That we might not render the figure two complicated Fig. 37. we have performed this process upon only one of these points P; it is readily seen, however, that the process is applicable to the other three. 93. PROBLEM. To construct the intersection of two cones. Suppose that the proposed cones have their bases in the same plane, or (which is the same thing) that we know the curvilinear section made in each of these cones by one of the co-ordinate planes, the horizontal, for example. The problem may always be reduced to this state. Now imagine a plane passing through the line which joins the summits of the proposed cones, and turning about this line; this plane in each of the positions where it meets the cones will cut each of them in two rectilinear elements of their surfaces; and as these are all in one plane, the two which belong to the first cone, will meet those which belong to the second; and these points of meeting will be points in the intersection required. Let (S', S"), (s', s"), (fig. 38) be the summits of the Fig. 38. cones, F'F' and f'f' the curves which are their intersections with the horizontal plane, E the point where the line joining their summits meets the horizontal plane; it is evident that the cutting plane, in all its positions, will pass through this point. We now draw the straight line E'F' at pleasure, but in such a manner that it will meet the bases of the cones, and consider this line as the horizontal trace of the cutting plane. We next construct the projections of lines drawn from the points F' to the summit of the first cone, and from the points f' to the summit of the second; these lines are respectively the projections of rectilinear elements of the conical surfaces proposed, situated in a plane which passes through the straight line which joins the summits of the cones, and through the straight line F'F'; their points of meeting marked upon each of the co-ordinate planes by the figures 1, 2, 3, 4, will be points in the intersection required. The figure represents a case in which one of the cones entirely penetrates the other. Two of the four points found by the preceding construction, belong to that part of the curve where the penetrating cone enters the other, and the other two points belong to the part where the first cone emerges from the second. 94. To find the intersection of a cone and cylinder, we imagine a straight line passing through the summit of the cone parallel to the axis of the cylinder ; then planes passing through this line will cut the proposed conical and cylindrical surfaces in their recti-linear ele ments. 95. To find the intersection of two cylinders, it will be necessary to cut these bodies by planes parallel to their axes; if the cylinders have their bases upon the same plane, the construction will be analogous to that given above for the cones. We first determine the horizontal traces of the cutting planes, which we do by passing through any point two straight lines parallel respectively to the axes of the two cylinders; the horizontal trace of the plane of these lines will of course be parallel to the horizontal traces of the cutting planes. We may draw as many of these traces as we wish; the points where they meet the periphery of the base, will be points in the rectilinear elements according to which these surfaces are supposed to be cut by these auxiliary planes; these right lines being constructed will give, by their mutual intersections, points in the required intersection. SECTION V. Linear Perspective. 96. Perspective is a science which teaches us to represent upon any surface whatever, the outline of objects, such as they appear when viewed from any given point. Light in passing through a homogeneous medium moves in straight lines; and objects become visible by means of the rays of light which proceed from their surfaces to the eye. These rays, by their inclinations among themselves, determine the images of bodies. Thus we perceive the contour, or apparent outline, of Fig. 39. the quadrilateral ABCD (fig. 39) because from each point of it a ray of light is conveyed to the eye. It is manifest that these rays, taken together, constitute the pyramid formed by the lines drawn from the different Fig. 39. points of the object to the eye. * Let O-ABCD represent this pyramid of rays, the summit O being the place of the eye. Each of these rays or rectilinear elements of this pyramid, must appear to the eye but a single point. If, therefore, this pyramid be cut by a plane or by any other surface, this section will exhibit to the eye at O the same outline as the quadrilateral ABCD. It is not necessary, therefore, in order to give us the sensation produced in us by the organ of vision, that the object itself should be presented to the eye; it is sufficient for this purpose, to determine an assemblage of rays disposed in the same manner respectively as those which pass to the eye from the different points in the object.† Hence we can represent objects upon a plane; for if we conceive the pyramid formed by the assemblage of rays transmitted from different parts of the object to our eye, to be cut by a plane, an image would be formed which would represent the contour of the body and the relative position of its differents parts. It follows from what precedes that the determination of this image depends entirely upon finding the intersections of the lines proceeding from the eye to different conspicuous points of the object, with the plane or surface on which it is to be represented. This surface is called the picture or plane of delineation. The respective positions of the eye, the picture, and the object, must be determined, in order that the image may be determined. The knowledge of the true form and dimensions of * This supposes the object to be either white or colored, but not black; for in that case it would be perceived only by the absence of light; thus we might say that the pyramid was determined by the absence of rays from the space occupied by the quadrilateral. + It is evident that a perspective of the object would also be formed, supposing the visual rays produced beyond it and extended until they meet the plane situated behind it; the image in this case would be greater than the object. the body which we wish to represent, will give us the projections of the conspicuous points which determine its contour, and the situation of the parts which compose it. The problem will then be reduced to finding upon the plane of delineation, the image of each of these points, or in other words, the meeting of a given straight line with a given plane. We shall discuss some of the different cases which the problem presents. 97. PROBLEM. To find upon the plane of the picture, situated in any manner whatever, the appearance or the perspective of a point given in space. Take the vertical projection of the proposed point, on a plane perpendicular to the common intersection of the plane of the picture with the horizontal plane. Let Fig. 40 TAT" (fig. 40) be the plane of the picture; O' and O' the projections of the eye O; P' and P" those of the point P which is to be put in perspective; O'P' and O"P" will be the projections of the visual ray OP. The meeting p of this line with the plane of the picture, will determine the perspective sought, which may be found by article 33; but as this point must be constructed upon the plane of the picture, the projections p and p" are not sufficient. This meeting of the visual ray with the plane of delineation is called the perspective of the point from which the ray emanates, and is really an oblique projection of this point upon the plane of the picture. As the lines by which the several points in the object should be projected upon this plane, must converge to the point in which the eye is situated, this projection is called a perspective projection. We therefore draw p'p through the horizontal projection of the required point p, and perpendicular to the horizontal trace of the plane of the picture. We now have the distances A p and Ap" of the required point from the two lines AT" and AT" perpendicular to each other in the plane of the picture. The line AT', which is ths intersection of the plane of the picture with the horizontal plane, is called in perspective the base line; it is considered as limiting the bottom of the picture, and as exhibiting the ground on which the original object stands. 98. When the plane of delineation is perpendicnlar to Fig. 40. the horizontal planc, as the plane T'A t", then the projections O'P' and O"P" themselves determine, by their intersections with the lines TA and t"A, the distances A q' and A q′′ of the perspective q from each of these straight lines. We take for example a pyramid (fig. 41) of which Fig. 41. the four triedral angles have their summits projected at the extremity of the rays drawn from the points O' and O". The construction of the perspective of one of these summits is designated by the same letters as in figure 40. In the case where the plane of delineation is vertical, the construction is very much simplified by taking the plane of the picture itself for the co-ordinate vertical plane. The eye, being supposed behind the plane of the picture, (fig. 42) has its horizontal projection in O′; that Fig. 42. of the point in question is in P', and p is the perspective of this point. 99. Remark. If the object to be represented is terminated by straight lines and planes, we may construct its image by seeking the perspectives of the vertices of the polyedral angles by which it is terminated; and in order to this, it will only be necessary to repeat the process which has just been indicated. Two points will determine a straight line, and the faces of the proposed object will be formed by a certain number of lines. The When the object is terminated by curve surfaces, no particular point is presented by which we may determine its form; we must first find its visible limit. visible limit of a body is the curve which separates the part which is seen from that which is not seen; it is evidently formed by the series of points in each of which the visual ray merely touches the surface of the body. If we conceive a conical surface having its summit placed at the eye, enveloping the proposed body, by touching it, the curve of contacts will be precisely that of the visible limit. If we cut this cone by planes drawn through the eye, in any manner whatever, each of them will form in the proposed body a section to which two of the right line elements of the cone will be tangent. From this results a general method of constructing the visible limit of a curve surface. Let us suppose this surface to be cut by a series of ver |