The Theory and Practice of Perspective Part 17

In the same manner we can draw a cubic figure (Fig. 154)--a box, for instance--at any required angle. In this case, besides the scale _AS_, _OS_, we have made use of the vanishing lines _DV_, _BV_, to corroborate the scale, but they can be dispensed with in these simple objects, or we can use a scale on each side of the figure as _aoS_, should both vanishing points be inaccessible. Let it be noted that in the scale _AOS_, _AO_ is made equal to _BC_, the height of the box.

[Ill.u.s.tration: Fig. 154.]

By a similar process we draw these two figures, one on the square, the other on the circle.

[Ill.u.s.tration: Fig. 155.]

[Ill.u.s.tration: Fig. 156.]

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POINTS IN s.p.a.cE

The chief use of these figures is to show how by means of diagonals, horizontals, and perpendiculars almost any figure in s.p.a.ce can be set down. Lines at any slope and at any angle can be drawn by this descriptive geometry.

The student can examine these figures for himself, and will understand their working from what has gone before. Here (Fig. 157) in the geometrical square we have a vertical plane _AabB_ standing on its base _AB_. We wish to place a projection of this figure at a certain distance and at a given angle in s.p.a.ce. First of all we transfer it to the side of the cube, where it is seen in perspective, whilst at its side is another perspective square lying flat, on which we have to stand our figure. By means of the diagonal of this flat square, horizontals from figure on side of cube, and lines drawn from point of sight (as already explained), we obtain the direction of base line _AB_, and also by means of lines _aa_ and _bb_ we obtain the two points in s.p.a.ce _ab_. Join _Aa_, _ab_ and _Bb_, and we have the projection required, and which may be said to possess the third dimension.

[Ill.u.s.tration: Fig. 157.]

In this other case (Fig. 158) we have a wedge-shaped figure standing on a triangle placed on the ground, as in the previous figure, its three corners being the same height. In the vertical geometrical square we have a ground-plan of the figure, from which we draw lines to diagonal and to base, and notify by numerals 1, 3, 2, 1, 3; these we transfer to base of the horizontal perspective square, and then construct shaded triangle 1, 2, 3, and raise to the height required as shown at 1, 2, 3. Although we may not want to make use of these special figures, they show us how we could work out almost any form or object suspended in s.p.a.ce.

[Ill.u.s.tration: Fig. 158.]

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THE SQUARE AND DIAGONAL APPLIED TO CUBES AND SOLIDS DRAWN THEREIN

[Ill.u.s.tration: Fig. 159.]

As we have made use of the square and diagonal to draw figures at various angles so can we make use of cubes either in parallel or angular perspective to draw other solid figures within them, as shown in these drawings, for this is simply an amplification of that method. Indeed we might invent many more such things. But subjects for perspective treatment will constantly present themselves to the artist or draughtsman in the course of his experience, and while I endeavour to show him how to grapple with any new difficulty or subject that may arise, it is impossible to set down all of them in this book.

[Ill.u.s.tration: Fig. 160.]

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TO DRAW AN OBLIQUE SQUARE IN ANOTHER OBLIQUE SQUARE WITHOUT USING VANISHING POINTS

It is not often that both vanishing points are inaccessible, still it is well to know how to proceed when this is the case. We first draw the square _ABCD_ inside the parallel square, as in previous figures. To draw the smaller square _K_ we simply draw a smaller parallel square _h h h h_, and within that, guided by the intersections of the diagonals therewith, we obtain the four points through which to draw square _K_.

To raise a solid figure on these squares we can make use of the vanishing scales as shown on each side of the figure, thus obtaining the upper square 1 2 3 4, then by means of the diagonal 1 3 and 2 4 and verticals raised from each corner of square _K_ to meet them we obtain the smaller upper square corresponding to _K_.

It might be said that all this can be done by using the two vanishing points in the usual way. In the first place, if they were as far off as required for this figure we could not get them into a page unless it were three or four times the width of this one, and to use shorter distances results in distortion, so that the real use of this system is that we can make our figures look quite natural and with much less trouble than by the other method.

[Ill.u.s.tration: Fig. 161.]

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SHOWING HOW A PEDESTAL CAN BE DRAWN BY THE NEW METHOD

This is a repet.i.tion of the previous problem, or rather the application of it to architecture, although when there are many details it may be more convenient to use vanishing points or the centrolinead.

[Ill.u.s.tration: Fig. 162.]

[Ill.u.s.tration: Fig. 163. Honfleur.]

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SCALE ON EACH SIDE OF THE PICTURE

As one of my objects in writing this book is to facilitate the working of our perspective, partly for the comfort of the artist, and partly that he may have no excuse for neglecting it, I will here show you how you may, by a very simple means, secure the general correctness of your perspective when sketching or painting out of doors.

Let us take this example from a sketch made at Honfleur (Fig. 163), and in which my eye was my only guide, but it stands the test of the rule.

First of all note that line _HH_, drawn from one side of the picture to the other, is the horizontal line; below that is a wall and a pavement marked _aV_, also going from one side of the picture to the other, and being lower down at _a_ than at _V_ it runs up as it were to meet the horizon at some distant point. In order to form our scale I take first the length of _Ha_, and measure it above and below the horizon, along the side to our left as many times as required, in this case four or five. I now take the length _HV_ on the right side of the picture and measure it above and below the horizon, as in the other case; and then from these divisions obtain dotted lines crossing the picture from one side to the other which must all meet at some distant point on the horizon. These act as guiding lines, and are sufficient to give us the direction of any vanishing lines going to the same point. For those that go in the opposite direction we proceed in the same way, as from _b_ on the right to _V_ on the left. They are here put in faintly, so as not to interfere with the drawing. In the sketch of Toledo (Fig. 164) the same thing is shown by double lines on each side to separate the two sets of lines, and to make the principle more evident.

[Ill.u.s.tration: Fig. 164. Toledo.]

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THE CIRCLE

If we inscribe a circle in a square we find that it touches that square at four points which are in the middle of each side, as at _a b c d_. It will also intersect the two diagonals at the four points _o_ (Fig. 165).

If, then, we put this square and its diagonals, &c., into perspective we shall have eight guiding points through which to trace the required circle, as shown in Fig. 166, which has the same base as Fig. 165.

[Ill.u.s.tration: Fig. 165.]