The Theory and Practice of Perspective Part 2

We perceive objects by means of the visual rays, which are imaginary straight lines drawn from the eye to the various points of the thing we are looking at. As those rays proceed from the pupil of the eye, which is a circular opening, they form themselves into a cone called the +Optic Cone+, the base of which increases in proportion to its distance from the eye, so that the larger the view which we wish to take in, the farther must we be removed from it. The diameter of the base of this cone, with the visual rays drawn from each of its extremities to the eye, form the angle of vision, which is wider or narrower according to the distance of this diameter.

Now let us suppose a visual ray _EA_ to be directed to some small object on the floor, say the head of a nail, _A_ (Fig. 17). If we interpose between this nail and our eye a sheet of gla.s.s, _K_, placed vertically on the floor, we continue to see the nail through the gla.s.s, and it is easily understood that its perspective appearance thereon is the point _a_, where the visual ray pa.s.ses through it. If now we trace on the floor a line _AB_ from the nail to the spot _B_, just under the eye, and from the point _o_, where this line pa.s.ses through or under the gla.s.s, we raise a perpendicular _oS_, that perpendicular pa.s.ses through the precise point that the visual ray pa.s.ses through. The line _AB_ traced on the floor is the horizontal trace of the visual ray, and it will be seen that the point _a_ is situated on the vertical raised from this horizontal trace.

[Ill.u.s.tration: Fig. 17.]

V

TRACE AND PROJECTION

If from any line _A_ or _B_ or _C_ (Fig. 18), &c., we drop perpendiculars from different points of those lines on to a horizontal plane, the intersections of those verticals with the plane will be on a line called the horizontal trace or projection of the original line.

We may liken these projections to sun-shadows when the sun is in the meridian, for it will be remarked that the trace does not represent the length of the original line, but only so much of it as would be embraced by the verticals dropped from each end of it, and although line _A_ is the same length as line _B_ its horizontal trace is longer than that of the other; that the projection of a curve (_C_) in this upright position is a straight line, that of a horizontal line (_D_) is equal to it, and the projection of a perpendicular or vertical (_E_) is a point only.

The projections of lines or points can likewise be shown on a vertical plane, but in that case we draw lines parallel to the horizontal plane, and by this means we can get the position of a point in s.p.a.ce; and by the a.s.sistance of perspective, as will be shown farther on, we can carry out the most difficult propositions of descriptive geometry and of the geometry of planes and solids.

[Ill.u.s.tration: Fig. 18.]

The position of a point in s.p.a.ce is given by its projection on a vertical and a horizontal plane--

[Ill.u.s.tration: Fig. 19.]

Thus _e_ is the projection of _E_ on the vertical plane _K_, and _e_ is the projection of _E_ on the horizontal plane; _fe_ is the horizontal trace of the plane _fE_, and _ef_ is the trace of the same plane on the vertical plane _K_.

VI

SCIENTIFIC DEFINITION OF PERSPECTIVE

The projections of the extremities of a right line which pa.s.ses through a vertical plane being given, one on either side of it, to find the intersection of that line with the vertical plane. _AE_ (Fig. 20) is the right line. The projection of its extremity _A_ on the vertical plane is _a_, the projection of _E_, the other extremity, is _e_. _AS_ is the horizontal trace of _AE_, and _ae_ is its trace on the vertical plane.

At point _f_, where the horizontal trace intersects the base _Bc_ of the vertical plane, raise perpendicular _fP_ till it cuts _ae_ at point _P_, which is the point required. For it is at the same time on the given line _AE_ and the vertical plane _K_.

[Ill.u.s.tration: Fig. 20.]

This figure is similar to the previous one, except that the extremity _A_ of the given line is raised from the ground, but the same demonstration applies to it.

[Ill.u.s.tration: Fig. 21.]

And now let us suppose the vertical plane _K_ to be a sheet of gla.s.s, and the given line _AE_ to be the visual ray pa.s.sing from the eye to the object _A_ on the other side of the gla.s.s. Then if _E_ is the eye of the spectator, its projection on the picture is _S_, the point of sight.

If I draw a dotted line from _E_ to little _a_, this represents another visual ray, and _o_, the point where it pa.s.ses through the picture, is the perspective of little _a_. I now draw another line from _g_ to _S_, and thus form the shaded figure _gaPo_, which is the perspective of _aAag_.

Let it be remarked that in the shaded perspective figure the lines _aP_ and _go_ are both drawn towards _S_, the point of sight, and that they represent parallel lines _Aa_ and _ag_, which are at right angles to the picture plane. This is the most important fact in perspective, and will be more fully explained farther on, when we speak of retreating or so-called vanis.h.i.+ng lines.

RULES

VII

THE RULES AND CONDITIONS OF PERSPECTIVE

The conditions of linear perspective are somewhat rigid. In the first place, we are supposed to look at objects with one eye only; that is, the visual rays are drawn from a single point, and not from two. Of this we shall speak later on. Then again, the eye must be placed in a certain position, as at _E_ (Fig. 22), at a given height from the ground, _SE_, and at a given distance from the picture, as _SE_. In the next place, the picture or picture plane itself must be vertical and perpendicular to the ground or horizontal plane, which plane is supposed to be as level as a billiard-table, and to extend from the base line, _ef_, of the picture to the horizon, that is, to infinity, for it does not partake of the rotundity of the earth.

We can only work out our propositions and figures in s.p.a.ce with mathematical precision by adopting such conditions as the above. But afterwards the artist or draughtsman may modify and suit them to a more elastic view of things; that is, he can make his figures separate from one another, instead of their outlines coming close together as they do when we look at them with only one eye. Also he will allow for the unevenness of the ground and the roundness of our globe; he may even move his head and his eyes, and use both of them, and in fact make himself quite at his ease when he is out sketching, for Nature does all his perspective for him. At the same time, a knowledge of this rigid perspective is the sure and unerring basis of his freehand drawing.

[Ill.u.s.tration: Fig. 22.]

[Ill.u.s.tration: Fig. 23. Front view of above figure.]

RULE 1

All straight lines remain straight in their perspective appearance.[4]

[Footnote 4: Some will tell us that Nature abhors a straight line, that all long straight lines in s.p.a.ce appear curved, &c., owing to certain optical conditions; but this is not apparent in short straight lines, so if our drawing is small it would be wrong to curve them; if it is large, like a scene or diorama, the same optical condition which applies to the line in s.p.a.ce would also apply to the line in the picture.]

RULE 2

Vertical lines remain vertical in perspective, and are divided in the same proportion as _AB_ (Fig. 24), the original line, and _ab_, the perspective line, and if the one is divided at _O_ the other is divided at _o_ in the same way.

[Ill.u.s.tration: Fig. 24.]

It is not an uncommon error to suppose that the vertical lines of a high building should converge towards the top; so they would if we stood at the foot of that building and looked up, for then we should alter the conditions of our perspective, and our point of sight, instead of being on the horizon, would be up in the sky. But if we stood sufficiently far away, so as to bring the whole of the building within our angle of vision, and the point of sight down to the horizon, then these same lines would appear perfectly parallel, and the different stories in their true proportion.

RULE 3

Horizontals parallel to the base of the picture are also parallel to that base in the picture. Thus _ab_ (Fig. 25) is parallel to _AB_, and to _GL_, the base of the picture. Indeed, the same argument may be used with regard to horizontal lines as with verticals. If we look at a straight wall in front of us, its top and its rows of bricks, &c., are parallel and horizontal; but if we look along it sideways, then we alter the conditions, and the parallel lines converge to whichever point we direct the eye.

[Ill.u.s.tration: Fig. 25.]

[Ill.u.s.tration: Fig. 26.]

This rule is important, as we shall see when we come to the consideration of the perspective vanis.h.i.+ng scale. Its use may be ill.u.s.trated by this sketch, where the houses, walls, &c., are parallel to the base of the picture. When that is the case, then objects exactly facing us, such as windows, doors, rows of boards, or of bricks or palings, &c., are drawn with their horizontal lines parallel to the base; hence it is called parallel perspective.

RULE 4

All lines situated in a plane that is parallel to the picture plane diminish in proportion as they become more distant, but do not undergo any perspective deformation; and remain in the same relation and proportion each to each as the original lines. This is called the front view.

[Ill.u.s.tration: Fig. 27.]