0.5

1 cm

82

ENGINEER AND MACHINIST’S DRAWING-BOOK.

Die perpendicular, describe a circle equal to the base, and

from the point A' draw the lines A' b' and A’ c, touching

this circle ; these are the outlines of the shadow cast upon

the horizontal plane. Then, from the centre A', draw the

radii A' If and A' (7, parallel to a b' and a c ; these

radii are the horizontal projections of the lines of shade,

the former of which, transferred to B D, is alone visible in

the elevation. But in order to trace the outline of that

portion of the shadow which is thrown upon the vertical

plane, it is necessary to project the point C' to C, from

which, by a construction which will be manifest from

inspection of the figures, we derive the point c, and the

line g d as part of the cast shadow of the line O' A'. The

rest of the outline of the vertical portion of the cast shadow,

is derived from the circumference of the base, as in Fig. 2.

Plate LIII., Fig. 6.—To determine the shadow cast by

a cylinder upon a hexagonal prism, the axes of the tivo

solids coinciding.

This problem resolves itself into an application of pre-

ceding problems. Draw, from the angular points a', c, e,

lines parallel to the light, intersecting the circumference

of the base of the cylinder in A', C', and E'. These points,

projected, will enable us to trace the lines A a, G c, and E e,

which give, by their intersections with the edges of the

prism, the limiting points a, c, e, &c. of the required shadow.

Fig. 1 represents a cylinder upon which a shadow is

thrown by a rectangular prism, of which the sides are paral-

lel to the planes of projection. The shadow, in this case, is

derived from the edges A' D' and A' E'; the first of which,

being perpendicular to the plane of projection, gives, accord-

ing to principles already laid down, a straight line at an

angle of 45° for the outline of its shadow, whereas, the side

A' E' being parallel to that plane, its shadow is determined

by a portion of a circle a b c, described from the centre o.

Figs. 2, 3.—If we suppose the prism to be pentagonal,

or that a cylinder be substituted for it, the mode of con-

struction remains the same. But it should be observed

that it is best in all such cases to commence by finding the

points which indicate the main direction of the outline.

To ascertain the point a, at which the shadow commences,

draw from a' the line a' A, at an angle of 45°, which is then

to be projected vertically to a A. Then the highest point, b,

fig. 3, should be determined by the intersection of the

radius 0 B' (drawn parallel to the ray), with the circum-

ference of the base of the cylinder on which the required

shadow is cast; and, finally, the point c, where the outline

of the cast shadow intersects the line of shade, should be

determined by a similar process.

Figs. 4, 5. — Other varieties of the problem are here

illustrated, and the constructions are made in the same

way as explained in the other cases.

Figs. 7,8.—To define the shadows cast upon the interior

of a hollow cylinder in section, by itself and by a cir-

cular piston fitted into it.

The example shows a steam cylinder, A, in section, by a

plane passing through its axis, with its piston and rod

in full.

Conceive, in the first instance, the piston P to be re-

moved; the shadow cast into the interior of the cylinder

will then consist, obviously, of that projected by the ver-

tical edge B C, and by a portion of the horizontal edge B A.

To find the first, draw through B', a line B' b', at an angle

of 45° with B' A; the point b', where this line meets the

interior surface of the cylinder, being projected upwards

to fig. 1, gives the line b f as the outline of the shadow

sought. Then, parallel to the direction of the light, draw

a tangent at F' to the inner circle of the base; its point of

contact being projected to F in the elevation, marks the

commencement of the outline of the shadow cast by the

upper edge of the cylinder. The point b, where it termi-

nates, will obviously be the intersection of the straight

line / b, already determined, with a ray B b from the upper

extremity of the edge B C; and any intermediate point

in the curve, as e, may be found by taking a point E', be-

tween If and b', projecting it to E, and causing rays

E e, E' e, to pass through these points. The outline of the

shadow required will then be the curve Feb, and the

straight line bf. Suppose now the piston P, and its rod T,

to be inserted into the cylinder, as shown. The lower sur-

face of the piston will then cast a shadow upon the interior

surface of the cylinder, of which the outline D d ho, may

be formed in the same way, as will be obvious from

inspection of the figures and comparison of the letters of

reference. The piston-rod T being cylindrical and ver-

tical, it casts also its shadow into the interior of the cylin-

der; it will obviously consist of a rectangle ijl k, drawn

parallel to the axis, and of which the sides ij and k l, are

determined by the tangents I' i' and K' k\

Figs. 11,12.—This example consists of a hollow cylinder,

surmounted by a circular disc, or cover, sectioned through

the centre, where it is also penetrated by a cylindrical

aperture. The construction necessary for finding the

outlines of the cast shadow, will obviously be the same as

already laid down. In this case, however, it is proper to

know beforehand what parts of the upper and lower edges

of the central aperture cast their shadows into the interior

of the cylinder; if, then, we take the trouble to construct

the shadows of each of these edges separately, we shall

find that that of the upper edge is a curve b cf and that

of the lower, a similar curve ace, cutting the former in c.

This point limits the parts of each curve which are actually

visible; namely, the portion.be of the first, and the por-

tion e c of the second; hence, it follows that, in order

to avoid unnecessary work, we should first determine the

position of the point of intersection, c, of the two curves,

which is, in fact, the cast shadow of the lowest point C in

the curve D C, previously laid down in the circular open-

ing of the cover, in the manner indicated in fig. 7.

Figs. 13, 14.—Conceive a cylinder in section to be set

at an inclination to the horizontal plane. To find the out-

line of the shadow cast into its interior, describe upon the

prolongation of the axis of the cylinder a semicircle A'a B',

representing its interior surface, and then, in any conve-

nient part of the paper, construct a square mn op (fig. 183,

page 82), and draw the diagonal m o; from one of the

extremities o, draw the line o r, parallel to A B', and

ENGINEER AND MACHINIST’S DRAWING-BOOK.

Die perpendicular, describe a circle equal to the base, and

from the point A' draw the lines A' b' and A’ c, touching

this circle ; these are the outlines of the shadow cast upon

the horizontal plane. Then, from the centre A', draw the

radii A' If and A' (7, parallel to a b' and a c ; these

radii are the horizontal projections of the lines of shade,

the former of which, transferred to B D, is alone visible in

the elevation. But in order to trace the outline of that

portion of the shadow which is thrown upon the vertical

plane, it is necessary to project the point C' to C, from

which, by a construction which will be manifest from

inspection of the figures, we derive the point c, and the

line g d as part of the cast shadow of the line O' A'. The

rest of the outline of the vertical portion of the cast shadow,

is derived from the circumference of the base, as in Fig. 2.

Plate LIII., Fig. 6.—To determine the shadow cast by

a cylinder upon a hexagonal prism, the axes of the tivo

solids coinciding.

This problem resolves itself into an application of pre-

ceding problems. Draw, from the angular points a', c, e,

lines parallel to the light, intersecting the circumference

of the base of the cylinder in A', C', and E'. These points,

projected, will enable us to trace the lines A a, G c, and E e,

which give, by their intersections with the edges of the

prism, the limiting points a, c, e, &c. of the required shadow.

Fig. 1 represents a cylinder upon which a shadow is

thrown by a rectangular prism, of which the sides are paral-

lel to the planes of projection. The shadow, in this case, is

derived from the edges A' D' and A' E'; the first of which,

being perpendicular to the plane of projection, gives, accord-

ing to principles already laid down, a straight line at an

angle of 45° for the outline of its shadow, whereas, the side

A' E' being parallel to that plane, its shadow is determined

by a portion of a circle a b c, described from the centre o.

Figs. 2, 3.—If we suppose the prism to be pentagonal,

or that a cylinder be substituted for it, the mode of con-

struction remains the same. But it should be observed

that it is best in all such cases to commence by finding the

points which indicate the main direction of the outline.

To ascertain the point a, at which the shadow commences,

draw from a' the line a' A, at an angle of 45°, which is then

to be projected vertically to a A. Then the highest point, b,

fig. 3, should be determined by the intersection of the

radius 0 B' (drawn parallel to the ray), with the circum-

ference of the base of the cylinder on which the required

shadow is cast; and, finally, the point c, where the outline

of the cast shadow intersects the line of shade, should be

determined by a similar process.

Figs. 4, 5. — Other varieties of the problem are here

illustrated, and the constructions are made in the same

way as explained in the other cases.

Figs. 7,8.—To define the shadows cast upon the interior

of a hollow cylinder in section, by itself and by a cir-

cular piston fitted into it.

The example shows a steam cylinder, A, in section, by a

plane passing through its axis, with its piston and rod

in full.

Conceive, in the first instance, the piston P to be re-

moved; the shadow cast into the interior of the cylinder

will then consist, obviously, of that projected by the ver-

tical edge B C, and by a portion of the horizontal edge B A.

To find the first, draw through B', a line B' b', at an angle

of 45° with B' A; the point b', where this line meets the

interior surface of the cylinder, being projected upwards

to fig. 1, gives the line b f as the outline of the shadow

sought. Then, parallel to the direction of the light, draw

a tangent at F' to the inner circle of the base; its point of

contact being projected to F in the elevation, marks the

commencement of the outline of the shadow cast by the

upper edge of the cylinder. The point b, where it termi-

nates, will obviously be the intersection of the straight

line / b, already determined, with a ray B b from the upper

extremity of the edge B C; and any intermediate point

in the curve, as e, may be found by taking a point E', be-

tween If and b', projecting it to E, and causing rays

E e, E' e, to pass through these points. The outline of the

shadow required will then be the curve Feb, and the

straight line bf. Suppose now the piston P, and its rod T,

to be inserted into the cylinder, as shown. The lower sur-

face of the piston will then cast a shadow upon the interior

surface of the cylinder, of which the outline D d ho, may

be formed in the same way, as will be obvious from

inspection of the figures and comparison of the letters of

reference. The piston-rod T being cylindrical and ver-

tical, it casts also its shadow into the interior of the cylin-

der; it will obviously consist of a rectangle ijl k, drawn

parallel to the axis, and of which the sides ij and k l, are

determined by the tangents I' i' and K' k\

Figs. 11,12.—This example consists of a hollow cylinder,

surmounted by a circular disc, or cover, sectioned through

the centre, where it is also penetrated by a cylindrical

aperture. The construction necessary for finding the

outlines of the cast shadow, will obviously be the same as

already laid down. In this case, however, it is proper to

know beforehand what parts of the upper and lower edges

of the central aperture cast their shadows into the interior

of the cylinder; if, then, we take the trouble to construct

the shadows of each of these edges separately, we shall

find that that of the upper edge is a curve b cf and that

of the lower, a similar curve ace, cutting the former in c.

This point limits the parts of each curve which are actually

visible; namely, the portion.be of the first, and the por-

tion e c of the second; hence, it follows that, in order

to avoid unnecessary work, we should first determine the

position of the point of intersection, c, of the two curves,

which is, in fact, the cast shadow of the lowest point C in

the curve D C, previously laid down in the circular open-

ing of the cover, in the manner indicated in fig. 7.

Figs. 13, 14.—Conceive a cylinder in section to be set

at an inclination to the horizontal plane. To find the out-

line of the shadow cast into its interior, describe upon the

prolongation of the axis of the cylinder a semicircle A'a B',

representing its interior surface, and then, in any conve-

nient part of the paper, construct a square mn op (fig. 183,

page 82), and draw the diagonal m o; from one of the

extremities o, draw the line o r, parallel to A B', and