804 PROJECTION plan of the section. If from E 1( Fj lines bo drawn perpendicular to AB, these will determine the points E, F on the developed face in which the plane a cuts it ; hence also the line EF. Similarly on the other faces. Of course BF must be the same length on BP and on BQ. If the plane o be rabatted to the plan, we get the real shape of the section as shown in the figure in EFGH. This is done easily by making F F = OF 2 , &c. If the figure representing the development of the pyramid, or better a copy of it, is cut out, and if the lateral faces be bent along the lines AB, BC, &c., we get a model of tho pyramid with the section marked on its faces. This may be placed on its plan ABCD and the plane of elevation bent about the axis x. The pyramid stands then in front of its eleva tions. If next the plane a with a hole cut out representing the true section be bent along the trace a till its edge coincides with a", the edges of the hole ought to coincide with the lines EF, FG, &c., on the faces. 47. Polyhedra like the pyramid in 46 are represented by the projections of their edges and vertices. But solids bounded by curved surfaces, or surfaces themselves, cannot be thus represented. For a surface we may use, as in case of the plane, its traces that is, the curves in which it cuts the planes of projection. We may also project points and curves on the surface. A ray cuts the surface generally in more than one point ; hence it will happen that some of the rays touch the surface, if two of these points coincide. The points of contact of these rays will form some curve on the surface and this will appear from the centre of pro jection as the boundary of the surface or of part of the surface. The outlines of all surfaces of solids which we see about us are formed by the points at which rays through our eye touch the surface. The projections of these contours are therefore best adapted to give an idea of the shape of a surface. Thus the tangents drawn from any finite centre to a sphere form a right circular cone, and this will be cut by any plane in a conic. It is often called the projection of a sphere, but it is better called the contour-line of the sphere, as it is the boundary of the projec tions of all points on the sphere. If the centre is at infinity the tangent cone becomes a right circular cylinder touching the sphere along a great circle, and if the projection is, as in our case, orthographic, then the section of this cone by a plane of projection will be a circle equal to the great circle of the sphere. We get such a circle in the plan and another in the elevation, their centres being plan and elevation of the centre of the sphere. Similarly the rays touching a cone of the second order will lie in two planes which pass through the vertex of the cone, the contour-line of the projection of the cone consists therefore of two lines meeting in the projection of the vertex. These may, however, be invisible if no real tangent rays can be drawn from the centre of projection ; and this happens when the ray projecting the centre of the vertex lies within the cone. In this case the traces of the cone are of importance. Thus in representing a cone of revolution with a vertical axis we get in the plan a circular trace of the surface whose centre is the plan of the vertex of the cone, and in the elevation the contour, consisting of a pair of lines intersecting in the elevation of the vertex of the cone. The circle in the plan and the pair of lines in the elevation do not determine the surface, for an infinite number of surfaces might be conceived which pass through the circular trace and touch two planes through the contour lines in the vertical plane. The surface becomes only completely defined if we write down to the figure that it shall represent a cone. The same holds for all surfaces. Even a plane is fully represented by its traces only under the silent understanding that the traces are those of a plane. 48. Some of the simpler problems connected with the repre sentation of surfaces are the determination of plane sections and of the curves of intersection of two such surfaces. The former is constantly used in nearly all problems concerning surfaces. Its solution depends of course on the nature of the surface. To determine the curve of intersection of two surfaces, we take a plane and determine its section with each of the two surfaces, rabatting this plane if necessary. This gives two curves which lie in the same plane and whose intersections will give us points on both surfaces. It must here be remembered that two curves in space do not necessarily intersect, hence that the points in which their projections intersect are not necessarily the projections of points common to the two curves. This will, however, be the case if the two curves lie in a common plane. By taking then a number of plane sections of the surfaces we can get as many points on their curve of intersection as we like. These planes have, of course, to be selected in such a way that the sections are curves as simple as the case permits of, and such that they can be easily and accurately drawn. Thus when possible the sections should be straight lines or circles. This not only saves time in drawing but determines all points on the sections, and therefore also the points where the two curves meet, with equal accuracy. 49. We give afew examples how these sections have to be selected. A cone is cut by every plane through the vertex in lilies, and if it is a cone of revolution by planes perpendicular to the axis in circles. A cylinder is cut by every plane parallel to the axis in lines, and if it is a cylinder of revolution by planes perpendicular to the axis in circles. A sphere is cut by every plane in a circle. Hence in case of two cones situated anywhere in space we take sections through both vertices. These will cut both cones in lines. Similarly in case of two cylinders we may take sections parallel to the axis of both. In case of a sphere and a cone of revolution with vertical axis, horizontal sections will cut both surfaces in circles whose plans are circles and whose elevations are lines, whilst vertical sections through the vertex of the cone cut the latter in lines and the sphere in circles. To avoid drawing the projections of these circles, which would in general be ellipses, we rabatt the plane and then draw the circles in their real shape. And so on in other cases. Special attention should in all cases be paid to those points in which the tangents to the projection of the curve of intersection are parallel or perpendicular to the axis x, or where these projections touch the contour of one of the surfaces. PERSPECTIVE. 50. We have seen that, if all points in a figure be pro jected from a fixed centre to a plane, each point on the pro jection will be the projection of all points on the projecting ray. A complete representation by a single projection is therefore possible only when there is but one point to be projected on each ray. This is the case by projecting from one plane to another, but it is also the case if we project the visible parts of objects in nature ; for every ray of light meeting the eye starts from that point in which the ray, if we follow its course from the eye, backward meets for the first time any object. Thus, if we project from a fixed centre the visible part of objects to a plane or other surface, then the outlines of the projection would give the same impression to the eye as the outlines of the things projected, provided that one eye only be used and that this be at the centre of projection. If at the same time the light emanating from the different points in the picture could be made to be of the same kind that is, of the same colour and intensity and of the same kind of polarization as that coming from the objects themselves, then the projection would give sensibly the same impres sion as the objects themselves. The art of obtaining this result constitutes a chief part of the technique of a painter, who includes the rules which guide him under the name of perspective, distinguishing between linear and aerial perspective, the former relating to the projection, to the drawing of the outlines, the latter to the colouring and the shading off of the colours in order to give the appearance of distance. We have to deal only with the former, which is in fact a branch of geometry consisting in the applications of the rules of projection. 51. Our problem is the following: There is given a figure in space, the plane of a picture, and a point as centre of projection ; it is required to project the figure from the point to the plane. From what has been stated about projection in general it follows at once that the projection of a point is a point, that of a line a line. Further, the projection of a point at infinity in a line is in general a finite point. Hence parallel lines are projected into a pencil of lines meeting at some finite point. This point is called the vanishing point of the direction to which it belongs. To find it, we project the point at infinity in one of the parallel lines ; that is, we draw through the eye a line in the given direction. This cuts the picture plane in the point required. Similarly all points at infinity in a plane are projected to a line ( 6) which is called the vanishing line of the plane and which is common to all parallel planes. All lines parallel to a plane have their vanishing points in a line, viz., in the vanishing line of the plane. All lines parallel to the picture plane have their vanishing points at infinity in the picture plane ; hence parallel lines which are parallel to the picture plane appear in tlie projection as parallel lines in their true direction. The projection of a line is determined by the projection of two points in it, these being very often its vanishing point and its trace on the picture plane. The projection of a point is determined by the pro
jection of two lines through it.