tain the generatrix, im, i'm', whose traces are i and q'; and farther, it must be a tangent to the cylinder at every point of this generatrix. Now the tangent plane to a surface contains the tangents to every curve traced upon this surface, and passing through the point of contact; if :. we draw the tangent ti, to the curve aec, it will be in the required plane, and since it is in the horizontal plane, it is the horizontal trace of the tangent plane. Produce it to meet xy in o; draw og"; o q′′ will be the vertical trace of the plane. PROBLEM 2. To draw a tangent plane to a cylinder, through a point without it. 42. The tangent plane to the cylinder contains a generatrix, and in the preceding problem we have seen that its horizontal trace is a tangent to that of the cylinder: if, therefore, we suppose through the given point a line to be drawn parallel to the generatrixes, it will be altogether in the tangent plane; and if, after having constructed the horizontal and vertical traces of this parallel, we draw through the former, tangents to the horizontal trace of the cylinder, these tangents will be the horizontal traces of tangent planes, passing through the given point. Afterwards, to find their vertical traces, join the vertical trace of the same parallel to the points where the ground line is intersected by the horizontal traces of the planes. The details of the construction are represented in the next figure. The projections of the cylinder are traced, as in the preceding proposition; the projections of the given point are m and m'; those of the parallel to the generatrix are mt and m't'; the trace of the parallel is t and t': therefore the tangents ta, tß, drawn to the horizontal trace of the cylinder, will be the horizontal traces of the tangent planes, and aa', Ba', will be its vertical traces. Р To verify the constructions, we have drawn through the given point lines parallel to ta and tß: the points r' and s', where the parallels meet the vertical plane, should be on aa' and Ba'. The generatrixes in which the planes touch the cylinder, serve the same purpose. One of them, which has its projections ip and i'p', meets the vertical plane in a point p': and the other, which has projections R, and k'R', meets it in R'. Now it is evident that the trace a a' must pass through the point p', and the trace Ba' through the point R'. PROBLEM 3. To draw a tangent plane to a cylinder, which shall be parallel to a given line. 43. A tangent plane, parallel to a given line, ought to contain a generatrix of the cylinder, and a parallel to the given line. If, therefore, we draw through a point of this line a parallel to the generatrixes, the tangent plane will be parallel to the plane determined by these two lines. Consequently, we construct the traces of this latter plane, and draw, parallel to its horizontal trace, tangents to the horizontal trace of the cylinder; these tangents will be the horizontal traces of the tangent planes, which satisfy the conditions of the problem. The vertical traces may be constructed, since we know the points where they cut the ground line, and a line to which they are parallel. In the fig., the point o o', taken upon the given line, is that through which the parallel to the generatrixes of the cylinder is drawn; and the traces of the plane passing through these two lines are yv and yv': the tangents ra and sẞ are drawn parallel to the horizontal trace yv, and ar' and ẞs', parallel to yv': the planes rar', sßs', are the tangent planes required. To verify the constructions, we observe that the generatrixes ip, ip', and kq, k'q', ought to meet the vertical plane in the traces ar' and ẞs'. When the point p p', is very distant, it would be inconvenient to determine k'q'. In this case take in the generatrix im, i'm', the point m,m', and through it conceive a line parallel to ra: its projections are mt, parallel to ra, and m't' parallel to xy: and since it is in the tangent plane, its vertical trace t is on the vertical trace of the plane. Hence the point can serve to determine the latter trace, or to verify it when it is known. With the same view we could 777 γ 31 draw a line parallel to the generatrixes, through any point in the horizontal trace ra. If we draw a second tangent plane to the cylinder, its intersection with the former ought to be parallel to the generatrixes, since each tangent plane contains a generatrix, and the generatrixes are parallel each to each. |