to the horizontal plane may be found by constructing the triangle abB. In like manner the true length and the inclination of AB to the vertical plane may be found by constructing the triangle a'b'A,, in which a'A, is equal to the difference between the distances of A and B from the vertical plane. The inclination of a line to the horizontal plane is usually denoted by the Greek letter 0 (theta) and its inclination to the vertical plane by the Greek letter (phi). Notice that is the letter O with a horizontal line through it, while is the same letter with a vertical line through it. The true length of a line AB and its inclinations to the planes of projection may also be found as follows. Referring to Fig. 295, through b draw a,bb, parallel to XY. With centre b and radius ba describe the arc aa,, cutting a,bb, at a1. Draw aa perpendicular to XY to meet a Comparing the constructions shown in Figs. 294 and 295, it will be seen that in both a quadrilateral is drawn having a base equal to one of the projections of the line, and in Fig. 294 this base is made to coincide with that projection while in Fig. 295 the base is made to coincide with XY. FIG. 295. ib When the inclination of a line is mentioned without reference to any particular plane, inclination to the horizontal plane is generally understood. The trace of a line on a surface is a point where the line intersects the surface. When the traces of a straight line are mentioned without reference to any particular surface, the points in which the line or the line produced cuts the planes of projection are understood. The horizontal trace of a line is the point where the line or the line produced cuts the horizontal plane of projection, and the vertical trace of a line is the point where the line or the line produced cuts the vertical plane of projection. After making the construction for finding the true length of a line, shown in Fig. 294, if BA, be produced to meet ba at h then h will be the horizontal trace of the line AB. In like manner if A,B, be produced to meet a'b' at v' then 'will be the vertical trace of AB. The correctness of these constructions is obvious from an inspection of Fig. 293. If it is only the traces of a line which are required, then it is only necessary to produce the elevation a'b' (Fig. 294) to meet XY at h' and then draw h'h perpendicular to XY to meet the plan ab produced at h in order to determine the horizontal trace. In like manner, by producing the plan ab to meet XY at v, and drawing vo' perpendicular to XY to meet the elevation a'b' produced at v', the vertical trace is determined. This construction fails however when the projections of the line are perpendicular to XY but the construction previously given will apply in this case also, provided that the plans and elevations of two points in the line are definitely marked and lettered. When the projections of a straight line are perpendicular to XY the line itself is perpendicular to XY although it may not meet XY. When a line is parallel to one of the planes of projection it has no trace on that plane. When a line is perpendicular to one of the planes of projection it has no trace on the other plane of projection, and its projection on the plane to which it is perpendicular is a point. X 139. True Form of a Plane Figure. -The projection of a plane figure on a plane will not have the same form or dimensions as the figure itself excepting when the plane of the figure is parallel to the plane of projection, in which case the figure and its projection will be exactly alike. To determine the true form of any plane figure, whose projections are given, it is necessary to know the true distances of a sufficient number of points in it from one another; now these distances may be found by one of the constructions given in the preceding article, TRUE FORM FIG. 296. An example is shown in Fig. 296, where abc and a'b'c' are the given projections of a triangle. The true lengths of the sides are found by the constructions shown, and the triangle which is the true form of the triangle whose projections are given is then drawn. If the figure given by its projections has more than three sides it should first be divided into triangles; the true forms of these triangles are then found and assembled to give the true form of the whole figure. In many cases the true form of a plane figure whose projections are given is best determined by the method of rabatment described in Art. 189, p. 221. The traces of the sides of a plane figure whose projections are given may be determined by one of the constructions of the preceding article, and it will be found that all the horizontal traces will lie in one straight line, and all the vertical traces will lie in another straight line, and these two straight lines will intersect on XY, excepting when they are parallel, in which case they will be parallel to XY. This matter will be referred to again in Chapter XVI. 140. To mark off a given Length on a given Line.-Let ab, a'b' (Fig. 297) be the projections of the line, it is required to find the projections of a point C in this line so that AC shall be a given length. Determine A,B, the true length of AB. Make A,C, equal to the given length. Through C, draw Ce perpendicular to ab to meet ab at c. Through c draw cc' perpendicular to XY to meet a'b' at c. c and care the projections required. If the given projections of the line are perpendicular to XY (Fig. 298), determine c as before then make the distance of c' from XY equal to C,c. 141. Given the True Length of a Line and the Distances of its Extremities from the Planes of Projection, to draw its Projections.- xa First determine the projections aa' (Fig. 299) cr C a FIG. 299, |