B2. From B2 draw the perpendicular Bob, to XY. Then since a'B, is the true length of the line and a1a' and b2B, are the distances of its extremities from the horizontal plane it follows (see Fig. 293, p. 172) that a,b, is the length of the plan of the line. Hence, if with centre a and radius equal to a,b, an arc be described to cut the parallel to XY through cat b then ab is the required plan of the line. A perpendicular to XY from b to meet the parallel to XY through c determines b', and a'b' is the required elevation of the line. b 142. Given the Projection of a Line on one of the Planes of Projection, its Inclination to that Plane, and the Distance of one end from it, to determine its other Projection. Let ab (Fig. 300) be the given projection, the inclination to the horizontal plane and let the distance of the end A of the line from the horizontal plane be given. The distance of a' from XY is equal to the X given distance of A from the horizontal plane. b At a make the angle baB, equal to 6 and draw bB, at right angles to ab to meet aB, at B1. The distance of b' from XY is equal to bB, plus the Bi distance of a' from XY. FIG. 300. a If the elevation a'b' is given and the inclination of the line to the vertical plane of projection and also the distance of A from the vertical plane of projection, the construction is similar to that just given and is shown in Fig. 300. 143. Given the Inclination of a Line to one of the Planes of Projection and the Angle which its Projection on that Plane makes with the Ground Line, to draw its Projections.-Let the line be inclined at an angle to the horizontal plane, and let its plan make an angle a with XY. From a point C in XY (Fig. 301) draw Cb' inclined at an angle to XY. Draw b'b at right angles to XY. Then bC is the length of the plan of the line whose true length is Co' and whose inclination to the horizontal plane is 0. With bas centre and bC as radius describe an arc, and from b draw ba to meet this arc at a and make an angle a with XY. ab is the required plan of the line. A perpendicular from a to XY determines a' and a'b' is the required elevation of the line. FIG. 301. 144, Given the Inclinations of a Line to the Planes of Projection, to determine its Projections.-Let the line be inclined at an angle to the horizontal plane and at an angle & to the vertical plane of projection. From a point C in XY (Fig. 302) draw Co' inclined at an angle to XY. Draw b'b at right angles to XY. Then bc is the length of the plan of the line whose true length is Co and whose inclination to the horizontal plane is 0. From b' draw 'D making the angle CD equal to 4. Draw CD perpendicular to b'D. Then b'D is the length of the elevation of the line whose true length is Cb' and whose inclination to the vertical plane of projection is p. With centre b' and radius b'D describe an arc to cut XY at a'. With centre b and radius bC describe the arc Ca to meet the perpendicular to XY from a' at a. ab is the plan and a'b' is the elevation required. Note. The sum of the angles and may vary between 0° and 90°. When 0 + 0° the projections of the line are parallel to XY, and when 0 + 90° the projections of the line are perpendicular to XY. = FIG. 302. 145. Projections of Parallel Lines.-The projections of parallel lines on to the same plane are parallel. Hence if it is required to draw the projections of a line which shall pass through a point whose projections pp' are given and which shall be parallel to another line whose projections ab, a'b' are given, draw through the plan p a line pq parallel to ab and through p' draw p'q' parallel to a'b'. Then pq and p'q' are the projections required. The projections on the same plane of equal parallel lines are equal. 146. Conditions that Two Lines whose Projections are given may intersect. If two lines intersect, their point of intersection is a point in each of the lines, therefore the plan of that point must be on the plan of each of the lines, and therefore the plans of the lines must intersect, and the point of intersection of the plans is the plan of the point of intersection of the lines. In like manner the elevations of the lines must intersect at a point which is the elevation of the point of intersection of the lines. But the plan and elevation of a point are in the same straight line at right angles to the ground line. Hence the conditions that two lines intersect is that their plans and elevations respectively intersect and that the points of intersection are in the same straight line at right angles to the ground line. There are exceptions to this rule. When the lines are perpendicular to the ground line and lie in the same vertical plane or when one of the lines only is perpendicular to the ground line the lines may or may not intersect. In such cases an auxiliary projection will show whether the lines intersect or not. 147. Angle between Two Intersecting Lines. Let AC and BC be two intersecting lines whose projections are given; it is required to find the true angle between these lines. Take any point D in AC and any point E in BC. Determine by Art. 139, p. 174, the true form of the triangle DCE. The angle C of this triangle is the angle required. The construction is simplified in many cases by taking for the points D and E the horizontal or vertical traces of the lines. In Fig. 303, D and E are the horizontal traces of the lines AC and BC respectively. dC1e is the true angle between AC and BC. N Fig. 303 also shows how to determine the projections of a line which bisects the angle ACB. Draw C1r bisecting the angle dC,e and intersecting de at r. Find the elevation of R. Join cr and c'r'. These are the projections of the line which bisects the angle between AC and BC. This also suggests the construction for finding the projections of a line which shall intersect the lines AC and BC and make a given angle with one of them. d a FIG. 303. 148. Angle between Two Nonintersecting Lines.-Let AB and CD be two non-intersecting lines whose projections are given; it is required to find the true angle between these lines. Through any point P in one of the lines, say AB, draw a line PQ parallel to the other line. The angle between these two intersecting lines PQ and AB is the angle required. Exercises XII 1. Show the projections of the following points, using the same ground line for all the projections, and making the projectors 0.5 inch apart. A, 1-2 inches in front of the V.P. and 1.8 inches above the H.P. C, 1.4 inches in front of the V.P. and in the H.P. D, 2 inches behind the V.P. and 1.6 inches above the H.P. E, 2 inches behind the V.P. and 1.6 inches below the H.P. 2. Show the projections of the following points as in the preceding exercise. A, 1.3 inches behind the V.P. and in the H.P. B, in the V.P. and in the H.P. C, 14 inches in front of the V.P. and 1.6 inches below the H.P. D, in the V.P. and 1-2 inches below the H.P. E, 1.5 inches behind the V.P. and 1.5 inches above the H.P. 3. State the exact positions of the points whose projections are given in Fig. 304 with reference to the planes of projection. 4. a'b', the elevation of a straight line, is 2 inches long and it is inclined at 30° to XY. The end A is in the horizontal plane and 2 inches from the vertical plane. The end B is in the vertical plane and the whole line is in the first dihedral angle. Draw the plan and elevation of the line. 5. A triangle ABC is in the first dihedral angle. A is in the V.P. and 2 inches above the H.P. B and C are in the H.P. ab makes 45° with XY and abc is an equilateral triangle of 2 inches side. Draw the plan and elevation of the triangle ABC. dd' FIG. 304. 6. The same as the preceding exercise except that while the side AB remains in the first dihedral angle the side AC is placed in the second dihedral angle. 7. ab, bc, and cd form three sides of a square of 1.5 inches side. a is on XY, ab is inclined at 30° to XY, and abcd is below XY. abcd is the plan of a piece of thin wire ABCD of which the parts AB, BC, and CD are straight. The heights of the points A, B, C, and D above the H.P. are 0·5, 0·9, 1·2, and 1·1 inches respectively. Draw the plan and elevation of the wire. 8. a'b', 2.5 inches long, is the elevation of a straight line which is parallel to the V.P. The end A is in the H.P. and 1 inch in front of the V.P. The end B is 1-5 inches above the H.P. Draw the plan and elevation of AB. 9. Draw the projections of the following lines, and then find their traces where possible. AB, 2 inches long, parallel to XY, 1 inch above the H.P. and 1.3 inches in front of the V.P. CD, 2.2 inches long, parallel to the H.P., inclined at 30° to the V.P., the end C to be in the V.P. and 1-2 inches above the H.P. EF, 1.5 inches long, perpendicular to the H.P., the end E to be 0.5 inch above the H.P. and 1 inch in front of the V.P. GH, 2 inches long, perpendicular to the V.P., 1 inch above the H.P., the end G to be 0.5 inch in front of the V.P. 10. The plan of a line is 2 inches long and it makes 35° with XY, the elevation makes 45° with XY, and the line intersects XY. Draw the plan and elevation of the line and then find its true length and its inclinations to the planes of projection. 11. Find the true length, the inclinations to the planes of projection, and the horizontal and vertical traces of each of the lines whose projections are given in In reproducing the above diagrams the sides of the small squares are to be taken equal to half an inch. Figs. 305-308. Show also in each case the projections of a point in the line whose true distance from the lower end of the line is 1 inch. 12. Determine the true form of each of the plane figures whose projections are given in Figs. 309-312. Find also for each figure the horizontal and vertical In reproducing the above diagrams the sides of the small squares are to be taken equal to half an inch. traces of each side, where possible, and show that the horizontal traces are in one straight line and the vertical traces in another. 13. A straight line 2.5 inches long has one end 0.3 inch in front of the vertical plane and 1.5 inches above the horizontal plane while the other end is 1.75 inches in front of the vertical plane and 0.6 inch above the horizontal plane. Draw the plan and elevation of the line. 14. A straight line inclined at 50° to the horizontal plane has one end 2 inches above the horizontal plane and 0.5 inch in front of the vertical plane of projection. The other end of the line is on the horizontal plane and 2 inches in front of the vertical plane of projection. Draw the projections of the line. 15. A straight line is 2.75 inches long. One end is in the horizontal plane and the other end is in the vertical plane of projection. The line is inclined at 30° to the horizontal plane and its plan makes an angle of 45° with XY. Draw the projections of the line. 16. a'b' (Fig. 313) is the elevation of a straight line. B is in the vertical plane of projection. The line is inclined at 35° to the horizontal plane. Draw the plan of the line. 17. cd (Fig. 314) is the plan of a straight line whose true length is 3 inches. The end C of the line is 0.5 inch below the horizontal plane and the end D is above the horizontal plane. Draw the elevation of the line. 18. abc (Fig. 315) is the plan of a triangle. The point A is on the horizontal plane. The point B is above the horizontal plane and is higher than the point C. The true length of AB is 2.5 inches and the inclination of BC to the horizontal plane is 40°. Draw the elevation of the triangle and find the true length of AC. FIG. 313. FIG. 314. FIG. 315. n FIG. 316. FIG. 317. In reproducing the above diagrams the sides of the small squares are to be taken equal to half an inch. 19. The middle point of a straight line 3 inches long is 1 inch above the horizontal plane and 1.25 inches in front of the vertical plane of projection. The line is inclined at 30° to the horizontal plane and 40° to the vertical plane of projection. Draw the projections of the line. 20. Find the real angle between the lines whose projections are given in Fig. 316. Show also the projections of the line bisecting the angle ABC. 21. Determine the angle MON of the triangle whose projections are given in Fig. 317, and draw the projections of the line which passes through the point M and intersects the line ON at right angles. 22. ABC is an equilateral triangle of 2.5 inches side. A is in the horizontal plane and B is in the vertical plane of projection. AB is inclined at 45° to the horizontal plane and 30° to the vertical plane of projection. BC is inclined at 35 to the horizontal plane. Draw the plan and elevation of the triangle. |