SHADING AND SHADOWS.

LIGHT is diffused through space in straight lines, and the lines of light are called rays. When the source of light is situated at a very great distance from the illuminated objects, as in the case of the sun with relation to the earth, the rays of light do not sensibly diverge, and may be regarded as exactly parallel to each other. Such is the case in mechanical drawings, where the objects to be represented are always regarded as illuminated by the solar light.

Light is called direct when it is transmitted to an object without the intervention of any opposing medium. But as all bodies subjected to the action of light possess, in a greater or less degree, the property of giving out a certain portion of it to the surrounding objects, this reflected light becomes in its turn, though with greatly diminished intensity, a source of illumination to those objects which are deprived of direct light.

Everything which tends to intercept or prevent the direct light from falling in upon a body, produces upon the surface of that body a degree of obscurity of greater or less intensity; this is called a shade or shadow. Such effects are usually classified as direct shadows and cast shadows.

The shade proper, or direct shadow, is that which occurs on that portion of the surface of a body which is situated opposite to the enlightened part, and is the natural result of the form of the body itself, and of its position with regard to the rays of light. The cast shadow, on the other hand, is that which is produced upon the surface of one body by the interposition of another between the former and the source of light; thus intercepting the rays which would otherwise illuminate that surface. An illustration of this distinction is afforded in the pyramid represented at fig. 1, Plate II, where the shade proper is shown upon that half of the figure which is denoted by the letters D' E' G' F' in the plan, while the cast shadow occupies the space comprised between the lines D'e and F'd on the horizontal plane of projection. Cast shadows may also obviously

be produced upon the surface of a body by the form of the body itself; as, for example, if it contain projecting or concave parts.

The limit of the direct shadow in any body, whatever may be its form or position, is a line of greater or less distinctness, termed the line of separation between light and shade; or, more shortly, the line of shade; this line is, of course, determined by the contact of the luminous rays with the surface of the body; and if these rays be prolonged till they meet a given surface, by joining all the points of intersection with that surface, we obtain the outline of the shadow cast upon it by the part of the body which is deprived of light.

The rays of light being regarded as parallel to each other, it is obvious that in the delineation of shadows, it is only necessary to know the direction of one of them; and as that direction is arbitrary, we have adopted the usual and confessedly the most convenient mode of regarding the rays. as in all cases falling in the direction of the diagonal of a cube, of which the sides are parallel to the planes of projection. The diagonal in projection upon the vertical and horizontal planes lies at an angle of 45° with the ground-line; and thus the light in both elevation and plan appears at the angle of 45°. In illustration, let R, R' (fig. 1, plate I) be the projections of a ray of light in elevation and plan; and let A, A', those of a point of which the shadows are required to be projected upon the vertical plane X Y. Draw the straight lines A a, A' a', parallel to the lines R, R', and from a', where the line A' a' meets the plane X Y, draw the perpendicular a' a to meet the oblique line A a; then the intersection a is the position of the shadow of the point A.

In the following illustrations, the same letter accented, is employed in the plan as in the elevation, to refer to the same point or object.

The projections of the diagonals of the imaginary cube which denote the direction of the rays of light being equal in both planes, it follows that in all cases, and whatever may be the form of the surface upon which the shadow is cast, the oblique lines joining the projections of the point which throws the shadow, and that which denotes it, are also equal. Thus the line A a in the elevation is equal to the line A' a' in the plan. Hence it will in some cases be found more convenient to use the compasses instead of a geometrical construction; as, for example, in place of projecting the point a' by a perpendicular to the ground-line, in order to obtain the position of the required shadow a, that point may be found by simply setting off upon the line A a a distance equal to A' a'.

Plate I, fig. 1.-Required to determine the shadow cast upon the vertical wall X Y by the straight line A B.

It is obvious that in this case the shadow itself will be a straight line; hence, to solve the problem, it is only necessary to find two points in that line. We have seen that the position of the shadow thrown by the point A is at a; by a similar process we can easily determine the point b, the position of shadow thrown by the opposite extremity B of the given line; the straight line a b, which joins these two points, is the shadow required.

It is evident from the construction of this figure, that the line ab is equal and parallel to the given line A B; this results from the circumstance that the latter is parallel to the vertical plane X Y. Hence, when a line is parallel to a plane, its shadow upon that plane is a line which is equal and parallel to it.

Suppose now that, instead of a mere line, a parallel slip of wood or paper, A B C D, be taken, which, for the sake of greater simplicity, we shall conceive as having no thickness. The shadow cast by this object upon the same vertical plane X Y is a rectangle a b c d, equal to that which represents the projection of the slip, because all the edges of the latter are parallel to the plane upon which the shadow is thrown. Hence, in general, when any surface, whatever may be its form, is parallel to a plane, its shadow thrown upon that plane is a figure similar to it, and similarly situated. This principle facilitates the delineation of shadows in many cases. In the present example, an idea may be formed of its utility; for, after having determined the position of any one of the points a, b, c, d, the figure may be completed by drawing lines equal and parallel to the sides of the slip, without requiring to go through the operations in detail.

Fig. 2.-When the object is not parallel to the given plane, the cast shadow is no longer a figure equal and similarly placed; the method of determining it remains, however, unchanged; thus, take the portion A E of the slip A B, which throws its shadow on the plane X Y; draw the lines A a, Ee, Cc, Ff, and A' a', E' e', parallel to the rays of light; make A a and Cc equal to A' a'; and Ee and Ff equal to E'e'; connect a efc, and we have the outline of the shadow of the slip A E.

By an exactly similar construction we have the shadow of the portion E B on the plane Y Z, which being inclined to the plane of projection in a direction contrary to X Y, necessarily causes the shadow to be broken, and the part e d to lie in a contrary direction to a f.

Fig. 3 still further illustrates the determination of the shadow of the slip upon a moulding placed on the plane X Y parallel to the slip.

Fig. 4. To find the shadow cast by a straight line A B upon a curved surface, either convex or concave, whose horizontal projection is represented by the line X e' Y.