The Theory and Practice of Perspective Part 13

LIX

HOW TO DIVIDE A DIAGONAL VANIs.h.i.+NG LINE INTO ANY NUMBER OF EQUAL OR PROPORTIONAL PARTS

In a previous figure (Fig. 116) we have shown how to find a measuring point when the exact measure of a vanis.h.i.+ng line is required, but if it suffices merely to divide a line into a given number of equal parts, then the following simple method can be adopted.

We wish to divide _ab_ into five equal parts. From _a_, measure off on the ground line the five equal s.p.a.ces required. From 5, the point to which these measures extend (as they are taken at random), draw a line through _b_ till it cuts the horizon at _O_. Then proceed to draw lines from each division on the base to point _O_, and they will intersect and divide _ab_ into the required number of equal parts.

[Ill.u.s.tration: Fig. 118.]

[Ill.u.s.tration: Fig. 119.]

The same method applies to a given line to be divided into various proportions, as shown in this lower figure.

[Ill.u.s.tration: Fig. 120.]

[Ill.u.s.tration: Fig. 121.]

LX

FURTHER USE OF THE MEASURING POINT O

One square in oblique or angular perspective being given, draw any number of other squares equal to it by means of this point _O_ and the diagonals.

Let _ABCD_ (Fig. 120) be the given square; produce its sides _AB_, _DC_ till they meet at point _V_. From _D_ measure off on base any number of equal s.p.a.ces of any convenient length, as 1, 2, 3, &c.; from 1, through corner of square _C_, draw a line to meet the horizon at _O_, and from _O_ draw lines to the several divisions on base line. These lines will divide the vanis.h.i.+ng line _DV_ into the required number of parts equal to _DC_, the side of the square. Produce the diagonal of the square _DB_ till it cuts the horizon at _G_. From the divisions on line _DV_ draw diagonals to point _G_: their intersections with the other vanis.h.i.+ng line _AV_ will determine the direction of the cross-lines which form the bases of other squares without the necessity of drawing them to the other vanis.h.i.+ng point, which in this case is some distance to the left of the picture. If we produce these cross-lines to the horizon we shall find that they all meet at the other vanis.h.i.+ng point, to which of course it is easy to draw them when that point is accessible, as in Fig. 121; but if it is too far out of the picture, then this method enables us to do without it.

Figure 121 corroborates the above by showing the two vanis.h.i.+ng points and additional squares. Note the working of the diagonals drawn to point _G_, in both figures.

LXI

FURTHER USE OF THE MEASURING POINT O

Suppose we wish to divide the side of a building, as in Fig. 123, or to draw a balcony, a series of windows, or columns, or what not, or, in other words, any line above the horizon, as _AB_. Then from _A_ we draw _AC_ parallel to the horizon, and mark thereon the required divisions 5, 10, 15, &c.: in this case twenty-five (Fig. 122). From _C_ draw a line through _B_ till it cuts the horizon at _O_. Then proceed to draw the other lines from each division to _O_, and thus divide the vanis.h.i.+ng line _AB_ as required.

[Ill.u.s.tration: Fig. 122 is a front view of the portico, Fig. 123.]

[Ill.u.s.tration: Fig. 123.]

In this portico there are thirteen triglyphs with twelve s.p.a.ces between them, making twenty-five divisions. The required number of parts to draw the columns can be obtained in the same way.

LXII

ANOTHER METHOD OF ANGULAR PERSPECTIVE, BEING THAT ADOPTED IN OUR ART SCHOOLS

In the previous method we have drawn our squares by means of a geometrical plan, putting each point into perspective as required, and then by means of the perspective drawing thus obtained, finding our vanis.h.i.+ng and measuring points. In this method we proceed in exactly the opposite way, setting out our points first, and drawing the square (or other figure) afterwards.

[Ill.u.s.tration: Fig. 124.]

Having drawn the horizontal and base lines, and fixed upon the position of the point of sight, we next mark the position of the spectator by dropping a perpendicular, _S ST_, from that point of sight, making it the same length as the distance we suppose the spectator to be from the picture, and thus we make _ST_ the station-point.

To understand this figure we must first look upon it as a ground-plan or bird's-eye view, the line V2V1 or horizon line representing the picture seen edgeways, because of course the station-point cannot be in the picture itself, but a certain distance in front of it. The angle at _ST_, that is the angle which decides the positions of the two vanis.h.i.+ng points V1, V2, is always a right angle, and the two remaining angles on that side of the line, called the directing line, are together equal to a right angle or 90. So that in fixing upon the angle at which the square or other figure is to be placed, we say 'let it be 60 and 30, or 70 and 20', &c. Having decided upon the station-point and the angle at which the square is to be placed, draw TV1 and TV2, till they cut the horizon at V1 and V2. These are the two vanis.h.i.+ng points to which the sides of the figure are respectively drawn. But we still want the measuring points for these two vanis.h.i.+ng lines. We therefore take first, V1 as centre and V1T as radius, and describe arc of circle till it cuts the horizon in M1, which is the measuring point for all lines drawn to V1. Then with radius V2T describe arc from centre V2 till it cuts the horizon in M2, which is the measuring point for all vanis.h.i.+ng lines drawn to V2. We have now set out our points. Let us proceed to draw the square _Abcd_. From _A_, the nearest angle (in this instance touching the base line), measure on each side of it the equal lengths _AB_ and _AE_, which represent the width or side of the square.

Draw EM2 and BM1 from the two measuring points, which give us, by their intersections with the vanis.h.i.+ng lines AV1 and AV2, the perspective lengths of the sides of the square _Abcd_. Join _b_ and V1 and dV2, which intersect each other at _C_, then _Adcb_ is the square required.

This method, which is easy when you know it, has certain drawbacks, the chief one being that if we require a long-distance point, and a small angle, such as 10 on one side, and 80 on the other, then the size of the diagram becomes so large that it has to be carried out on the floor of the studio with long strings, &c., which is a very clumsy and unscientific way of setting to work. The architects in such cases make use of the centrolinead, a clever mechanical contrivance for getting over the difficulty of the far-off vanis.h.i.+ng point, but by the method I have shown you, and shall further ill.u.s.trate, you will find that you can dispense with all this trouble, and do all your perspective either inside the picture or on a very small margin outside it.

Perhaps another drawback to this method is that it is not self-evident, as in the former one, and being rather difficult to explain, the student is apt to take it on trust, and not to trouble about the reasons for its construction: but to show that it is equally correct, I will draw the two methods in one figure.

LXIII

TWO METHODS OF ANGULAR PERSPECTIVE IN ONE FIGURE

[Ill.u.s.tration: Fig. 125.]

It matters little whether the station-point is placed above or below the horizon, as the result is the same. In Fig. 125 it is placed above, as the lower part of the figure is occupied with the geometrical plan of the other method.

In each case we make the square _K_ the same size and at the same angle, its near corner being at _A_. It must be seen that by whichever method we work out this perspective, the result is the same, so that both are correct: the great advantage of the first or geometrical system being, that we can place the square at any angle, as it is drawn without reference to vanis.h.i.+ng points.

We will, however, work out a few figures by the second method.

LXIV

TO DRAW A CUBE, THE POINTS BEING GIVEN

As in a previous figure (124) we found the various working points of angular perspective, we need now merely transfer them to the horizontal line in this figure, as in this case they will answer our purpose perfectly well.

[Ill.u.s.tration: Fig. 126.]