DOI Seite / Zitierlink:
https://doi.org/10.11588/diglit.25888#0056
40
ENGINEER AND MACHINIST’S DRAWING-BOOK.
describe a circle, cutting M N in A' and D\ From these
points with the same radius, draw four arcs of circles,
cutting the primary circle in four points. These six
points being joined by straight lines, will form the figure
A' B' O' D' E' F', which is the base of the pyramid ; and
the lines A' D', B' E', and C' F', will represent the projec-
tions of its edges fore-shortened as they would appear in
the plan. If this operation has been correctly performed,
the opposite sides of the hexagon should be parallel to
each other, and to one of the diagonals; this should be
tested by the application of the square or other instru-
ment proper for the purpose.
By the help of the plan obtained as above described,
the vertical projection of the pyramid may be easily con-
structed. Since its base rests upon the horizontal plane,
it must be projected vertically upon the ground line;
therefore, from each of the angles at A', B', C', and D',
raise perpendiculars to that line. The points of intersec-
tion A, B, C, and D, are the true positions of all the angles
of the base; and it only remains to determine the height
of the pyramid, which is to be set off from the point G to
S, and to draw S A, S B, SC, and S D, which are the
only edges of the pyramid visible in the elevation. Of
these it is to be remarked that S A and S D alone, being
parallel to the vertical plane, are seen in their true length;
and moreover, that from the assumed position of the solid
under examination, the points F and E' being situated in
the lines B B' and C G, the lines S B and S C are each
the projections of two edges of the pyramid.
Figs. 3 and 4.—To construct the projections of the same
pyramid, having its base set in an inclined position,
but with its edges S A and S D still parallel to the ver-
tical plane.
It is evident, that, with the exception of the inclina-
tion, the vertical projection of this solid is precisely the
same as in the preceding example, and it is only necessary
to copy Fig. 1. For this purpose, after having fixed the
position of the point D, upon the ground line, and at the
most suitable distance from the vertical Z Z', draw through
this point an indefinite straight line D A, making with
L T an angle equal to the desired inclination of the base
of the pyramid. Then set off the distance DA, Fig. 1,
from D to A, Fig. 3 ; and, since the axis must still, as in
the former case, pass perpendicularly through the centre
of the base, draw two circular arcs from the points D and
A as centres, and with the distance D A as radius ; join
their points of intersection, and set off G S equal to the
height of the pyramid. Transfer also from Fig. 1 the dis-
tance B G or C G to the corresponding points in Fig. 3,
and complete the figure by drawing the straight lines
A S, B S, C S and D S, representing, as before, the outer
edges of the pyramid.
In proceeding to construct the horizontal projection of
the pyramid in this position, it is first to be remarked
that since the edges S A and S D are still parallel to the
vertical plane, and the point D remains unaltered, the
projection of the point A will still be in the line M N.
Its position at A', Fig. 4, is determined by the intersec-
tion of the perpendicular A A' with that line. The re-
maining points B', O', &c., in the projection of the base,
are found in a similar manner, by the intersections of
perpendiculars let fall from the corresponding points in
the elevation, with lines drawn parallel to M N, at a dis-
tance, (set off at o, p), equal to the width of the base. By
joining all the contiguous points, we obtain the figure
A' B' G D' E' F', representing the horizontal projection
of the base, two of its sides, however, being dotted, as
they must be supposed to be concealed by the body of
the pyramid. The vertex S having been similarly pro-
jected to S', and joined by straight lines to the several
angles of the base, the projection *of the solid is completed.
Figs. 5 and 6.—To find the horizontal projection of a
transverse section of the same pyramid, made by a
plane perpendicular to the vertical, but inclined at
an angle to the horizontal plane of projection ; and let
all the sides of the base be at an angle with the ground
line.
Having drawn, as before, the vertical S S', the centre
line of the figures, its point of intersection with the line
MN is the centre of the plan. From this point, then, as
a centre, draw, as in Fig. 2, a circle, and inscribe in it a
regular hexagon; but since none of the sides of the base
are to be parallel with the ground line, draw a diameter
A' D' making the required angle with that line, and from
the points A' and D', proceed to set out the angular points
of the hexagon as already explained. Then, in order to
obtain the projections of the edges of the pyramid, join
the angular points which are diametrically opposite ; and,
following the method pointed out in reference to Fig. 1,
project the figure thus obtained upon the vertical plane,
as shown at Fig. 5.
Now, if the cutting plane be represented by the line
a d, in the elevation, it is obvious that it will expose, as
the section of the pyramid, a polygon whose angular
points, being the intersections of the various edges with
the cutting plane, will be projected in perpendiculars
drawn from the points where it meets these edges respec-
tively. If, therefore, from the points a, f, b, &c., we let
fall the perpendiculars a a ,f f, b b , &c., and join their con-
tiguous points of intersection with the lines A' D', F C',
B' G, &c., we shall form a six-sided figure, which shall
represent the section required. This hexagon, it is almost
unnecessary to remark, is irregular, being formed by a
plane which is inclined to the base of the pyramid. The
edges F S and E S being concealed in the elevation, but
necessary for the construction of the plan, have been
expressed in dotted lines, as also the portion of the pyra-
mid situated above the cutting plane, which, though sup-
posed to be removed, is necessary in order to draw the
lines representing the edges. We have here, also, intro-
duced the ordinary method of expressing sections in purely
line-drawings, and in engraving on metal and wood; and
to which we shall adhere throughout this work, namely,
by filling up the spaces comprised within their outlines,
with a quantity of parallel straight lines drawn at equal
distances.
ENGINEER AND MACHINIST’S DRAWING-BOOK.
describe a circle, cutting M N in A' and D\ From these
points with the same radius, draw four arcs of circles,
cutting the primary circle in four points. These six
points being joined by straight lines, will form the figure
A' B' O' D' E' F', which is the base of the pyramid ; and
the lines A' D', B' E', and C' F', will represent the projec-
tions of its edges fore-shortened as they would appear in
the plan. If this operation has been correctly performed,
the opposite sides of the hexagon should be parallel to
each other, and to one of the diagonals; this should be
tested by the application of the square or other instru-
ment proper for the purpose.
By the help of the plan obtained as above described,
the vertical projection of the pyramid may be easily con-
structed. Since its base rests upon the horizontal plane,
it must be projected vertically upon the ground line;
therefore, from each of the angles at A', B', C', and D',
raise perpendiculars to that line. The points of intersec-
tion A, B, C, and D, are the true positions of all the angles
of the base; and it only remains to determine the height
of the pyramid, which is to be set off from the point G to
S, and to draw S A, S B, SC, and S D, which are the
only edges of the pyramid visible in the elevation. Of
these it is to be remarked that S A and S D alone, being
parallel to the vertical plane, are seen in their true length;
and moreover, that from the assumed position of the solid
under examination, the points F and E' being situated in
the lines B B' and C G, the lines S B and S C are each
the projections of two edges of the pyramid.
Figs. 3 and 4.—To construct the projections of the same
pyramid, having its base set in an inclined position,
but with its edges S A and S D still parallel to the ver-
tical plane.
It is evident, that, with the exception of the inclina-
tion, the vertical projection of this solid is precisely the
same as in the preceding example, and it is only necessary
to copy Fig. 1. For this purpose, after having fixed the
position of the point D, upon the ground line, and at the
most suitable distance from the vertical Z Z', draw through
this point an indefinite straight line D A, making with
L T an angle equal to the desired inclination of the base
of the pyramid. Then set off the distance DA, Fig. 1,
from D to A, Fig. 3 ; and, since the axis must still, as in
the former case, pass perpendicularly through the centre
of the base, draw two circular arcs from the points D and
A as centres, and with the distance D A as radius ; join
their points of intersection, and set off G S equal to the
height of the pyramid. Transfer also from Fig. 1 the dis-
tance B G or C G to the corresponding points in Fig. 3,
and complete the figure by drawing the straight lines
A S, B S, C S and D S, representing, as before, the outer
edges of the pyramid.
In proceeding to construct the horizontal projection of
the pyramid in this position, it is first to be remarked
that since the edges S A and S D are still parallel to the
vertical plane, and the point D remains unaltered, the
projection of the point A will still be in the line M N.
Its position at A', Fig. 4, is determined by the intersec-
tion of the perpendicular A A' with that line. The re-
maining points B', O', &c., in the projection of the base,
are found in a similar manner, by the intersections of
perpendiculars let fall from the corresponding points in
the elevation, with lines drawn parallel to M N, at a dis-
tance, (set off at o, p), equal to the width of the base. By
joining all the contiguous points, we obtain the figure
A' B' G D' E' F', representing the horizontal projection
of the base, two of its sides, however, being dotted, as
they must be supposed to be concealed by the body of
the pyramid. The vertex S having been similarly pro-
jected to S', and joined by straight lines to the several
angles of the base, the projection *of the solid is completed.
Figs. 5 and 6.—To find the horizontal projection of a
transverse section of the same pyramid, made by a
plane perpendicular to the vertical, but inclined at
an angle to the horizontal plane of projection ; and let
all the sides of the base be at an angle with the ground
line.
Having drawn, as before, the vertical S S', the centre
line of the figures, its point of intersection with the line
MN is the centre of the plan. From this point, then, as
a centre, draw, as in Fig. 2, a circle, and inscribe in it a
regular hexagon; but since none of the sides of the base
are to be parallel with the ground line, draw a diameter
A' D' making the required angle with that line, and from
the points A' and D', proceed to set out the angular points
of the hexagon as already explained. Then, in order to
obtain the projections of the edges of the pyramid, join
the angular points which are diametrically opposite ; and,
following the method pointed out in reference to Fig. 1,
project the figure thus obtained upon the vertical plane,
as shown at Fig. 5.
Now, if the cutting plane be represented by the line
a d, in the elevation, it is obvious that it will expose, as
the section of the pyramid, a polygon whose angular
points, being the intersections of the various edges with
the cutting plane, will be projected in perpendiculars
drawn from the points where it meets these edges respec-
tively. If, therefore, from the points a, f, b, &c., we let
fall the perpendiculars a a ,f f, b b , &c., and join their con-
tiguous points of intersection with the lines A' D', F C',
B' G, &c., we shall form a six-sided figure, which shall
represent the section required. This hexagon, it is almost
unnecessary to remark, is irregular, being formed by a
plane which is inclined to the base of the pyramid. The
edges F S and E S being concealed in the elevation, but
necessary for the construction of the plan, have been
expressed in dotted lines, as also the portion of the pyra-
mid situated above the cutting plane, which, though sup-
posed to be removed, is necessary in order to draw the
lines representing the edges. We have here, also, intro-
duced the ordinary method of expressing sections in purely
line-drawings, and in engraving on metal and wood; and
to which we shall adhere throughout this work, namely,
by filling up the spaces comprised within their outlines,
with a quantity of parallel straight lines drawn at equal
distances.