PROJECTION 805 These are the general rules which we now apply. We suppose the picture plane to be vertical. 52. Let (fig. 21) S be the centre of projection, where the eye is situated, and which in perspective is called the point of sight, AJ3KL the picture plane, ABMN a horizontal plane on. which we suppose the ob jects 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 some times the geometrical plane. It cuts the picture plane in a horizontal line AB called the ground line or base line or funda mental line of the picture. A horizontal line SV, A Fig. 21. 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 hori zontal plane, is got by drawing the projecting rays from S to the points at infinity in the plane in other words, by drawing all hori zontal 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 con tains 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 deter mined ; it is the vertical trace of a line parallel to it through the eye (comp. 35). 53. To have construction in a single plane, we suppose the pic ture 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. 22). 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 AjBj, 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 R. B, A Q. W, Fig. 22. 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. 54. Let it now be required to find the perspective P of a point P! (figs. 21 and 22) in the ground plane. We draw through P l two lines of which the projection can easily be found. The most con venient 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 plan are PjQj and PjHj. The first cuts the picture at Qj or at Q, and has the vanishing point V; hence QV is its perspective. The other cuts the picture in R lt or rather in R, and has the vanish ing 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 = Q 1 R 1 = Q 1 P 1 gives the distance of the point P behind the pic ture plane. Hence if we know the point Q where a perpendicular from a point to the picture plane cuts the latter, and also the dis tance of the point behind the picture plane, we can find its perspec tive. 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 dis tance 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 Tith part of the distance and join it to a point E, so that VE equals the nth part of VD. Thus if QW = JQR 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 ; 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 per pendicular draivnfrom it to the picture plane, tJiat 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. 23) A l 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 (% 51). If then the line VAj cuts the picture plane at Q, and we erect at Q a perpendicular in the picture plane to its base and set off on it QA 2 equal to the real height of the point A above the ground plane" then the point A 2 is the elevation of A and hence the line A 2 V will pass through the point A. The latter thus is deter mined by the intersection of the vertical line through A x and the line A 2 V. -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 hori zontal 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. 55. The process is the same if we know the coordinates of the point, viz. , we take in the base line a point as origin, and we take the base line, the line 0V, and the perpendicular OZ as axes of coordinates. If we then know the coordinates x, y, z measured in these directions, we make OQ = a?, set off on QV a distance QA such that its real length QR = j/, make QA 2 = z, and 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 0V a scale obtained by projecting the scale on OQ from D to the line 0V. 56. The methods explained give the perspective of any point in space. If lines have to be found, we may determine the perspec tive 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 pro
jection of a conic is a conic, or in special cases a line. The