DOI Seite / Zitierlink:
https://doi.org/10.11588/diglit.25888#0059
DRAWING OR MACHINERY BY ORDINARY GEOMETRICAL PROJECTION.
43
Let Z' Z', Fig. 6, denote the position of the cutting
plane, as seen in horizontal projection: then the section
of the cone exhibited in the elevation will be seen in its
actual dimensions. Now it is obvious that the very same
method of construction which we have already explained
In reference to the two preceding examples is also appli-
cable to the present; therefore we shall only indicate the
application in a single instance.
Having drawn any horizontal line E F at pleasure,
bisect it, and from the centre S', with half of that line as
radius describe a circle, which will be intersected by the
line Z' Z' at the points G' and Hj these being projected
to G and H, on the line E F, determine two points in the
curve. The vertex I of the curve, which in this case is a
hyperbola, will be obtained by describing from the centre
S' with the distance S' I', a circle intersecting A B in
K', projecting this latter point to K, and drawing the
horizontal line K L. Should the cutting plane pass through
the axis of the cone, the section obviously will resolve
itself into an isosceles triangle.
SECTION Y.
Penetrations or Intersections of Solids.
Having thus minutely explained the mode of projecting
the conic sections, it will be quite unnecessary to enter
into any detail with regard to the sections of the cylinder.
Supposing it to stand in an upright position, a plane cut-
ting it obliquely will obviously form an ellipse, of which
the minor axis is the diameter of the cylinder, and the
major axis may easily be determined by the methods
already laid down in treating of the cone.
On examining the minor details of most machines, we
shall find numerous examples of cylindrical, and other
forms, fitted to, and even appearing to pass through each
other in a great variety of ways. We shall, therefore, pro-
ceed to examine attentively the outlines formed by the
penetrations of various solid bodies; our examples are
grouped in Plate YI11, and are selected with the view of
exhibiting those cases which are of most frequent occur-
rence, and at the same time, of elucidating the general
principles which are applicable in every case.
Penetrations of Cylinders.—Plate IV.
#
Figs. 1 and 2 represent the projections of two cylinders
of unequal diameters, meeting each other at right angles ;
one of which is denoted by the rectangle A B E D in the
vertical, and by the circle A' H' B' in the horizontal pro-
jections ; while the other which is supposed to be hori-
zontal, is indicated in the former by the circle L P I N,
and in the latter by the figure L' I' K' M'. From the
position of these two solids, it is evident that the curves
formed by their junction will be projected in the circles
A' FI' B' and LPIN; and further, that such would also
be the case even although their axes did not intersect
each other.
But if we suppose the position of these bodies to be
changed into that represented at Figs. 3 and 4, the lines
of their intersection will assume in the vertical projection,
a totally different aspect, and may be accurately deter-
mined by the following construction.
Through any point taken upon the plan, Fig. 4, draw
a horizontal line a b', which is to be considered as indi-
cating a plane cutting both cylinders, parallel to their
axes ; this plane would cut the vertical cylinder in lines
drawn perpendicularly through the points c and d'. To
find the vertical projection of its intersection with the
other cylinder, conceive its base I' L', after being trans-
ferred to I2 L2, to be turned over parallel to the horizontal
plane; this is expressed by simply drawing a circle of the
diameter I2 L2; and producing the line a' b' to a? ; then
set off the distance a2 e, on each side of the axis I K, and
draw straight lines through these points parallel to it.
These lines ab, g h denote the intersection of the plane
a b' with the horizontal cylinder, and therefore the points
c, d, m, o, where they cut the perpendiculars c c', d d' are
points in the curve required. By laying down other
planes, similar to a b', and operating as before, any number
of points may be obtained. The vertices i and k, of the
curves are obviously projected directly ; and their extreme
points are determined by the intersections of the outlines
of both cylinders. When the cylinders are of unequal
diameters, as in the present case, the curves of penetration
are hyperbolas.
Figs. 5 and 6.—When the diameters of the cylinders
are equal, and when they cut each other at right angles,
the curves of penetration are projected vertically in straight
lines perpendicular to each other. For, if we proceed to
apply the method given above, we shall soon discover that
the various points in these curves are situated in two planes
at right angles to each other, and to the vertical plane,
the sections formed by them being, in fact, ellipses equal
and similar to each other. We need not enter into any
details in illustration of this case, further than to call at-
tention to the figures, where the projections of some of the
points are indicated, in elevation and plan, by the same
letters of reference.
Figs. 7 and 8.—To delineate the intersections of tivo
cylinders of equal diameters at right angles, when one
of the cylinders is inclined to the vertical plane.
Supposing the two preceding figures to have been drawn,
we may easily ascertain the projection c, of any point such
as c', by observing that it must be situated in the perpen-
dicular c c, and that, since the distance of this point, (pro-
jected at c in Fig. 5), from the horizontal plane remains
unaltered, it must also be in the horizontal line c c. TJpon
these principles all the points indicated by literal refer-
ences in Fig. 7 are determined; the curves of penetration
resulting therefrom intersecting each other at two points
projected upon the axial line L K, of which that marked
q, alone is seen. The ends of the horizontal cylinder are
represented by ellipses, the construction of which will also
be obvious on referring to the figures ; and they do not
require further consideration here.
43
Let Z' Z', Fig. 6, denote the position of the cutting
plane, as seen in horizontal projection: then the section
of the cone exhibited in the elevation will be seen in its
actual dimensions. Now it is obvious that the very same
method of construction which we have already explained
In reference to the two preceding examples is also appli-
cable to the present; therefore we shall only indicate the
application in a single instance.
Having drawn any horizontal line E F at pleasure,
bisect it, and from the centre S', with half of that line as
radius describe a circle, which will be intersected by the
line Z' Z' at the points G' and Hj these being projected
to G and H, on the line E F, determine two points in the
curve. The vertex I of the curve, which in this case is a
hyperbola, will be obtained by describing from the centre
S' with the distance S' I', a circle intersecting A B in
K', projecting this latter point to K, and drawing the
horizontal line K L. Should the cutting plane pass through
the axis of the cone, the section obviously will resolve
itself into an isosceles triangle.
SECTION Y.
Penetrations or Intersections of Solids.
Having thus minutely explained the mode of projecting
the conic sections, it will be quite unnecessary to enter
into any detail with regard to the sections of the cylinder.
Supposing it to stand in an upright position, a plane cut-
ting it obliquely will obviously form an ellipse, of which
the minor axis is the diameter of the cylinder, and the
major axis may easily be determined by the methods
already laid down in treating of the cone.
On examining the minor details of most machines, we
shall find numerous examples of cylindrical, and other
forms, fitted to, and even appearing to pass through each
other in a great variety of ways. We shall, therefore, pro-
ceed to examine attentively the outlines formed by the
penetrations of various solid bodies; our examples are
grouped in Plate YI11, and are selected with the view of
exhibiting those cases which are of most frequent occur-
rence, and at the same time, of elucidating the general
principles which are applicable in every case.
Penetrations of Cylinders.—Plate IV.
#
Figs. 1 and 2 represent the projections of two cylinders
of unequal diameters, meeting each other at right angles ;
one of which is denoted by the rectangle A B E D in the
vertical, and by the circle A' H' B' in the horizontal pro-
jections ; while the other which is supposed to be hori-
zontal, is indicated in the former by the circle L P I N,
and in the latter by the figure L' I' K' M'. From the
position of these two solids, it is evident that the curves
formed by their junction will be projected in the circles
A' FI' B' and LPIN; and further, that such would also
be the case even although their axes did not intersect
each other.
But if we suppose the position of these bodies to be
changed into that represented at Figs. 3 and 4, the lines
of their intersection will assume in the vertical projection,
a totally different aspect, and may be accurately deter-
mined by the following construction.
Through any point taken upon the plan, Fig. 4, draw
a horizontal line a b', which is to be considered as indi-
cating a plane cutting both cylinders, parallel to their
axes ; this plane would cut the vertical cylinder in lines
drawn perpendicularly through the points c and d'. To
find the vertical projection of its intersection with the
other cylinder, conceive its base I' L', after being trans-
ferred to I2 L2, to be turned over parallel to the horizontal
plane; this is expressed by simply drawing a circle of the
diameter I2 L2; and producing the line a' b' to a? ; then
set off the distance a2 e, on each side of the axis I K, and
draw straight lines through these points parallel to it.
These lines ab, g h denote the intersection of the plane
a b' with the horizontal cylinder, and therefore the points
c, d, m, o, where they cut the perpendiculars c c', d d' are
points in the curve required. By laying down other
planes, similar to a b', and operating as before, any number
of points may be obtained. The vertices i and k, of the
curves are obviously projected directly ; and their extreme
points are determined by the intersections of the outlines
of both cylinders. When the cylinders are of unequal
diameters, as in the present case, the curves of penetration
are hyperbolas.
Figs. 5 and 6.—When the diameters of the cylinders
are equal, and when they cut each other at right angles,
the curves of penetration are projected vertically in straight
lines perpendicular to each other. For, if we proceed to
apply the method given above, we shall soon discover that
the various points in these curves are situated in two planes
at right angles to each other, and to the vertical plane,
the sections formed by them being, in fact, ellipses equal
and similar to each other. We need not enter into any
details in illustration of this case, further than to call at-
tention to the figures, where the projections of some of the
points are indicated, in elevation and plan, by the same
letters of reference.
Figs. 7 and 8.—To delineate the intersections of tivo
cylinders of equal diameters at right angles, when one
of the cylinders is inclined to the vertical plane.
Supposing the two preceding figures to have been drawn,
we may easily ascertain the projection c, of any point such
as c', by observing that it must be situated in the perpen-
dicular c c, and that, since the distance of this point, (pro-
jected at c in Fig. 5), from the horizontal plane remains
unaltered, it must also be in the horizontal line c c. TJpon
these principles all the points indicated by literal refer-
ences in Fig. 7 are determined; the curves of penetration
resulting therefrom intersecting each other at two points
projected upon the axial line L K, of which that marked
q, alone is seen. The ends of the horizontal cylinder are
represented by ellipses, the construction of which will also
be obvious on referring to the figures ; and they do not
require further consideration here.