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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
C0??0S.
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 ex-
ample. 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 rectilin-
ear 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
cones, FE" and fif the curves which are their inter-
sections 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'E' 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 cut-
ting plane.
We next construct the projections of lines drawn from
the points F to the summit of the first cone, and from
the points fo 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 FR" ; their
points of meeting marked upon each of the co-ordinate
planes by the figures 1, 2, 3, 4, will be points in the in-
tersection 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
14

Fig. 38. 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 elementS.

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

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 joi

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.f 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.

Fig. 40.

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 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 pop 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 A p" 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 plane, as the plane T'A t”, then the projections O'P' and O"P" themselves determine, by their intersections with the lines TA and to 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 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 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.

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. The 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

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Fig. 4 I.

Fig. 42.

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