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to prosecute the claimant for perjury. Besides the enactments above referred to there are also a number of provisions for dealing with the personation of sailors, soldiers, pensioners and owners of stock in the public funds or shares in joint-stock companies, and of persons who falsely acknowledge in the name of another recognizances, deeds or instruments, before a court or person authorized to take the acknowledgment.

PERSPECTIVE (Lat. perspicere, to see through), in mathematics the name given to the art of representing solid objects by a plane drawing which affects the eye as does the object itself. In the article Projection it is shown that if all points in a figure be projected from a fixed centre to a plane, each point on the projection 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 impression 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. Here we deal only with the former, which is in fact a branch of geometry consisting in the applications of the rules of projection.

§ 1. 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 (q.v.) 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 (see Projection: § 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 the projection as parallel lines in their true direction.

Fig. 1

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 projection of two lines through it.

These are the general rules which we now apply. We suppose the picture plane to be vertical.

§ 2. Let (fig. 1) S be the centre of projection, where the eye is situated, and which in perspective is called the point of sight, ABKL the picture plane, ABMN a horizontal plane on which we suppose the objects to rest of which a perspective drawing is to be made. The lowest plane which contains points that are to appear in the picture is generally selected for this purpose, and is therefore called the ground plane, or sometimes the geometrical plane. It cuts the picture plane in a horizontal line AB called the ground line or base line or fundamental line of the picture. A horizontal line SV, drawn through the eye S perpendicular to the picture, cuts the latter at a point V called the centre of the picture or the centre of vision. The distance SV of the eye from the picture is often called the distance simply, and the height ST of the eye above the ground the height of the eye.

The vanishing line of the ground plane, and hence of every horizontal plane, is got by drawing the projecting rays from S to the points at infinity in the plane—in other words, by drawing all horizontal rays through S. These lie in a horizontal plane which cuts the picture plane in a horizontal line DD′ through the centre of vision V. This line is called the horizon in the picture. It contains the vanishing points of all horizontal lines, the centre of vision V being the vanishing point of all lines parallel to SV, that is perpendicular to the picture plane. To find the vanishing point of any other line we draw through S the ray projecting the point at infinity in the line; that is, we draw through S a ray parallel to the line, and determine the point where this ray cuts the picture plane. If the line is given by its plan on the ground plane, and its elevation on the picture plane, then its vanishing point can at once be determined; it is the vertical trace of a line parallel to it through the eye (cf. Geometry: § Descriptive, § 6).

§ 3. To have construction in a single plane, we suppose the picture plane turned down into the ground plane, but before this is done the ground plane is pulled forward till, say, the line MN takes the place of AB, and then the picture plane is turned down. By this we keep the plan of the figure and the picture itself separate. In this new position the plane of the picture will be that of the paper (fig. 2). On it are marked the base line AB, the centre of vision V, and the horizon DD′, and also the limits ABKL of the actual picture. These, however, need not necessarily be marked. In the plan the picture plane must be supposed to pass through A₁B₁, and to be perpendicular to the ground plane. If we further suppose that the horizontal plane through the eye which cuts the picture plane in the horizon DD′ be turned down about the horizon, then the centre of sight will come to the point S, where VS equals the distance of the eye.

To find the vanishing point of any line in a horizontal plane, we have to draw through S a line in the given direction and see where it cuts the horizon. For instance to find the vanishing points of the two horizontal directions which make angles of 45° with the horizon, we draw through S lines SD and SD′ making each an angle of 45° with the line DD′. These points can also be found by making VD and VD′ each equal to the distance SV. The two points D, D′ are therefore called the distance points.

§ 4. Let it now be required to find the perspective P of a point P₁ (figs. 1 and 2) in the ground plane. We draw through P₁ two lines of which the projection can easily be found. The most convenient lines are the perpendicular to the base line, and a line making an angle of 45° with the picture plane. These lines in the ground plane are P₁Q₁ and P₁R₁. The first cuts the picture at Q₁ or at Q, and has the vanishing point V; hence QV is its perspective. The other cuts the picture in R₁, or rather in R, and has the vanishing point D; its perspective is RD. These two lines meet at P, which is the point required. It will be noticed that the line QR＝Q₁R₁＝Q₁P₁ gives the distance of the point P behind the picture plane. Hence if we know the point Q where a perpendicular from a point to the picture plane cuts the latter, and also the distance of the point behind the picture plane, we can find its perspective. We join Q to V, set off QR to the right equal to the distance of the point behind the picture plane, and join R to the distance point to the left; where RD cuts QV is the point P required. Or we set off QR′ to the left equal to the distance and join R′ to the distance point D′ to the right.

If the distance of the point from the picture should be very great, the point R might fall at too great a distance from Q to be on the drawing. In this case we might set off QW equal to the nth part of the distance and join it to a point E, so that VE equals the nth part of VD. Thus if QW＝1/3QR and VE＝1/3VD, then WE will again pass through P. It is thus possible to find for every point in the ground plane, or in fact in any horizontal plane, the perspective;