DOI Seite / Zitierlink:
https://doi.org/10.11588/diglit.25888#0055
DRAWING OF MACHINERY BY ORDINARY GEOMETRICAL PROJECTION.
39
meet, ought to be lightly drawn,—namely a b or a' b',
a d, and a' /'. Again, the lateral planes represented by
b c, c d, b' e, and e /, are obviously in the shade, as no light
falls upon them directly: and these lines are strengthened,
to express the distinction.
In (Figs. 174 and 175), the lines composing the interior
and exterior contours in the elevation are parallel, and thus
contrast well. It is obvious that the portion of the exte-
rior from b by c to cZ is in the shade, while the rest is light;
and the inverse is the case with the inner edges. A pecu-
liarity, however, occurs at d, for here the edges, inner and
outer, are parallel to the direction of the light. It is plain
that the surfaces which come up to these edges will be in
a medium shade, and that the lines at d should be of
medium thickness.
(Figs. 176 and 177) represent a hollow cylinder in projec-
tion. In the plan, two lines a, c, drawn parallel to the direc-
tion of the light, and touching the exterior of the cylinder,
define the semicircular outline, a b' c, which is thrown in
the shade, and ought to be strengthened. The outlines at
a and c are, like the edges at d (Fig. 174), parallel to the
light, and the contour on each side gradually recedes and
advances to the light. The thickness of the line should,
therefore, be rather gradually reduced at the points a, c,
and it would besides appear awkward to stop the shade
abruptly on a circle. In the elevation, the base-line d f
should be shaded, and b d but half-shaded as it lies in a
curve surface.
If, again, the cylinder be hollow, presenting in plan the
interior contour circle e h, then the semicircle e g h ex-
presses the shady side of the interior, the light striking-
directly upon the opposite semicircle.
These examples illustrate every case of shade-lining
that occurs in outline drawings. The effect is enhanced
by proportioning the thickness of the lines to the depth
of the surfaces to which they belong, below the original
surfaces from which the shadows arise.
The system of shading above described, according to
which the light is supposed in plan to strike towards the
right hand upper corner, falling as it were, in front of the
objects, is mostly peculiar to French practice. In England,
the shadows are, for simplicity, thrown all one way, in ele-
vation and plan, towards the right hand and foot of the
sheet. The illustrative Plates contain examples of shading
in both ways.
The constructions of the following problems in projection
are made independently of the use of the T square. The
subjects of the earlier problems are exhibited in Plate VIII.
SECTION IV.
PROJECTIONS OF SIMPLE BODIES.
Projections of a* Pyramid.—Plate I.
The first subject for projection is a regular hexagonal
pyramid, shown by Figs. 1 and 2, Plate VIII. On in-
specting the figure, it would appear that two distinct
geometrical views are necessary to convey a complete idea
of the form of the object; namely, an elevation to repre-
sent the sides of the body, and to express its height; and a
plan of the upper surface, to express the form horizontally.
It is to be observed that this body has an imaginary
axis or centre-line, about which the same parts are equally
distant. This is an essential characteristic of all symme-
trical figures, or such as may be supposed to consist of two
halves of the same form joined together. A cone, for in-
stance, may be cut in two halves of the same form, down
through the axis or centre line, and so also may the
pyramid.
In the first place, a horizontal straight line L T is to
be drawn through the centre of the sheet; this line will
represent the ground line. Then draw a perpendicular
Z Z' through the middle of the ground line: this may be
done by the geometrical methods pointed out; but, in
case the compasses should not be capable of opening far
enough to admit of the arcs at Z and Z' being described
from the extremities of the ground line, the points m and
n, equidistant from these extremities, may be assumed as
centres, from which, with the largest convenient radius,
portions of circles are to be struck, above and below the
ground line, and their points of intersection joined by the
straight line Z Z', which will be the perpendicular re-
quired. It is to these two straight lines L T and Z Z'
that all the other lines in the drawings are to be referred;
those parallel to the former being considered as horizontal,
and those parallel to the latter as vertical. We should
next set out the border lines of the drawing, that is, the
rectangular space within which all the figures are to be
contained. Having determined its dimensions, take half
the breadth as a radius, and from any points as L and T,
in the ground line describe arcs above and below it, and
draw tangents X U and V Y to these arcs. Then from
the points g and li, where these lines intersect the per-
pendicular Z Z', set off, on each side of the latter, half
the length of the rectangular space required, and draw
XY, U V.
The next step is to draw the axes or centre lines of the
various figures to be represented. The vertical centre
line S S', Figs. 1 and 2, is obtained by describing from
the points g and h, at any convenient distance from the
line Z Z', arcs intersecting the horizontal border lines in
x and y, and joining x y. The other vertical axes may be
set out in the same manner. For the sake of preserving
the symmetry of the drawing, the centres of the lower
range of figures are all in the same straight line M N ;
this is to be drawn parallel to the ground line, by mark-
ing off, from L to M, and from T to N, its proper distance
from that line.
Plate I.—Figs. 1, 2.—In delineating the pyramid, it is
necessary, in the first place, to construct the plan, or hori-
zontal projection. The point S', where the centre line S S'
intersects the line M N, is to be taken as the centre of the
figure, and from this point, with a radius equal to the side
of the hexagon which forms the base of the pyramid,
39
meet, ought to be lightly drawn,—namely a b or a' b',
a d, and a' /'. Again, the lateral planes represented by
b c, c d, b' e, and e /, are obviously in the shade, as no light
falls upon them directly: and these lines are strengthened,
to express the distinction.
In (Figs. 174 and 175), the lines composing the interior
and exterior contours in the elevation are parallel, and thus
contrast well. It is obvious that the portion of the exte-
rior from b by c to cZ is in the shade, while the rest is light;
and the inverse is the case with the inner edges. A pecu-
liarity, however, occurs at d, for here the edges, inner and
outer, are parallel to the direction of the light. It is plain
that the surfaces which come up to these edges will be in
a medium shade, and that the lines at d should be of
medium thickness.
(Figs. 176 and 177) represent a hollow cylinder in projec-
tion. In the plan, two lines a, c, drawn parallel to the direc-
tion of the light, and touching the exterior of the cylinder,
define the semicircular outline, a b' c, which is thrown in
the shade, and ought to be strengthened. The outlines at
a and c are, like the edges at d (Fig. 174), parallel to the
light, and the contour on each side gradually recedes and
advances to the light. The thickness of the line should,
therefore, be rather gradually reduced at the points a, c,
and it would besides appear awkward to stop the shade
abruptly on a circle. In the elevation, the base-line d f
should be shaded, and b d but half-shaded as it lies in a
curve surface.
If, again, the cylinder be hollow, presenting in plan the
interior contour circle e h, then the semicircle e g h ex-
presses the shady side of the interior, the light striking-
directly upon the opposite semicircle.
These examples illustrate every case of shade-lining
that occurs in outline drawings. The effect is enhanced
by proportioning the thickness of the lines to the depth
of the surfaces to which they belong, below the original
surfaces from which the shadows arise.
The system of shading above described, according to
which the light is supposed in plan to strike towards the
right hand upper corner, falling as it were, in front of the
objects, is mostly peculiar to French practice. In England,
the shadows are, for simplicity, thrown all one way, in ele-
vation and plan, towards the right hand and foot of the
sheet. The illustrative Plates contain examples of shading
in both ways.
The constructions of the following problems in projection
are made independently of the use of the T square. The
subjects of the earlier problems are exhibited in Plate VIII.
SECTION IV.
PROJECTIONS OF SIMPLE BODIES.
Projections of a* Pyramid.—Plate I.
The first subject for projection is a regular hexagonal
pyramid, shown by Figs. 1 and 2, Plate VIII. On in-
specting the figure, it would appear that two distinct
geometrical views are necessary to convey a complete idea
of the form of the object; namely, an elevation to repre-
sent the sides of the body, and to express its height; and a
plan of the upper surface, to express the form horizontally.
It is to be observed that this body has an imaginary
axis or centre-line, about which the same parts are equally
distant. This is an essential characteristic of all symme-
trical figures, or such as may be supposed to consist of two
halves of the same form joined together. A cone, for in-
stance, may be cut in two halves of the same form, down
through the axis or centre line, and so also may the
pyramid.
In the first place, a horizontal straight line L T is to
be drawn through the centre of the sheet; this line will
represent the ground line. Then draw a perpendicular
Z Z' through the middle of the ground line: this may be
done by the geometrical methods pointed out; but, in
case the compasses should not be capable of opening far
enough to admit of the arcs at Z and Z' being described
from the extremities of the ground line, the points m and
n, equidistant from these extremities, may be assumed as
centres, from which, with the largest convenient radius,
portions of circles are to be struck, above and below the
ground line, and their points of intersection joined by the
straight line Z Z', which will be the perpendicular re-
quired. It is to these two straight lines L T and Z Z'
that all the other lines in the drawings are to be referred;
those parallel to the former being considered as horizontal,
and those parallel to the latter as vertical. We should
next set out the border lines of the drawing, that is, the
rectangular space within which all the figures are to be
contained. Having determined its dimensions, take half
the breadth as a radius, and from any points as L and T,
in the ground line describe arcs above and below it, and
draw tangents X U and V Y to these arcs. Then from
the points g and li, where these lines intersect the per-
pendicular Z Z', set off, on each side of the latter, half
the length of the rectangular space required, and draw
XY, U V.
The next step is to draw the axes or centre lines of the
various figures to be represented. The vertical centre
line S S', Figs. 1 and 2, is obtained by describing from
the points g and h, at any convenient distance from the
line Z Z', arcs intersecting the horizontal border lines in
x and y, and joining x y. The other vertical axes may be
set out in the same manner. For the sake of preserving
the symmetry of the drawing, the centres of the lower
range of figures are all in the same straight line M N ;
this is to be drawn parallel to the ground line, by mark-
ing off, from L to M, and from T to N, its proper distance
from that line.
Plate I.—Figs. 1, 2.—In delineating the pyramid, it is
necessary, in the first place, to construct the plan, or hori-
zontal projection. The point S', where the centre line S S'
intersects the line M N, is to be taken as the centre of the
figure, and from this point, with a radius equal to the side
of the hexagon which forms the base of the pyramid,