related as to render some other process more convenient. The revolution of the plane itself may be effected by finding the distance between any two points, one in each trace, whether they be in a plane perpendicular to one of the traces or not : and then upon the axis of revolution constructing a triangle whose sides shall be that distance so found, and the intercepted portion of the other trace. See Case 2 of the next proposition for illustration. PROPOSITION XVIII. To construct the eidographic angle of inclination of two given straight lines. The general principle that runs through every construction of this problem is—to exhibit the two lines after revolution on one of the coordinate planes, the axis of revolution being one of the traces of the plane which contains them. (1.) Let the lines be given by means of their horizontal traces Bi,yu and the projections of their point of intersection an, dą. Join Bi Yu, and draw a, p perpendicular to it and a p' parallel. Make a, p' equal to aag, and describe a circle about p with pp' as a radius to cut a, p in a'. Then, drawing B, a', y, a', the angle B, a' y, is B, the required inclination of the given lines. (2.) Let the lines be given by means of their traces on both coordinate planes. In this case let a, B2, 7, d, be the lines exhibited by their traces. Through the traces draw the traces of the plane which contains them, meeting the axis in P. Draw do od perpendicular to the axis, and 88, perpendicular to the trace a P, which produce. From P as centre, describe arcs d, 8 and B2B, the former meeting dâ, in 8', and the latter cutting Põ in Bi. Then Sand B, are the traces Bę and d, on the horizontal coordinate plane after revolution. Whence the lines d' Y, and b, a, being drawn, they will be the given lines, and the angle at f, their intersection, is that which we were required to construct. In reference to both the preceding cases, we may remind the student that he may draw any two lines parallel to the given ones, through the same point, so as to bring the work into the desired position. The angle contained by these is equal to the angle of the two original lines (Pls. 1. 7). (3.) If the given lines be not considerably remote from parallelism to the axis, the traces of them will fall beyond the limits of the drawing. In this case we must proceed somewhat differently. The following is the simplest process that has been devised for the purpose. Cut them by profile planes (as in Case 3, Prop. VII.), and find the traces of the plane which contains them. Turn this plane upon one of the coordinate planes, and likewise two points of each of the given lines (those used in the profile will be most convenient in practice) : then these lines give the inclination sought, as also will any parallels to them, which will sometimes suggest an advantageous construction. PROPOSITION XIX. Given the orthograph traces of two planes, to construct their eidographic Let the given planes, P, Q in the eidograph intersect in the line an Be, and let a plane be drawn perpendicular to it through any point A, cutting the horizontal coordinate plane in 7, 8, ; and the given planes in Ay, Ad, and the projecting plane a, B. B, in Aa'. Then it is required in the problem to construct an angle equal to Yi Adi. Now since the plane Y, Aồ, is perpendicular to the line Q, Ba, this line is perpendicular to AY, Aa,, and A&. (Pls. 11. 2). Again, since a B, is perpendicular to the plane Y. A&, the plane a, B. By through it is perpendicular to Y, A8, (Pls. 11. 16); and since it is the projecting plane upon the horizontal plane of a, B2, it is perpendicular also to the horizontal coordinate plane. Whence the plane a, B. B, is perpendicular to y, d, the intersection of these planes; and a'A in it is perpendicular also to rei. It follows from this, that if the plane 7, A8, be turned upon the horizontal plane about you, the point A will be situated somewhere in the line a, a,; and it only remains to find the eidographic length of a'A by a plane construction. Two general cases will arise here. (1.) When the traces of the line a B, can be employed, that is, when they both fall within the limits of the picture. Draw any line y, ô, to cut the horizontal traces of the given planes in ridi, and the horizontal projection a B, of the line, perpendicularly at a'. Make Bia = B, C, and B, a" = B,a'; join B, a and draw the perpendicular a"A": then make a'A' = a"A", and join Y A', d, A'. The angle Y, A'8, is that required. For, evidently, the triangle Ba B, a is the eidographic triangle Bz B, an revolved on the vertical ; and a"A" the perpendicular from a' upon the line B, dz. It thus fulfils the condition already deduced. In certain cases this construction may be a little simplified, viz. : (a.) When the figure will admit of it, the point A may be taken at Ba, in which case the perpendicular from B, upon B, a takes the place of the perpendicular A"a" of this construction. (6.) A plane parallel to one of the given planes may be used instead of that plane itself; or, even if the work would fall more conveniently, planes parallel to both of them may be used instead of them. (c.) The case when the two given planes intersect the axis in the same point renders a plane parallel to one of indispensable use, in order to avoid a long and complex construction dependent on a separate investigation. (2.) Let the traces of the lines of section be situated so as to be incapable of being employed in the construction. Find the projections of two points in the line of section (Prop. VII., Case 3), and draw its horizontal and vertical projections; also let an, u, be the horizontal projections of one of these points. Through this point draw a plane perpendicular to the line thus found (Prop. XVI.), and let it cut the horizontal traces of the planes, as before, and find the vertical projection of the point a' in which it cuts the horizontal projection of the line of section. Lastly. Find the distance of the two points a a, and a'a': this is the perpendicular from a' upon Q, B2, and the angle is found as before. PROPOSITION XX. Through a given line to trace a plane parallel to another given line. Through the given line &, , to draw a plane parallel to the given line a, Ba. Take any point a, a, in 12, and draw a line parallel to a Ba, viz., y, dz. Then Y., &, and done will give the traces of the plane sought. SCHOLIUM. In nearly the same way we may trace a plane through a given point parallel to two given lines which are not in one plan e. Let m n be the lines, and a, a, the point. Through a, a, draw the lines parallel to the given ones respectively. Let as B, be the traces of the line parallel to ni ng, and 8, those of the line parallel to m, mę. Then the traces of the plane through a, a, and which is parallel to m, m, and n, nq are a, Yu, Bde. PROPOSITION XXI. To construct the eidographic inclination of a given line to a given plane. 1. Through any point in the given line draw a plane perpendicular to the given plane (Prop. xvi.). 2. Find the intersection of this plane with the given plane (Prop.vii.). 3. Find the angle made by this line of intersection and the given line (Prop. xvIII.). This is the angle required. PROPOSITION XXII. Through a given point to draw a line perpendicular to a given line. 1. Find the traces of the plane through the given point and given line (Prop. xiv.). 2. Find the traces of a plane perpendicular to the given line which shall also contain the given point (Prop. X., Scholium 2.). 3. The intersection of these two planes being formed, it is evidently the perpendicular required. PROPOSITION XXIII. To draw a line which shall be perpendicular to two given lines which are not in the same plane. 1. Through each of the given lines draw a plane parallel to the other (Prop. xx). These planes will be parallel. 2. Through each line draw a plane perpendicular to that which has been made to pass through the other line (Prop. XXII.). 3. Find the intersection of these perpendicular planes (Prop. vii.). This line will be perpendicular to both the given lines. EXERCISES. 1. Construct the traces of a plane which passes through the three points (-1, 2, 3), (1, 3, 5), (2, 0, 3). 2. Given the horizontal projection of an equilateral triangle, and the vertical projection of one of its sides to construct the vertical projection of the entire triangle. 3. Given the horizontal and vertical projections of a tetrahedron to assign the conditions that must be fulfilled amongst the parts of the projection to render it a regular tetrahedron. 4. Through a given point to draw a plane parallel to two given lines. In what case does this problem become indeterminate? 5. Find a line equal to the distance between two given parallel planes. 6. Given the projections of a point and a line, to construct those of a line drawn through the given point perpendicular to the given line. 7. Through a given point to draw a line parallel to two given planes. 8. Draw a plane which shall make equal dihedral angles with two given planes : (1.) The plane to be drawn through the intersection of the given ones; (2.) Through any given point, 9. Given the projections of a finite line to divide it in any given ratio. 10. Find the angle contained between the traces of a given plane. 11. Draw a plane through a given line which shall make a given angle with either plane of projection; and another which shall make equal angles with both planes. 12. Given a trihedral angle and a point; to show the supplementary trihedral angle whose vertex is in the point. 13. Given the projections of any one radius of a circle to find the circle after revolution on the vertical plane. 14. Given the projections of three points in the circumference of a circle to construct its radius. 15. Through a given point to draw a line which shall meet two given lines. 16. Through a given point in a given plane to draw that line in the given plane which shall have the greatest inclination to the horizon, and that which shall have the least inclination to the vertical plane. 17. A building is to be erected on a declivity, but its foundation is to be horizontal, having the plan of the foundation, to trace on the surface of the ground the outline of the cutting. 18. A triangle on a plane is given : to find the position of a plane upon which if it be projected orthographically it shall become similar to a given triangle. |