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mt and m'ť ; 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.
To verify the constructions, we have drawn through the given point lines parallel to ta and tß: the points rand 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 i p. and i'p', meets the vertical plane in a point p': and the other, which has projections kr, 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'.
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 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 oo', 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 r a and sß are drawn parallel to the horizontal trace yv, and ar' and Bs', parallel to yv': the planes rar', sßs', are the tangent planes required.
To verify the constructions, we observe that the generatrixes i p, i'p', and k q, k'd, ought to meet the vertical plane in the traces ar' and B8.
When the point p p', is very distant, it would be inconvenient to determine k'd. 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'ť parallel to xy: and since it is in the tangent plane, its vertical trace ť is on the vertical trace of the plane. Hence the point t can serve to determine the latter trace, or to verify it when it is known. With the same view we could
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.
PROBLEM 4. Given the horizontal trace of a cone and the projections of its
vertex, to find the tangent plane to a point in the surface.
44. Let acd be the base or horizontal trace of the cone, 0,0' the projections of the vertex.
27 To find the outline of the cone, or the limits between which all the generatrixes are projected, draw in the horizontal plane the tangents oa and ob; then the tangents cc', dd', _ to xy: and the lines o'd', o'd', in the vertical plane.
First, to determine the points of the surface which correspond with a given vertical projection, m'. Draw the line o'm', cutting xy in e', and e'e 1 to xy; then o'e', e'e, are the traces of a plane + to the vertical plane, which contains the required points, and which passes through the vertex of the cone. This plane can only meet the cone in its generatrixes; and as its horizontal trace ee', meets the base acd, in points e and f, it is evident that oe and of are the horizontal projections of these generatrixes; thence it is easy to find the horizontal projections m and n of the required points.
Suppose that the tangent plane to the point mm', be required. We observe that the plane ought to contain the generatrix oe, oʻé', and touch the cone in the whole extent of that line. Next, draw the tangent ex, to the point e, join & p', p' being the point where the generatrix meets the vertical plane, then ex and xp' will be the traces of the tangent plane.
To draw a tangent plane to a cone, through a point without
45. Every tangent plane to a cone passes through its vertex, and its horizontal trace is a tangent to its base.
Draw, therefore, a line joining the given point and the vertex of the cone; and from the point where this line meets the horizontal plane, draw tangents to the base of the
These tangents will be the horizontal traces of the tangent planes required. To find their vertical traces, join the vertical trace of the same line to the points where the tangents to the base meet the ground line.