We shall confine ourselves for the present, to the three preceding examples, because they are sufficient fur all the surfaces, 31 hose generation we have defined. hi the course of this work, we shall have occasion to investigate the gene rations of tribes of surfitees, infinitely more nuinerous: and as they present themselves. we shall apply the same method to the determination of their tangent planes, and of their normal:. At present we are about to propound a question, to the solution of which the eonsideration of the tangent plane may be appropriately and usefilly applied.
•• Fourth Queeion.—Figare 15. Two right lines being given, by their horizontal projections, A B, c a, and by their vertical projections, a b, c d; to construct the projections, P N, p a, of their shortest distance; that is to say, of the line that is at one and the same time perpendicular to both ; and to find the quantity of this distance.
" Solation.—From the first of the two given right lines, conceive a plane parallel to the second ; which is always possible, because if' from any point whatever of the first, a line be drawn parallel to the second, and if this third line be conceived to move parallel to itself along the first, it will generate the plane spoken of. Imagine, also, a cylindric surface, with a circular base, having the second given right line for its axis, and the distance required for its radius ; this surface Will be touched by the plane in a line parallel to the axis, and will cut the first right line in a point. It from this point, a perpendicular to the plane be drawn, it will be the line required ; for it will pass, in fact, through :t point of the first given right line, and will be perpendicular to it, as it would to a plane passing along this right line ; it will also intersect the second right line perpendicularly, because it will be a radius of the cylinder, of which such second line is the axis.
It remains, then, only to construct, successively, all the parts of' this solution.
" (I.) To construct the traces of the plane drawn through the first right line parallel to the second, we must first find the point, a, wherein this first line meets the horizontal and which will be a point of the horizontal trace. To effect this, produce the vertical projection b a till it cut the line L ar in the point a. draw a a perpendicular to L and
its intersection with the horizontal projection a u will determine the point A. Through the point in which the first right line cuts the vertical plane, whose projections are a 1), conceive a right line parallel to the second given right line, and construct its projections by drawing, indefinitely. n E parallel to c D, and b e parallel to c d. In like manner, con struct the point, E, of CC/incidence of this parallel with the horizontal plane, by drawing e E perpendicular to L at ; and the point a will be a second point of the horizontal trace of the plane. Then, if' the right line A E be drawn, and pro duced till it cut, in the point F, the line L m, it will give the horizontal trace; and it is evident, that if through the points F and 1), the right line F b be drawn, we shall have the trace on the vertical plane.
"(2.) To construct the line of contact of the plane with the cylindric surface ; from any point of the second right line, which is the axis of the cylinder (as from the point c, for example, in which it meets the horizontal plane) drop a normal, that is, a perpendicular upon the tangent Iolane: and the thot of such normal will be a point of a line of contact.
"To find this foot, according the method already laid down in Figure 6, first construct the indefinite projections of the normal, by drawing through the point c, the line u G perpendicular to the trace a E, and through the point c, the line c a, perpendicular to the trace F b ; then having produced II G till it Meet a E in the point G, and L M in IL 111*0 jent the point G in g, and the point rt in h, on the trace F b ; draw the line a h, whose intersection with c x N% ill determine the vertical projection, i, of the foot of the normal ; and we shall have, on a it, the horizontal prcjection of the same point by letting r i fall perpendicularly on L N. The projections, i, of' the foot of the normal being found, draw r s through the point t, parallel to c D, and i a, parallel to c (1, and wP shall have the projections of the line of contact of the plane with the cylindric surface. Lastly, the points N, a. in which these projections meet those of the first given right line. will be the projections of the point of such line, through which the common perpendicular required will pass.