The Theory and Practice of Perspective Part 3

RULE 5

All horizontals which are at right angles to the picture plane are drawn to the point of sight.

Thus the lines _AB_ and _CD_ (Fig. 28) are horizontal or parallel to the ground plane, and are also at right angles to the picture plane _K_. It will be seen that the perspective lines _Ba_, _Dc_, must, according to the laws of projection, be drawn to the point of sight.

This is the most important rule in perspective (see Fig. 7 at beginning of Definitions).

An arrangement such as there indicated is the best means of ill.u.s.trating this rule. But instead of tracing the outline of the square or cube on the gla.s.s, as there shown, I have a hole drilled through at the point _S_ (Fig. 29), which I select for the point of sight, and through which I pa.s.s two loose strings _A_ and _B_, fixing their ends at _S_.

[Ill.u.s.tration: Fig. 28.]

[Ill.u.s.tration: Fig. 29.]

As _SD_ represents the distance the spectator is from the gla.s.s or picture, I make string _SA_ equal in length to _SD_. Now if the pupil takes this string in one hand and holds it at right angles to the gla.s.s, that is, exactly in front of _S_, and then places one eye at the end _A_ (of course with the string extended), he will be at the proper distance from the picture. Let him then take the other string, _SB_, in the other hand, and apply it to point _b'_ where the square touches the gla.s.s, and he will find that it exactly tallies with the side _b'f_ of the square _ab'fe_. If he applies the same string to _a_, the other corner of the square, his string will exactly tally or cover the side _ae_, and he will thus have ocular demonstration of this important rule.

In this little picture (Fig. 30) in parallel perspective it will be seen that the lines which retreat from us at right angles to the picture plane are directed to the point of sight _S_.

[Ill.u.s.tration: Fig. 30.]

RULE 6

All horizontals which are at 45, or half a right angle to the picture plane, are drawn to the point of distance.

We have already seen that the diagonal of the perspective square, if produced to meet the horizon on the picture, will mark on that horizon the distance that the spectator is from the point of sight (see definition, p. 16). This point of distance becomes then the measuring point for all horizontals at right angles to the picture plane.

Thus in Fig. 31 lines _AS_ and _BS_ are drawn to the point of sight _S_, and are therefore at right angles to the base _AB_. _AD_ being drawn to _D_ (the distance-point), is at an angle of 45 to the base _AB_, and _AC_ is therefore the diagonal of a square. The line 1C is made parallel to _AB_, consequently A1CB is a square in perspective. The line _BC_, therefore, being one side of that square, is equal to _AB_, another side of it. So that to measure a length on a line drawn to the point of sight, such as _BS_, we set out the length required, say _BA_, on the base-line, then from _A_ draw a line to the point of distance, and where it cuts _BS_ at _C_ is the length required. This can be repeated any number of times, say five, so that in this figure _BE_ is five times the length of _AB_.

[Ill.u.s.tration: Fig. 31.]

RULE 7

All horizontals forming any other angles but the above are drawn to some other points on the horizontal line. If the angle is greater than half a right angle (Fig. 32), as _EBG_, the point is within the point of distance, as at _V'_. If it is less, as _ABV''_, then it is beyond the point of distance, and consequently farther from the point of sight.

[Ill.u.s.tration: Fig. 32.]

In Fig. 32, the dotted line _BD_, drawn to the point of distance _D_, is at an angle of 45 to the base _AG_. It will be seen that the line _BV'_ is at a greater angle to the base than _BD_; it is therefore drawn to a point _V'_, within the point of distance and nearer to the point of sight _S_. On the other hand, the line _BV''_ is at a more acute angle, and is therefore drawn to a point some way beyond the other distance point.

_Note._--When this vanishing point is a long way outside the picture, the architects make use of a centrolinead, and the painters fix a long string at the required point, and get their perspective lines by that means, which is very inconvenient. But I will show you later on how you can dispense with this trouble by a very simple means, with equally correct results.

RULE 8

Lines which incline upwards have their vanishing points above the horizontal line, and those which incline downwards, below it. In both cases they are on the vertical which pa.s.ses through the vanishing point (_S_) of their horizontal projections.

[Ill.u.s.tration: Fig. 33.]

This rule is useful in drawing steps, or roads going uphill and downhill.

[Ill.u.s.tration: Fig. 34.]

RULE 9

The farther a point is removed from the picture plane the nearer does its perspective appearance approach the horizontal line so long as it is viewed from the same position. On the contrary, if the spectator retreats from the picture plane _K_ (which we suppose to be transparent), the point remaining at the same place, the perspective appearance of this point will approach the ground-line in proportion to the distance of the spectator.

[Ill.u.s.trations: Fig. 35.

Fig. 36.

The spectator at two different distances from the picture.]

Therefore the position of a given point in perspective above the ground-line or below the horizon is in proportion to the distance of the spectator from the picture, or the picture from the point.

[Ill.u.s.tration: Fig. 37.]

[Ill.u.s.trations: The picture at two different distances from the point.

Fig. 38.

Fig. 39.]

Figures 38 and 39 are two views of the same gallery from different distances. In Fig. 38, where the distance is too short, there is a want of proportion between the near and far objects, which is corrected in Fig. 39 by taking a much longer distance.

RULE 10

Horizontals in the same plane which are drawn to the same point on the horizon are parallel to each other.

[Ill.u.s.tration: Fig. 40.]

This is a very important rule, for all our perspective drawing depends upon it. When we say that parallels are drawn to the same point on the horizon it does not imply that they meet at that point, which would be a contradiction; perspective parallels never reach that point, although they appear to do so. Fig. 40 will explain this.

Suppose _S_ to be the spectator, _AB_ a transparent vertical plane which represents the picture seen edgeways, and _HS_ and _DC_ two parallel lines, mark off s.p.a.ces between these parallels equal to _SC_, the height of the eye of the spectator, and raise verticals 2, 3, 4, 5, &c., forming so many squares. Vertical line 2 viewed from _S_ will appear on _AB_ but half its length, vertical 3 will be only a third, vertical 4 a fourth, and so on, and if we multiplied these s.p.a.ces _ad infinitum_ we must keep on dividing the line _AB_ by the same number. So if we suppose _AB_ to be a yard high and the distance from one vertical to another to be also a yard, then if one of these were a thousand yards away its representation at _AB_ would be the thousandth part of a yard, or ten thousand yards away, its representation at _AB_ would be the ten-thousandth part, and whatever the distance it must always be something; and therefore _HS_ and _DC_, however far they may be produced and however close they may appear to get, can never meet.

[Ill.u.s.tration: Fig. 41.]

Fig. 41 is a perspective view of the same figure--but more extended. It will be seen that a line drawn from the tenth upright _K_ to _S_ cuts off a tenth of _AB_. We look then upon these two lines _SP_, _OP_, as the sides of a long parallelogram of which _SK_ is the diagonal, as _cefd_, the figure on the ground, is also a parallelogram.

The student can obtain for himself a further ill.u.s.tration of this rule by placing a looking-gla.s.s on one of the walls of his studio and then sketching himself and his surroundings as seen therein. He will find that all the horizontals at right angles to the gla.s.s will converge to his own eye. This rule applies equally to lines which are at an angle to the picture plane as to those that are at right angles or perpendicular to it, as in Rule 7. It also applies to those on an inclined plane, as in Rule 8.

[Ill.u.s.tration: Fig. 42. Sketch of artist in studio.]

With the above rules and a clear notion of the definitions and conditions of perspective, we should be able to work out any proposition or any new figure that may present itself. At any rate, a thorough understanding of these few pages will make the labour now before us simple and easy. I hope, too, it may be found interesting. There is always a certain pleasure in deceiving and being deceived by the senses, and in optical and other illusions, such as making things appear far off that are quite near, in making a picture of an object on a flat surface to look as if it stood out and in relief by a kind of magic. But there is, I think, a still greater pleasure than this, namely, in invention and in overcoming difficulties--in finding out how to do things for ourselves by our reasoning faculties, in originating or being original, as it were. Let us now see how far we can go in this respect.

VIII

A TABLE OR INDEX OF THE RULES OF PERSPECTIVE