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In the figure (see last fig.) the projections of the vertex are 00'; those of the given point are rr'; those of the line passing through these two points are or and oʻm; the traces of this line t and u', and the tangent planes, are u'ß and tß.
To verify the constructions, we may find the vertical traces of the generatrixes of contact, and also the vertical traces of the parallels, drawn through the given point, to the horizontal traces of the two tangent planes.
To draw a tangent plane to a cone, parallel to a given line.
46. The tangent plane, which passes through the vertex, must contain a line drawn from the vertex parallel to the given line. This will be sufficient to shew that, after having traced the projections ot, o'ť', of this parallel, the constructions will be the same as in the preceding problem.
CURVES AND THEIR TANGENTS.
A LINE is defined and determined, either by some property which characterises it, or as being produced by the motion of a point, or as the intersection of two surfaces, each of which contains it. In the greater number of cases, in the practical application of descriptive geometry, lines have to be considered as being produced by the intersection of surfaces ; and it is chiefly under this point of view that they will be treated of in this part. Nevertheless, the helix and the epicycloid will be investigated as examples of curves of the two former kinds. In the other examples the question to be resolved will always be comprised in this general enunciation : “ Two surfaces being given, to determine the curce formed by their mutual intersection and to draw the tangent to it at any proposed point. When one of the surfaces is a plane, it is further necessary to construct the curve of intersection, of its true magnitude, and if one of the surfaces is developable, the curve formed by the developement of the intersection must also be drawn.
Let the intersection of a plane and a curved surface be first considered: the mode of generation of the surface being known, the generatrix may be defined in its different positions and its intersection with the plane determined. By this means the projections of as many points of the required section may be obtained as may be considered necessary, and the projections of the curve must be drawn, continuously, through them.
When it is the intersection of two curved surfaces that is required, the following is the general mode of proceeding. The surfaces are considered to be cut by a series of parallel planes, each of which determines by its intersection with the given surfaces two curves, the projections of which must be constructed, the points common to these curves obviously belong to the curve sought. It might at first be supposed that four curves would have to be drawn for each cutting plane, two on each plane of projection : but since the cutting planes, may be assumed in any position, they may always be taken as parallel to one plane of projection, consequently the projections of the sections made by these planes with the surfaces on the other plane of projection will be straight lines parallel to the ground line, or will be the trace of each cutting plané. The points on this straight line common to the intersection are immediately obtained from the projections of the curves on the other plane of projection.
In certain cases of frequent occurrence, the choice of these auxiliary sectional planes is determined by the nature of the surfaces: thus, if one of them be a cylinder, the planes should be assumed as parallel to the axis of the solid, because in that case the intersections with the surface will be right lines, and if the two given surfaces are both cylindrical the cutting planes may be assumed parallel to the generatrixes, or axes, of both cylinders.
If one surface be a cone, and the other a cylinder, the planes should be assumed as passing through the vertex of the former, and as parallel to the axis of the latter solid.
If one of the given surfaces be a surface of revolution, the auxiliary sectional planes should be assumed as perpendicular to the axis of revolution, and the sections with this solid will, consequently, be circles (37).
It must be borne in mind that the tangent-plane at any point of a curved surface, contains all the tangents which can be drawn at that point to any curve lying in the surface, passing through the point. Hence it follows, that to draw a tangent at any point of a curve, resulting from the intersection of a given plane with a curved surface, the tangent-plane to the surface at that point must be constructed, and the intersection of the first
mentioned plane with this tangent-plane, will be the tangent to the curve. And, generally, if a line be the intersection of any two surfaces the intersection of the two tangent planes drawn, one to each surface at any point of the intersection of the surfaces, will be the tangent to that intersection at the point in question.
The tangent to a curve may be also determined from other principles. The normals (Def., p. 33) drawn to two surfaces from any point of their common intersection, are respectively perpendicular to the two tangent-planes at that point: hence the required tangent to that intersection at that point, will be perpendicular to the plane passing through the normals.
This mode of proceeding is preferable to the former when the surfaces are surfaces of revolution, for in this case the normals are more readily drawn than the tangentplanes.
The projection of a tangent to a curve, is always a tangent to the projection of that curve ; for if the tangent be considered as the prolongation of an element of the curve, the projection of that tangent will be the prolongation of the projection of that element.
It must be observed that in those cases, when from the particular conditions of the case the foregoing principles cannot be applied, the required tangent can only be drawn mechanically to touch the projection of the curve at the given point.
§ 1. SECTIONS OF Curved SURFACES BY PLANES.
PROBLEM 1. Intersection of a cylinder perpendicular to the horizontal
plane by a plane perpendicular to the vertical plane. 47. The construction will consist of three distinct parts. First, To determine the projections of the intersection
and those of its tangents. Secondly, To construct this intersection of its true mag
nitude, and the tangents to it at the same points. Thirdly, To develop the cylinder and to describe the
curve which will represent the section developed,
and to draw the tangents to this developed curve. 1°. The horizontal trace of the cylinder may be any curve at pleasure abc ... and its generatrixes are perpendicular to the plane by the hypothesis. The vertical trace of the cutting plane may be any straight line qp', while its horizontal trace 9p, must be at right angles to the ground line xy. Since every line on the cylindrical surface has abc... for its horizontal projection, and every line on the cutting plane has p'q for its vertical projection, the two projections of the intersection are determined.
Take any point m in abc... at pleasure, and through it draw mm" m' perpendicular to xy to cut p’q in m'; it is obvious that m" m' is the vertical projection of the generatrix which passes through m, and that the point in which this generatrix cuts the plane pap' has m, m' for its projections. m, m' are, therefore, the projections of a point in the curve of intersection. By repeating this construction, the projections of as many points may be obtained as are requisite.