SECTION IV. THE CONE, CYLINDER, AND SPHERE. These are the only surfaces which are usually brought into engineering practice, except on very special occasions. When such occur, a more extended series of studies of the Descriptive Geometry must be entered upon, and the engineer must have recourse to treatises expressly and solely devoted to the subject. The problems here given will amply display the general character of this class of researches ; and it will be seen that they differ but little, as to general method, from those already developed at length. It will be desirable, however, to prefix a few notes on the mode of designating surfaces to suit this method of construction. (1.) As has been already shown, a plane is designated usually (because most conveniently) by its traces on two coordinate planes. Indeed one trace and a point on the other is sufficient; and when the plane is given by means of other specific conditions, it is still the best way of proceeding to find the traces of that plane as the preliminary step. (2.) The cylinder being the extreme case of the cone (where the vertex is infinitely distant and the edges of the cone consequently parallel), the two surfaces will be designated by nearly similar methods, and the constructions will partake in a great degree of the same character. (a) The cone and cylinder may each be defined (that is, their elements represented) by means of the projections of two sections parallel to either of the coordinate planes. Thus: When these two sections are equal, the figure is a cylinder ; and when unequal, a cone. They are necessarily similar whether equal or not(Prop. xx. Chap. 1. Pls. and Sols.). (6) The most usual way of defining the cone and cylinder is by means of the horizontal trace of the surface and the projections of the axis. The horizontal trace (the vertical might have been taken instead) of the cone or cylinder is most frequently taken as a circle, in which case the centre of the circle is a point in the axis; and only one point more is required for fixing the axis. In the cone, that one point is invariably the centre of the surface or vertex; in the cylinder, any point whatever. Even when the cone or cylinder is given by the first method, it is most convenient, and sometimes necessary, to reduce the con ditions to the second method of definition. (3.) The sphere is defined by means of its radius given in magnitude, and the projections of its centre given in position. Sometimes it may happen that when the sphere, instead of being directly given, results from prescribed constructions, it may present itself by means of some other elements. Those elements can however always be, and indeed always are, reduced to the condition stated above, anterior to any further constructions which involve it. Of the ellipsoid, which is the surface next to these as regards elementary character, nothing can be given in this course, and hence the mode of defining it may be properly omitted. PROPOSITION I. Given one trace and the projections of the axis of a cone to find the extreme limits of its projections on both planes. Let v, v, be the vertex, e, the centre of the circular trace, and the circle as in the figure. Draw the tangents a, a, b, b perpendicular to the axis; and join av., bvg. The space av, b comprehends the projections on the vertical plane of all the points of the cone between the vertex and trace inclusive. If points beyond these limits be taken into consideration, they will still lie between the prolongations of these lines taken suitably to the case. Draw the tangents v, Cu, v, d, to the trace. Then the horizontal projections of all the points in the cone between the vertex and the base inclusive will lie in the space bounded by the circular arc c, a, d, and the tangents C, vi, d, ej. The extensions of the tangentsc, v dv, include the projections of all points in the extended cone, as in the former case. In the cylinder the process is precisely similar, except that the limiting lines are parallel instead of converging to a point, as in the unlettered figure. SCHOLIUM. The figures contain also the construction for the projections of the axis, which the student will explain. PROPOSITION II. Given the elements of a cone or cylinder and the projections of a point, to find whether that point be on the surface or not. (1.) If the projections a, a, of the point do not both lie within the regions prescribed by the preceding problem, the point cannot be on the surface. (2.) If they be situated within the limits, each will be the projection of some two points of the surface (since a line may cut the surface in two points); the horizontal projection being that of one pair of such points, and the vertical that of another pair of such points. Suppose, then, that a, is the horizontal projection of a point on the surface: we are required to find whether a, is the vertical projection of the same point. Draw v, a, (v. v, being the vertex) to meet the trace in Pu 91. This is the horizontal projection of the edge of the cone through that point; and pu, q, are the traces of the two edges, either of which might contain the point in question. Find, as usual, their vertical projections pvz, qve, and let them cut a, a, in r and s. Then, if either r or s coincide with a, a, will be the vertical projection of the point on the surface whose horizontal projection is a,: if not, the point a, a, is not upon the given surface. SCHOLIUM. If a, fall between r and s, the point a, a, is within the cone or cylinder ; if without, without. In this process we have solved the problem: given one projection of a point situated on a given cone or cylinder to find the other. There is however another case left for the student: when the vertical projection a, is given, to find az. PROPOSITION III. Through a given point to draw a tangent plane to a given cone or cylinder. (1.) Let viv, be the given vertex of the cone, and a, a, the given point. Both these are in the tangent plane, and the tangent plane contains both the traces of the line through them. Let the horizontal trace a and the vertical B, be found as usual. Again, since the plane touches the cone, its horizontal trace is a tangent to the horizontal trace of the cone. Whence drawing a, P, Q, Q to touch the horizontal trace of the cone in P, Qı, these will be the horizontal traces of the two planes through a, a, touching the cone. Lastly, as the plane also passes through By, we have only to draw PB, QB, which will be the vertical traces required. (2). When the point a, a, is in the surface of the cone, the two tangent planes coalesce. The process, though slightly modified, is very nearly the same; the difference merely arising from a, falling in the trace of the cone. The tangent to the trace of the cone at that point is the horizontal trace of the tangent plane; and the vertical trace is found as before. (3.) For the cylinder, the only difference in the construction is, that instead of drawing the line a, a,, V, V, to a given point, we must draw it parallel to the given axis, or one of the given generatrices. PROPOSITION IV. To draw a tangent plane to a given cone, which shall make a given angle with the horizontal plane of projection. Using the former notation for the data, draw v, r parallel to the axis, and make v, n = vv2, and the angle v, rs the complement of the given inclination ; with centre v, and radius vis describe a circle; and to this and the trace draw a common tangent P. P. This will be the horizontal trace required, but the proof is left as an exercise for the student. To find the vertical trace, we have only to recollect that the plane (P) passes through v V2, and that we have found the horizontal trace P, P. The construction is therefore reduced to a former problem; and is, moreover, suggested by the lines drawn in the figure. SCHOLIUM. When the circle about v, falls wholly without the trace of the cone, there will be four solutions; when it touches externally, there will be three; when it cuts, there will be two; when it touches internally, there will be one; and when it falls wholly within, there will be none. The construction for the cylinder will only differ from the preceding in minor particulars; which, however, the student is required to point out. PROPOSITION V. A cone and a line through its vertex are given, to draw through the line a tangent plane to the cone. All planes through the given line (whether they are tangent to the cone or not) pass through the traces of that line. Let then vi ve be the vertex of the cone, and dig Be the traces of the given line through it. Draw a, P, P to touch the trace of the cone; this will be one of the traces of the plane. The other trace passing through B,, the line PB, is that trace itself. The plane is hence determined. SCHOlium. If the line , B, fall within the conical surface, no such tangent plane can be so drawn; if it fall on the surface, one only can be so drawn; and if wholly without, there can be two. There is no corresponding case in respect to the cylinder. |