DOI Seite / Zitierlink:
https://doi.org/10.11588/diglit.25888#0095
THE PROJECTION OF SHADOWS.
79
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 LII.,
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 effect of shadows upon any object would be to
render those parts upon which they fall totally obscure
and invisible, but for the influence of the reflected light,
which greatly modifies their tone and treatment. We
shall have occasion, in another part of this work, to con-
sider the effects of reflected light; for the present, there-
fore, to avoid complicating the subject, we shall set aside
this element in the problem, and proceed to point out the
methods of determining the form and position of shadows
as affected by the direct light of the sun only.
The limit of the direct shadow in any body, whatever
may be its form or position, is a line of greater or less dis-
tinctness, which we may term 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
we suppose these rays to 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 dia-
gonal 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 eleva-
tion and plan, appears at the angle of 45°. In illustration,
let R, R', Fig. 1, Plate L., be the projections of a ray of
light in elevation and plan ; and 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 pro-
jecting 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
fine A a-, a distance equal to Aa'.
Plate L. Fig. 1.—Required to determine the shadoiu
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 neces-
sary 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 the shadow thrown by the opposite extre-
mity 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 a b is equal and parallel to the given line A B ;
this results from the circumstance that the latter is paral-
lel to the vertical plane X Y. Hence we have this general
rule, that 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, we have a
parallel slip of wood, or paper, A B C D, 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 abed, 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 we conclude, in general,
that when a surface, whatever may be its form, is paral-
lel to a plane, its shadow thrown upon that plane is a
figure similar and equal 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, as will be observed by examining
Fig. 2, where the lines A a, A! a, &c., are still drawn at an
angle of 45°.
Fig. 3.—If, while the given plane remains parallel to
the vertical plane of projection, as in the first example,
the slip is placed at an inclination to it (preserving, how-
ever, its vertical position), the figure of the cast shadow
79
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 LII.,
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 effect of shadows upon any object would be to
render those parts upon which they fall totally obscure
and invisible, but for the influence of the reflected light,
which greatly modifies their tone and treatment. We
shall have occasion, in another part of this work, to con-
sider the effects of reflected light; for the present, there-
fore, to avoid complicating the subject, we shall set aside
this element in the problem, and proceed to point out the
methods of determining the form and position of shadows
as affected by the direct light of the sun only.
The limit of the direct shadow in any body, whatever
may be its form or position, is a line of greater or less dis-
tinctness, which we may term 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
we suppose these rays to 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 dia-
gonal 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 eleva-
tion and plan, appears at the angle of 45°. In illustration,
let R, R', Fig. 1, Plate L., be the projections of a ray of
light in elevation and plan ; and 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 pro-
jecting 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
fine A a-, a distance equal to Aa'.
Plate L. Fig. 1.—Required to determine the shadoiu
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 neces-
sary 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 the shadow thrown by the opposite extre-
mity 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 a b is equal and parallel to the given line A B ;
this results from the circumstance that the latter is paral-
lel to the vertical plane X Y. Hence we have this general
rule, that 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, we have a
parallel slip of wood, or paper, A B C D, 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 abed, 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 we conclude, in general,
that when a surface, whatever may be its form, is paral-
lel to a plane, its shadow thrown upon that plane is a
figure similar and equal 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, as will be observed by examining
Fig. 2, where the lines A a, A! a, &c., are still drawn at an
angle of 45°.
Fig. 3.—If, while the given plane remains parallel to
the vertical plane of projection, as in the first example,
the slip is placed at an inclination to it (preserving, how-
ever, its vertical position), the figure of the cast shadow