DOI Seite / Zitierlink:
https://doi.org/10.11588/diglit.25888#0098
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