0.5

1 cm

THE PROJECTION OF SHADOWS.

81

the shadow sought, while its minor axis is at once deter-

mined by a b, equal and parallel to A B.

Fig. 5 exhibits the case of a circle parallel to the vertical

plane of projection, throwing its shadow at once upon two

plane surfaces inclined to each other. To delineate this

shadow all that it is necessary specially to point out is

that the points cl and e are found by drawing from Y,

a line Y IT, parallel to the rays of light and projecting the

point D' to D and E.

Fig. 6 represents constructions similar to the foregoing,

for obtaining the form of the shadow cast by a horizontal

circle upon a vertical curved surface.

We may here remark that in every drawing where the

shadows are to be inserted, it is of the utmost importance

that the projections which represent the object whose

shadow is required should be exactly defined, as well as

the surface upon which this shadow is cast; it is therefore

advisable, in order to prevent mistakes, and to insure

accuracy, to draw the figures in China ink, and to erase

all pencil-marks before proceeding to the operations neces-

sary for finding the shadows.

Plate LII. Fig. 1.—To find the outline of the shadoiv

cast upon both planes of projection by a regular hexa-

gonal pyramid.

In these figures, it is at once obvious that the three sides

A' B' F', A' B' C' and A'C' D', alone receive the light; con-

sequently the edges A' F' and A D' are the lines of shade.

To solve this problem, then, we have only to determine

the shadow cast by these two lines, which is accomplished

by drawing, from the projections of the vertex of the

pyramid, the lines A b and A' a', parallel to the ray of

light; then raising from the point b a perpendicular to

the ground line, which gives at a' the shadow of the ver-

tex on the horizontal plane, and finally by joining this

last point a with the points D' and F'; the lines D a" and

F a' are the outlines of the required shadow on the hori-

zontal plane. But as the pyramid happens to be situated

sufficiently near the vertical plane to throw a portion of

its shadow, towards the vertex, upon it, this portion may

be found by raising from the point c where the line A' a

cuts the ground line, a perpendicular c a, intersecting the

line A b in a ; the lines a d and a e, joining this point

with those where the horizontal part of the shadow meets

the ground line, will be its outline upon the vertical plane.

Fig. 4 represents a hexagonal prism whose shadows cast

upon the two planes of projection have also been delineated.

The lines drawn on these figures are sufficient to indicate

the necessary construction without the help of further ex-

planations.

Fig. 2.—Required to determine the limit of shade in a

cylinder placed vertically, and likewise its shadow cast

upon the tivo planes of projection.

The lines of shade in a cylinder situated as indicated,

are at once found by drawing two tangents to its base,

parallel to the ray of light; and projecting, through the

points of contact, lines parallel to the axis of the cylinder.

Draw the tangents D'd' and C' c', parallel to the ray

Pd; these are the outlines of the shadow cast upon the

horizontal plane. Through the point of contact C draw the

vertical line C E; this line denotes the line of shade upon

the surface of the cylinder. It is obviously unnecessary

to draw the perpendicular from the opposite point D',

because it is altogether concealed in the vertical elevation

of the solid. In order to ascertain the points C' and D'

with greater accuracy, it is proper to draw, through the

centre O', a diameter perpendicular to the ray of light R'.

Had this cylinder been placed at a somewhat greater

distance from the vertical plane of projection, its shadow

would have been entirely cast upon the horizontal plane,

in which case it would have terminated in a semicircle

drawn from the centre o', with a radius equal to that of

the base. But as, in our example, a portion of the shadow

of the upper part is thrown upon the vertical plane, its

outline will be defined by an ellipse drawn in the manner

indicated in Fig. 2 of the preceding Plate.

Fig. 5.—When the cylinder is placed horizontally, and

at the same time, at an angle with the vertical plane, the

construction is the same as that explained above; namely,

lines are to be drawn parallel to the ray of light, and

touching the opposite points of either base of the cylinder;

and, through the points of contact A and C, the horizontal

lines A B and C D are to be drawn, denoting the limits

of the shade on the figure. The latter of these lines only

is visible in the elevation ; while, on the other hand, the

former, A B alone, is seen in the plan, where it may be

found by drawing a perpendicular from A meeting the

base F' G' in A'. The line A E' drawn parallel to the

axis of the cylinder is the line of shade required.

The example here given presents the particular case in

which the base of the cylinder is parallel to the direction

of the rays of light in the horizontal projection. This case

admits of a simpler solution than the preceding, in which

the necessity for drawing the vertical projection of the

figure is dispensed with. All that is required in order

to determine the line A' E' is to ascertain the angle which

the ray of light makes with the projection of the figure.

Draw a tangent to the circle F' A2 G' (which represents

the base of the cylinder laid down on the horizontal plane),

in such a manner as to make with F' G' an angle of 35° 16',

and through the point of contact A2 draw a line parallel

to the axis of the cylinder; this line E A will be the line

of shade as before.

Fig. 3.—To find the line of shade in a cone, and its

shado w cast upon the two planes of projection.

By a construction similar to that for Fig. 1, we find the

point a ; from this point draw tangents to the opposite

sides of the base; these two lines will denote the outlines

of the shadow cast upon the horizontal plane. Their points

of contact B' and C' joined to the centre A', will give the

lines A B' and A C' for the required lines of shade in

the plan ; of these, the first only will be visible at A B in

the elevation.

Fig. 6.—If the cone be situated in the reverse position,

as in the figure, the shade is determined in the following

manner:—From the centre A' of the base, draw a line

parallel to the light; from the point a where it intersects

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81

the shadow sought, while its minor axis is at once deter-

mined by a b, equal and parallel to A B.

Fig. 5 exhibits the case of a circle parallel to the vertical

plane of projection, throwing its shadow at once upon two

plane surfaces inclined to each other. To delineate this

shadow all that it is necessary specially to point out is

that the points cl and e are found by drawing from Y,

a line Y IT, parallel to the rays of light and projecting the

point D' to D and E.

Fig. 6 represents constructions similar to the foregoing,

for obtaining the form of the shadow cast by a horizontal

circle upon a vertical curved surface.

We may here remark that in every drawing where the

shadows are to be inserted, it is of the utmost importance

that the projections which represent the object whose

shadow is required should be exactly defined, as well as

the surface upon which this shadow is cast; it is therefore

advisable, in order to prevent mistakes, and to insure

accuracy, to draw the figures in China ink, and to erase

all pencil-marks before proceeding to the operations neces-

sary for finding the shadows.

Plate LII. Fig. 1.—To find the outline of the shadoiv

cast upon both planes of projection by a regular hexa-

gonal pyramid.

In these figures, it is at once obvious that the three sides

A' B' F', A' B' C' and A'C' D', alone receive the light; con-

sequently the edges A' F' and A D' are the lines of shade.

To solve this problem, then, we have only to determine

the shadow cast by these two lines, which is accomplished

by drawing, from the projections of the vertex of the

pyramid, the lines A b and A' a', parallel to the ray of

light; then raising from the point b a perpendicular to

the ground line, which gives at a' the shadow of the ver-

tex on the horizontal plane, and finally by joining this

last point a with the points D' and F'; the lines D a" and

F a' are the outlines of the required shadow on the hori-

zontal plane. But as the pyramid happens to be situated

sufficiently near the vertical plane to throw a portion of

its shadow, towards the vertex, upon it, this portion may

be found by raising from the point c where the line A' a

cuts the ground line, a perpendicular c a, intersecting the

line A b in a ; the lines a d and a e, joining this point

with those where the horizontal part of the shadow meets

the ground line, will be its outline upon the vertical plane.

Fig. 4 represents a hexagonal prism whose shadows cast

upon the two planes of projection have also been delineated.

The lines drawn on these figures are sufficient to indicate

the necessary construction without the help of further ex-

planations.

Fig. 2.—Required to determine the limit of shade in a

cylinder placed vertically, and likewise its shadow cast

upon the tivo planes of projection.

The lines of shade in a cylinder situated as indicated,

are at once found by drawing two tangents to its base,

parallel to the ray of light; and projecting, through the

points of contact, lines parallel to the axis of the cylinder.

Draw the tangents D'd' and C' c', parallel to the ray

Pd; these are the outlines of the shadow cast upon the

horizontal plane. Through the point of contact C draw the

vertical line C E; this line denotes the line of shade upon

the surface of the cylinder. It is obviously unnecessary

to draw the perpendicular from the opposite point D',

because it is altogether concealed in the vertical elevation

of the solid. In order to ascertain the points C' and D'

with greater accuracy, it is proper to draw, through the

centre O', a diameter perpendicular to the ray of light R'.

Had this cylinder been placed at a somewhat greater

distance from the vertical plane of projection, its shadow

would have been entirely cast upon the horizontal plane,

in which case it would have terminated in a semicircle

drawn from the centre o', with a radius equal to that of

the base. But as, in our example, a portion of the shadow

of the upper part is thrown upon the vertical plane, its

outline will be defined by an ellipse drawn in the manner

indicated in Fig. 2 of the preceding Plate.

Fig. 5.—When the cylinder is placed horizontally, and

at the same time, at an angle with the vertical plane, the

construction is the same as that explained above; namely,

lines are to be drawn parallel to the ray of light, and

touching the opposite points of either base of the cylinder;

and, through the points of contact A and C, the horizontal

lines A B and C D are to be drawn, denoting the limits

of the shade on the figure. The latter of these lines only

is visible in the elevation ; while, on the other hand, the

former, A B alone, is seen in the plan, where it may be

found by drawing a perpendicular from A meeting the

base F' G' in A'. The line A E' drawn parallel to the

axis of the cylinder is the line of shade required.

The example here given presents the particular case in

which the base of the cylinder is parallel to the direction

of the rays of light in the horizontal projection. This case

admits of a simpler solution than the preceding, in which

the necessity for drawing the vertical projection of the

figure is dispensed with. All that is required in order

to determine the line A' E' is to ascertain the angle which

the ray of light makes with the projection of the figure.

Draw a tangent to the circle F' A2 G' (which represents

the base of the cylinder laid down on the horizontal plane),

in such a manner as to make with F' G' an angle of 35° 16',

and through the point of contact A2 draw a line parallel

to the axis of the cylinder; this line E A will be the line

of shade as before.

Fig. 3.—To find the line of shade in a cone, and its

shado w cast upon the two planes of projection.

By a construction similar to that for Fig. 1, we find the

point a ; from this point draw tangents to the opposite

sides of the base; these two lines will denote the outlines

of the shadow cast upon the horizontal plane. Their points

of contact B' and C' joined to the centre A', will give the

lines A B' and A C' for the required lines of shade in

the plan ; of these, the first only will be visible at A B in

the elevation.

Fig. 6.—If the cone be situated in the reverse position,

as in the figure, the shade is determined in the following

manner:—From the centre A' of the base, draw a line

parallel to the light; from the point a where it intersects

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