V

SHADOWS.

It is a well-known physical fact that in a homogeneous medium light is propagated in straight lines. Hence when light from any luminous point falls on an opaque body there is a certain portion of space behind the body as well as part of the body itself which is deprived of the direct rays from the point. The line on the surface of the body which separates the illumined from the dark side is called the line of shade, and the dark space behind is called the shadow of the body. When any surface comes within the shadow so as to have the whole or part of it deprived of the direct rays from the luminous point, it is said to have a shadow cast upon it by the opaque body.

By assuming that the medium surrounding the bodies is homogeneous and the source of light a fixed point, the line of shade on any known surface and the shadow cast by it on any other known surface can be determined by Geometry, provided the relative positions of the two surfaces to one another and to the luminous point are given. For the shadow-surface, that is the surface separating the shadow from surrounding space, is generated by straight lines passing through the luminous point and touching the surface which casts the shadow. The points of contact determine the shade line, and the intersection of the shadow-surface with the other given surface is the outline of the cast-shadow. The outline of the cast-shadow is therefore the common section of two known surfaces. For example, in fig. 96, if V were the source of light, the line CPQD would be the line of shade on the given sphere, and the line of intersection of the right circular cone VCPQD with any other surface would be the shadow cast by the sphere on that surface. For instance, the horizontal trace of that cone would be the shadow cast by the sphere on the horizontal plane of projection. It would evidently be an ellipse.

In putting the shadows on Engineering Drawings it is the custom to take the rays of light parallel to one another,

which is equivalent to assuming the source of light at an infinite distance, so that the shadow surface is a cylinder, according to the definition in Chapter IV. Moreover the rays of light are always assumed to proceed forward and downward from the left-hand side-frequently in the direction of the diagonal of a cube having two of its faces coinciding with the planes of projection, so that the projections of the rays make angles of 45° with the ground line. This direction is found very convenient.

Exercises.

1. A right circular cylinder, 2" diameter, penetrates a regular hexagonal prism, the shortest diameter of which is 3": the axis of the cylinder meets the axis of the prism at right angles. Draw the projection of the line of intersection on a plane parallel to the axis of the cylinder and one of the faces of the prism.

2. If the prism of the last example penetrates a sphere of 4" diameter, the axis of the prism passing through the centre of the sphere, draw the projection of the line of intersection on a plane parallel to a face of the prism.

3. A right circular cylinder of 2" diameter penetrates another of 3′′ diameter, the axis passing" from one another, and at an angle of 60°. Draw the projection of the two solids on a plane parallel to their axes.

4. A cone of revolution, 3" diameter at base, and 4" high, has a circular cylindrical hole, 1" diameter, bored through it; the axes are at right angles to one another and "apart; the axis of the hole 1" from the base of the cone. Draw the plan and elevation of the cone when standing on the horizontal plane, the axis of the hole making an angle of 30° with the vertical plane of projection.

5. Draw the development of the conical surface of the last example, shewing the holes.

6. Draw the projections of the curve of intersection of a cone and a sphere

(1) When the cone is one of revolution and its axis passes through the centre of the sphere.

(2) When the cone is one of revolution but the axis does not pass through the centre of the sphere.

(3) When the cone is not one of revolution.

7. The longest diameter of the base of a regular octagonal prism is 2" and its height 4"; it is placed with its base on the horizontal plane, and its axis 2" from the vertical plane of projection. Find the shadow cast by it on the two planes of projection

(1) When the light proceeds from a luminous point 3" from the vertical plane of projection, 6" above the horizontal plane, and 6" from the axis, produced, of the pyramid.

(2) When the rays are parallel and their projections on both planes make angles of 45° with the ground line.

8. Draw a bolt 4" long, and 1" diameter, with a hexagonal head, greatest diameter 22", depth 1". Find the shadow cast by the head on the bolt, and by both on the horizontal plane when the bolt is vertical with head uppermost. Direction of light the same as in the second case of the last example.

9. A cylinder, 2" diameter and 3" high, stands on a table, and a sphere of 3" diameter rests on the top of it, with its centre in line with the axis of the cylinder. Draw the shadow cast by the two solids on the table when the rays are parallel to one another, and inclined at 45° to the table.

10. A cone of revolution, base 2′′ diameter, height 4′′, stands on a table; a sphere of 2" diameter rests on the same table, the distance between the axis of the cone and the centre of the sphere being 3". Find the shadow cast by the cone on the sphere when the rays are parallel to the plane containing the axis of the cone and centre of the sphere, and inclined at 45° to the table.