Perspective

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.

§ t. 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. 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. I) 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 cr 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 p lane, 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 A 1 B 1, 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 5, 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 P i (figs. i and 2) in the ground plane. We draw through P 1 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 P 1 Q 1 and P 1 R 1. The first cuts the picture at Qi or at Q, and has the vanishing point V; hence QV is its perspective. The other cuts the picture in R 1, 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=Qiri=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 = 3QR and VE _VD, 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; xxl. 9 A FIG. I.

for the construction will not be altered if the ground plane be replaced by any other horizontal plane. We can in fact now find the perspective of every point as soon as we know the foot of the perpendicular drawn from it to the picture plane, that is, if we know its elevation on the picture plane, and its distance behind it. For this reason it is often convenient to draw in slight outlines the elevation of the figure on the picture plane.

Instead of drawing the elevation of the figure we may also proceed as follows. Suppose (fig. 3) A 1 to be the projection of the plan of a point A. Then the point A lies vertically above A 1 because vertical lines appear in the perspective as vertical lines (§1). If then the line VA 1 cuts the figure plane at Q, and we erect at Q a perpendicular in the picture plane to its base and set off on it QA2 equal to the real height of the point A above the ground plane, H Q y R FIG. 3.

then the point A2 is the elevation of A and hence the line A2V will pass through the point A. The latter thus is determined by the intersection of the vertical line through A 1 and the line A2V.

This process differs from the one mentioned before in this that the construction for finding the point is not made in the horizontal plane in which it lies, but that its plan is constructed in the ground plane. But this has a great advantage. The perspective of a horizontal plane from the picture to the line at infinity occupies in the picture the space between the line where the plane cuts the picture and the horizon, and this space is the greater the farther the plane is from the eye, that is, the farther its trace on the picture plane lies from the horizon. The horizontal plane through the eye is projected into a line, the horizon; hence no construction can be performed in it. The ground plane on the other hand is the lowest horizontal plane used. Hence it offers most space for constructions, which consequently will allow of greater accuracy.

§ 5\. The process is the same if we know the co-ordinates of the point, viz. we take in the base line a point 0 as origin, and we take the base line, the line OV, and the perpendicular OZ as axes of co-ordinates. If we then know the co-ordinates x, y, z measured in these directions, we make OQ =x, set off on QV a distance QA such that its real length QR =y, make QA 2 =z, and we find A as before. This process might be simplified by setting off to begin with along OQ and OZ scales in their true dimensions and along OV a scale obtained by projecting the scale on OQ from D to the line OV.

§ 6\. The methods explained give the perspective of any point in space. If lines have to be found, we may determine the perspective of two points in them and join these, and this is in many cases the most convenient process. Often, however, it will be advantageous to determine the projection of a line directly by finding its vanishing point. This is especially to be recommended when a number of parallel lines have to be drawn.

The perspective of any curve is in general a curve. The projection of a conic is a conic, or in special cases a line. The perspective of a circle may be any conic, not necessarily an ellipse. Similarly the perspective of the shadow of a circle on a plane is some conic.

§ 7\. A few words must be said about the determination of shadows in perspective. The theory of their construction is very simple. We have given, say, a figure and a point L as source of light. We join the point L to any point of which we want to find the shadow and produce this line till it cuts the surface on which the shadow falls. These constructions must in many cases first be performed in plan and elevation, and then the point in the shadow has to be found in perspective. The constructions are different according as we take as the source of light a finite point (say, the flame of a lamp), or the sun, which we may suppose to be at an infinite distance. If, for instance, in fig. 3, A is a source of light, Ehgf a vertical wall, and C a point whose shadow has to be determined, then the shadow must lie on the line joining A to C. To see where this ray meets the floor we draw through the source of light and the point C a vertical plane. This will cut the floor in a line which contains the feet A 1, C 1 of the perpendiculars drawn from the points A, C to the floor, or the plans of these points. At C', where the line A1C1 cuts AC, will be the shadow of C on the floor. If the wall Ehgf prevents the shadow from falling on the floor, we determine the intersection K of the line A 1 C 1 with the base EF of the wall and draw a vertical through it, this gives the intersection of the wall with the vertical plane through A and C. Where it cuts AC is the shadow C" of C on the wall.

If the shadow of a screen CDD 1 C I has to be found we find the shadow D' of D which falls on the floor; then D 1 D' is the shadow of D 1 D and D'C' is the shadow on the floor of the line DC. The shadow of D 1 D, however, is intercepted by the wall at L. Here then the wall takes up the shadow, which must extend to D" as the shadow of a line on a plane is a line. Thus the shadow of the screen is found in the shaded part in the figure.

§ 8\. If the shadows are due to the sun, we have to find first the perspective of the sun, that is, the vanishing point of its rays. This will always be a point in the picture plane; but we have to distinguish between the cases where the sun is in the front of the picture, and so behind the spectator, or behind the picture plane, and so in front of the spectator. In the second case only does the vanishing point of the rays of the sun actually represent the sun itself. It will be a point above the horizon. In the other case the vanishing point of the rays will lie below the horizon. It is the point where a ray of the sun through the centre of sight S cuts the picture plane, or it will be the shadow of the eye on the picture. In either case the ray of the sun through any point is the line joining the perspective of that point to the vanishing point of the sun's rays. But in the one case the shadow falls away from the vanishing point, in the other it falls towards it. The direction of the sun's rays may be given by the plan and elevation of one ray.

For the construction of the shadow of points it is convenient first to draw a perpendicular from the point to the ground and to find its shadow on the ground. But the shadows of verticals from a point at infinity will be parallel; hence they have in perspective a vanishing point L 1 in the horizon. To find this point, we draw that vertical plane through the eye which contains a ray of the sun. This cuts the horizon in the required point L 1 and the picture plane in a vertical line which contains the vanishing point of the sun's rays themselves. Let then (fig. 4) L be the vanishing point FIG. 4.

of the sun's rays, L 1 be that of their projection in a horizontal plane, and let it be required to find the shadow of the vertical column AH. We draw AL 1 and EL; they meet at E', which is the shadow of E. Similarly we find the shadows of F, G, H. Then E'F'G'H' will be the shadow of the quadrilateral Efgh. For the shadow of the column itself we join E' to A, &c., but only mark the outlines; F'B, the shadow of BF, does not appear as such in the figure.

If the shadow E has to be found when falling on any other surface we use the vertical plane through E, determine its intersection with the surface, and find the point where this intersection is cut by the line EL. This will be the required shadow of E.

§ 9\. If the picture is not to be drawn on a vertical but on another plane - say, the ceiling of a room - the rules given have to be slightly modified. The general principles will remain true. But if the picture is to be on a curved surface the constructions become somewhat more complicated. In the most general case conceivable it would be necessary to have a representation in plan and elevation of the figure required and of the surface on which the projection has to be made. A number of points might also be found by calculation, using co-ordinate geometry. But into this we do not enter. As an example we take the case of a panorama, where the surface is a vertical cylinder of revolution, the eye being in the axis. The ray projecting a point A cuts the cylinder in two points on opposite sides of the eye, hence geometrically speaking every point has two projections; of these only the one lying on the half ray from the eye to the point can be used in the picture. But the other has sometimes to be used in constructions, as the projection of a line has to pass through both. Parallel lines have two vanishing points which are found by drawing a line of the given direction through the eye; it cuts the cylinder in the vanishing points required. This operation may be performed by drawing on the ground the plan of the ray through the foot of the axis, and through the point where it cuts the cylinder a vertical, on which the point required must lie. Its height above is easily found by making a drawing of a vertical section on a reduced scale.

Parallel planes have in the same manner a vanishing curve. This will be for horizontal planes a horizontal circle of the height of the eye above the ground. For vertical planes it will be a pair of generators of the cylinder. For other planes the vanishing curves will be ellipses having their centre at the eye.

The projections of vertical lines will be vertical lines on the cylinder. Of all other lines they will be ellipses with the centre at the eye. If the cylinder be developed into a plane, then these ellipses will be changed into curves of sines. Parallel lines are thus represented by curves of sines which have two points in common. There is no difficulty in making all the constructions on a small scale on the drawing board and then transferring them to the cylinder.

§ 10\. A variety of instruments have been proposed to facilitate perspective drawings. If the problem is to make a drawing from nature then a camera obscura or, better, Wollaston's camera lucida may be used. Other instruments are made for the construction of perspective drawings. It will often happen that the vanishing point of some direction which would be very useful in the construction falls at a great distance off the paper, and various methods have been proposed of drawing lines through such a point. For some of these see Stanley's Descriptive Treatise on Mathematical Drawing Instruments. (0. H.)