Thus, in descriptive geometry, in order to express the form and position of a curved surface, it is only necessary, for any point of such surfitee, of which the projections may be taken at pleasure, to give the manner rat' constructing the and vertical projections of two different generators, which pass that point.
" We shall now proceed to apply these general principles to the plane, of all surthees the most simple, and the most frequently in use.
" A plane is generated from the motion of a given right due, of a position, which moves so that all its points may describe lines parallel to a second given right line. If this second line be itself in the plane in question, it may be also said that the plane is generated by such second right line moving so that all its points may describe lines parahel to the first.
We have therefore an idea of the position of a plane, from an observation of the tea right lines. each of which may be considered as its generator. The position of these two right lines in the plane which they generate, is altogether inditrerent ; it is only necessary, therefore, tin. projections, to make choice of such as are cef the most simple construction. Hence, in descriptive geometry. the position of a plane is indicated by giving the two right lines along which it cuts the planes of projeetien. It is easy to recollect that these two right lines must meet the intersection of the two planes of projection in one and t he salute point. and that, this must be the point. in which they meet themselves.
" As we shall have frequent occasion to bring planes under our consideration, we shall, for the sake of bre% itv. adopt the term traces to describe those right lines, by which they cut the planes of projection, and by which their position is indicated.
" I laying settled these preliminaries. we now proceed to the solution of various questions. which will at once serve as exercises on the method of projections, and facilitate our farther progress in descriptive geometry.
cc First Question,— Figure 4. A point, whose projections are D, d, and a right line, whose are A n and a 6, being given ; to construct the projections of a second right line, drawn from the point given. parallel to the first.
" Solution.—The two horizontal projections of the given right line, and of the line. sought, must be parallel to each
other ; being the intersections of two vertical planes. parallel to a common plane. It is also the same with the vertical projections of similar right lines. Therefore, as the right line sought for must necessarily pass through the given point, its projections must also pass through those of the point respectively. lf, then, from the point n. E r be drawn parallel to A B. and if from the point d. e he drawn parallel to a L. the lines E F and e f will he the projections required.
Second Question.— Figure 5. A plane whose two traces are A 13, D c. and a point, whose projections arc n, p, being given. to construct the traces of a second plane, drawn from the given point, parallel to the first.
" Solution.—The traces of the plane sought for, must be parallel to the respective traces of the given plane, because these traces taken in pairs, arc the intersections of two planes parallel to a common plane. We have, therefere, only to find, for each of them, one of the points through which they respectively pass. To obtain this, from the given point, conceive a horizontal right line in the plane sought for this line will be parallel to the trace A D. cutting the vertical plane in a point. which will he one of those of the trace of the plane sought fie' on it : and we shall have its two pro jeetions, by drawing the indefinite horizontal p F from the point g, and the right line G 1 from the point c, parallel to A B. be produced to meet the intersection. L Mt, of the two planes of projection in the point 1, such point will be the hori zontal projection of the interaction of the horizontal right line with the vertical plane. This point of intersection, there fore, will be found upon the vertical line 1 r, drawn t'rom the point 1. Rut as it must also be found upon p F, it will he discovered at the point F, of intersection of the two latter right lines. Lastly, by drawing a line from F parallel to u c, we shall have upon the vertical plane the trace of the plane required : and it' this trace be produced till it meet L rat ill the piiiut an l E D he drawn parallel to A B, we shall have the trace of the same plane upon the horizontal plane.