0.5

1 cm

80

ENGINEER AND MACHINIST’S DRAWING-BOOK.

then becomes different from the preceding, but the mode

of construction is the same.

Fig. 4.—When the object is situated in the angle

formed by the junction of two vertical planes inclined to

each other, and to the vertical plane of projection, its

shadow will be thrown upon both surfaces, and will be

expressed by two distinct parallelograms having a com-

mon side ef in the line of intersection of the planes. The

construction necessary for determining these figures will

be easily understood by inspection of our illustration, by

which it will be observed that the only part of the slip

which casts a shadow upon the plane X Y is the rectangle

A C F E, and that the shadow of the rest is thrown upon

the plane Y Z, which being inclined to the plane of pro-

jection in a direction contrary to the first, necessarily

causes the shadow also to lie in the contrary direction to

the part a efc.

Figs. 5 and 6.—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

Xer Y.

We have already explained that the shadow of a point

upon any surface whatever is found by drawing a straight

line through that poinf, parallel to the direction of the

light, and marking its intersection with the given surface.

Therefore, through the projections A and A' of one of the

points in the given straight line, draw the lines A a, A a',

at an angle of 45°; and through the point a, where the

latter meets the projection of the given surface, raise a

perpendicular to the ground-line; its intersection with

the line A a, is the position of the shadow of the first point

taken; and so for all the remaining points in the line.

If it be required to delineate the entire shadow cast by

a slip A B C D, as before, upon the surfaces under consi-

deration, we shall be enabled, by the construction above

explained, to trace two equal and parallel curves a e b,

cfd, representing the shadows of the sides A B and C D;

while those of the remaining sides will be found denoted

by the vertical straight lines a c and b d, also equal and

parallel to each other, and to the corresponding sides of

the figure, seeing that these are themselves vertical and

parallel to the given surfaces.

Fig. 7.—When the slip is placed perpendicularly to a

given plane X Y, on which a projecting moulding, of any

form whatever, is situated, the shadow of the upper side

A' B' which is projected vertically in A, will be simply a

line A a, at an angle of 45°, traversing the entire surface

of the moulding, and prolonged unbroken beyond it. This

may easily be demonstrated by finding the position of the

shadow of any number of points such as D', taken at plea-

sure upon the straight line A' B'. The shadow of the

opposite side, projected in G, will follow the same rule,

and be denoted by the line C c, parallel to the former.

From this example we are led to state as a useful general

rule: that in all cases where a straight line is perpen-

dicular to a plane of projection, it throws a shadow

upon that plane, in a straight line, forming an angle of

45° with the ground-line.

Fig. 8 represents still another example of the shadow

cast by the slip in a new position; here it is supposed to

be set horizontally in reference to its own surface, and

perpendicularly to the given plane X Y. Here we see

that the shadow commences from the side D B, which is

in contact with this plane, and terminates in the horizon-

tal line a c, which corresponds to the opposite side A C of

the slip

Plate LI. Fig. 1.—Required to find the shadow cast

upon a vertical plane AY by a given circle parallel to it.

Let C, O' be the projections of the centre of the circle,

and R, R' those of the rays of light.

We have already shown that when a figure is parallel

to a plane, its shadow cast upon that plane is a figure in

every respect equal to, and symmetrical with it; therefore

the shadow cast by the circle now under consideration

will be expressed by another circle of equal radius ; con-

sequently, if we can find the position of the centre of this

new circle the problem will be solved. Now the position

of the shadow of the central point C, according to the

rules we have already folly developed, is easily fixed at c ;

from which point if we describe a circle equal to the

given circle, it will represent the outline of the required

shadow.

Fig. 2.—If the given circle be horizontal, its shadow

cast upon the vertical plane X Y becomes an ellipse which

must be constructed by means of points, as indicated by

the figures referred to above; that is to say, that in the

circumference of the circle a certain number of points are

to be taken, such as A', D', B', &c., which are to be pro-

jected successively to A, D, B, on the line A B, and through

each of these points, lines are to be drawn parallel to the

direction of the rays of light, and their intersection with

the given plane determined. The junction of all these

points will give the ellipse adb, which is the contour of

the required shadow.

Fig. 3.—When the circle is perpendicular to both planes

of projection, its projection upon each will obviously be

represented by the equal diameters A B and C' D', both

perpendicular to the ground-line. In this case, in order

to determine the cast shadow, we must describe the given

circle upon both planes, as indicated by the figures, and

divide the circumference of each into any number of equal

parts; then, having projected the points of division, as

A2, C2, E2, &c., to their respective diameters AB and C' IT,

draw from them lines parallel to the rays of light, which,

by their intersection with the given plane, will indicate so

many points in the outline of the cast shadow.

Fig. 4 represents a circle whose plane is situated perpen-

dicularly to the direction of the luminous rays. In this

example the method of constructing the cast shadow does

not differ from that pointed out in reference to Fig. 2,

provided that both projections are made use of. But it is

obvious that instead of laying down the entire horizontal

projection of this circle, all that is necessary is to set off

the diameter D' E', equal to A B; because the shadow of

this diameter, transferred in the usual way, gives the

major axis of the ellipse which constitutes the outline of

ENGINEER AND MACHINIST’S DRAWING-BOOK.

then becomes different from the preceding, but the mode

of construction is the same.

Fig. 4.—When the object is situated in the angle

formed by the junction of two vertical planes inclined to

each other, and to the vertical plane of projection, its

shadow will be thrown upon both surfaces, and will be

expressed by two distinct parallelograms having a com-

mon side ef in the line of intersection of the planes. The

construction necessary for determining these figures will

be easily understood by inspection of our illustration, by

which it will be observed that the only part of the slip

which casts a shadow upon the plane X Y is the rectangle

A C F E, and that the shadow of the rest is thrown upon

the plane Y Z, which being inclined to the plane of pro-

jection in a direction contrary to the first, necessarily

causes the shadow also to lie in the contrary direction to

the part a efc.

Figs. 5 and 6.—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

Xer Y.

We have already explained that the shadow of a point

upon any surface whatever is found by drawing a straight

line through that poinf, parallel to the direction of the

light, and marking its intersection with the given surface.

Therefore, through the projections A and A' of one of the

points in the given straight line, draw the lines A a, A a',

at an angle of 45°; and through the point a, where the

latter meets the projection of the given surface, raise a

perpendicular to the ground-line; its intersection with

the line A a, is the position of the shadow of the first point

taken; and so for all the remaining points in the line.

If it be required to delineate the entire shadow cast by

a slip A B C D, as before, upon the surfaces under consi-

deration, we shall be enabled, by the construction above

explained, to trace two equal and parallel curves a e b,

cfd, representing the shadows of the sides A B and C D;

while those of the remaining sides will be found denoted

by the vertical straight lines a c and b d, also equal and

parallel to each other, and to the corresponding sides of

the figure, seeing that these are themselves vertical and

parallel to the given surfaces.

Fig. 7.—When the slip is placed perpendicularly to a

given plane X Y, on which a projecting moulding, of any

form whatever, is situated, the shadow of the upper side

A' B' which is projected vertically in A, will be simply a

line A a, at an angle of 45°, traversing the entire surface

of the moulding, and prolonged unbroken beyond it. This

may easily be demonstrated by finding the position of the

shadow of any number of points such as D', taken at plea-

sure upon the straight line A' B'. The shadow of the

opposite side, projected in G, will follow the same rule,

and be denoted by the line C c, parallel to the former.

From this example we are led to state as a useful general

rule: that in all cases where a straight line is perpen-

dicular to a plane of projection, it throws a shadow

upon that plane, in a straight line, forming an angle of

45° with the ground-line.

Fig. 8 represents still another example of the shadow

cast by the slip in a new position; here it is supposed to

be set horizontally in reference to its own surface, and

perpendicularly to the given plane X Y. Here we see

that the shadow commences from the side D B, which is

in contact with this plane, and terminates in the horizon-

tal line a c, which corresponds to the opposite side A C of

the slip

Plate LI. Fig. 1.—Required to find the shadow cast

upon a vertical plane AY by a given circle parallel to it.

Let C, O' be the projections of the centre of the circle,

and R, R' those of the rays of light.

We have already shown that when a figure is parallel

to a plane, its shadow cast upon that plane is a figure in

every respect equal to, and symmetrical with it; therefore

the shadow cast by the circle now under consideration

will be expressed by another circle of equal radius ; con-

sequently, if we can find the position of the centre of this

new circle the problem will be solved. Now the position

of the shadow of the central point C, according to the

rules we have already folly developed, is easily fixed at c ;

from which point if we describe a circle equal to the

given circle, it will represent the outline of the required

shadow.

Fig. 2.—If the given circle be horizontal, its shadow

cast upon the vertical plane X Y becomes an ellipse which

must be constructed by means of points, as indicated by

the figures referred to above; that is to say, that in the

circumference of the circle a certain number of points are

to be taken, such as A', D', B', &c., which are to be pro-

jected successively to A, D, B, on the line A B, and through

each of these points, lines are to be drawn parallel to the

direction of the rays of light, and their intersection with

the given plane determined. The junction of all these

points will give the ellipse adb, which is the contour of

the required shadow.

Fig. 3.—When the circle is perpendicular to both planes

of projection, its projection upon each will obviously be

represented by the equal diameters A B and C' D', both

perpendicular to the ground-line. In this case, in order

to determine the cast shadow, we must describe the given

circle upon both planes, as indicated by the figures, and

divide the circumference of each into any number of equal

parts; then, having projected the points of division, as

A2, C2, E2, &c., to their respective diameters AB and C' IT,

draw from them lines parallel to the rays of light, which,

by their intersection with the given plane, will indicate so

many points in the outline of the cast shadow.

Fig. 4 represents a circle whose plane is situated perpen-

dicularly to the direction of the luminous rays. In this

example the method of constructing the cast shadow does

not differ from that pointed out in reference to Fig. 2,

provided that both projections are made use of. But it is

obvious that instead of laying down the entire horizontal

projection of this circle, all that is necessary is to set off

the diameter D' E', equal to A B; because the shadow of

this diameter, transferred in the usual way, gives the

major axis of the ellipse which constitutes the outline of