_PROJECTIONS._
In projecting, the lines in one view are used to mark those in other
views, and to find their shapes or curvature as they will appear in
other views. Thus, in Figure 225_a_ we have a spiral, wound around a
cylinder whose end is cut off at an angle. The pitch of the spiral is
the distance A B, and we may delineate the curve of the spiral looking
at the cylinder from two positions (one at a right-angle to the other,
as is shown in the figure), by means of a circle having a circumference
equal to that of the cylinder.
The circumference of this circle we divide into any number of
equidistant divisions, as from 1 to 24. The pitch A B of the spiral or
thread is then divided off also into 24 equidistant divisions, as marked
on the left hand of the figure; vertical lines are then drawn from the
points of division on the circle to the points correspondingly numbered
on the lines dividing the pitch; and where line 1 on the circle
intersects line 1 on the pitch is one point in the curve. Similarly,
where point 2 on the circle intersects line 2 on the pitch is another
point in the curve, and so on for the whole 24 divisions on the circle
and on the pitch. In this view, however, the path of the spiral from
line 7 to line 19 lies on the other side of the cylinder, and is marked
in dotted lines, because it is hidden by the cylinder. In the
right-hand view, however, a different portion of the spiral or thread is
hidden, namely from lines 1 to 13 inclusive, being an equal proportion
to that hidden in the left-hand view.
[Illustration: Fig. 225 _a_.]
The top of the cylinder is shown in the left-hand view to be cut off at
an angle to the axis, and will therefore appear elliptical; in the
right-hand view, to delineate this oval, the same vertical lines from
the circle may be carried up as shown on the right hand, and horizontal
lines may be drawn from the inclined face in one view across the end of
the other view, as at P; the divisions on the circle may be carried up
on the right-hand view by means of straight lines, as Q, and arcs of
circle, as at R, and vertical lines drawn from these arcs, as line S,
and where these vertical lines S intersect the horizontal lines as P,
are points in the ellipse.
Let it be required to draw a cylindrical body joining another at a
right-angle; as for example, a Tee, such as in Figure 226, and the
outline can all be shown in one view, but it is required to find the
line of junction of one piece, A, with the other, B; that is, find or
mark the lines of junction C. Now when the diameters of A and B are
equal, the line of junction C is a straight line, but it becomes a
curved one when the diameter of A is less than that of B, or _vice
versa_; hence it may be as well to project it in both cases. For this
purpose the three views are necessary. One-quarter of the circle of B,
in the end view, is divided off into any number of equal divisions; thus
we have chosen the divisions marked _a_, _b_, _c_, _d_, _e_, etc.; a
quarter of the top view is similarly divided off, as at _f_, _g_, _h_,
_i_, _j_; from these points of division lines are projected on to the
side view, as shown by the dotted lines _k_, _l_, _m_, _n_, _o_, _p_,
etc., and where these lines meet, as denoted by the dots, is in each
case a point in the line of junction of the two cylinders A, B.
[Illustration: Fig. 226.]
[Illustration: Fig. 227.]
Figure 227 represents a Tee, in which B is less in diameter than A;
hence the two join in a curve, which is found in a similar manner, as is
shown in Figure 227. Suppose that the end and top views are drawn, and
that the side view is drawn in outline, but that the curve of junction
or intersection is to be found. Now it is evident that since the centre
line 1 passes through the side and end views, that the face _a_, in the
end view, will be even with the face _a'_ in the side view, both being
the same face, and as the full length of the side of B in the end view
is marked by line _b_, therefore line _c_ projected down from _b_ will
at its junction with line _b'_, which corresponds to line _b_, give the
extreme depth to which _b'_ extends into the body A, and therefore, the
apex of the curve of intersection of B with A. To obtain other points,
we divide one-quarter of the circumference of the circle B in the top
view into four equal divisions, as by lines _d_, _e_, _f_, and from the
points of division we draw lines _j_, _i_, _g_, to the centre line
marked 2, these lines being thickened in the cut for clearness of
illustration. The compasses are then set to the length of thickened line
_g_, and from point _h_, in the end view, as a centre, the arc _g'_ is
marked. With the compasses set to the length of thickened line _i_, and
from _h_ as a centre, arc _i'_ is marked, and with the length of
thickened line _j_ as a radius and from _h_ as a centre arc _j'_ is
marked; from these arcs lines _k_, _l_, _m_ are drawn, and from the
intersection of _k_, _l_, _m_, with the circle of A, lines _n_, _o_, _p_
are let fall. From the lines of division, _d_, _e_, _f_, the lines _q_,
_r_, _s_ are drawn, and where lines _n_, _o_, _p_ join lines _q_, _r_,
_s_, are points in the curve, as shown by the dots, and by drawing a
line to intersect these dots the curve is obtained on one-half of B.
Since the axis of B is in the same plane as that of A, the lower half of
the curve is of the same curvature as the upper, as is shown by the
dotted curve.
[Illustration: Fig. 228.]
In Figure 228 the axis of piece B is not in the same plane as that of D,
but to one side of it to the distance between the centre lines C, D,
which is most clearly seen in the top view. In this case the process is
the same except in the following points: In the side view the line _w_,
corresponding to the line _w_ in the end view, passes within the line
_x_ before the curve of intersection begins, and in transferring the
lengths of the full lines _b_, _c_, _d_, _e_, _f_ to the end view, and
marking the arcs _b'_, _c'_, _d'_, _e'_, _f'_, they are marked from the
point _w_ (the point where the centre line of B intersects the outline
of A), instead of from the point _x_. In all other respects the
construction is the same as that in Figure 227.
[Illustration: Fig. 229.]
In these examples the axis of B stands at a right-angle to that of A.
But in Figure 229 is shown the construction where the axis of B is not
at a right-angle to A. In this case there is projected from B, in the
side view, an end view of B as at B', and across this end at a
right-angle to the centre line of B is marked a centre line C C of B',
which is divided as before by lines _d_, _e_, _f_, _g_, _h_, their
respective lengths being transferred from W as a centre, and marked by
the arcs _d'_, _e'_, _f'_, which are marked on a vertical line and
carried by horizontal lines, to the arc of A as at _i_, _j_, _k_. From
these points, _i_, _j_, _k_, the perpendicular lines _l_, _m_, _n_, _o_,
are dropped, and where these lines meet lines _p_, _q_, _r_, _s_, _t_,
are points in the curve of intersection of B with A. It will be observed
that each of the lines _m_, _n_, _o_, serves for two of the points in
the curve; thus, _m_ meets _q_ and _s_, while _n_ meets _p_ and _t_, and
_o_ meets the outline on each side of B, in the side view, and as _i_,
_j_, _k_ are obtained from _d_ and _e_, the lines _g_ and _h_ might have
been omitted, being inserted merely for the sake of illustration.
In Figure 230 is an example in which a cylinder intersects a cone, the
axes being parallel. To obtain the curve of intersection in this case,
the side view is divided by any convenient number of lines, as _a_, _b_,
_c_, etc., drawn at a right-angle to its axis A A, and from one end of
these lines are let fall the perpendiculars _f_, _g_, _h_, _i_, _j_;
from the ends of these (where they meet the centre line of A in the top
view), half-circles _k_, _l_, _m_, _n_, _o_, are drawn to meet the
circle of B in the top view, and from their points of intersection with
B, lines _p_, _q_, _r_, _s_, _t_, are drawn, and where these meet lines
_a_, _b_, _c_, _d_ and _e_, which is at _u_, _v_, _w_, _x_, _y_, are
points in the curve.
[Illustration: Fig. 230.]
[Illustration: Fig. 231.]
It will be observed, on referring again to Figure 229, that the branch
or cylinder B appears to be of elliptical section on its end face,
which occurs because it is seen at an angle to its end surface; now the
method of finding the ellipse for any given degree of angle is as in
Figure 231, in which B represents a cylindrical body whose top face
would, if viewed from point I, appear as a straight line, while if
viewed from point J it would appear in outline a circle. Now if viewed
from point E its apparent dimension in one direction will obviously be
defined by the lines S, Z. So that if on a line G G at a right angle to
the line of vision E, we mark points touching lines S, Z, we get points
1 and 2, representing the apparent dimension in that direction which is
the width of the ellipse. The length of the ellipse will obviously be
the full diameter of the cylinder B; hence from E as a centre we mark
points 3 and 4, and of the remaining points we will speak presently.
Suppose now the angle the top face of B is viewed from is denoted by the
line L, and lines S', Z, parallel to L, will be the width for the
ellipse whose length is marked by dots, equidistant on each side of
centre line G' G', which equal in their widths one from the other the
full diameter of B. In this construction the ellipse will be drawn away
from the cylinder B, and the ellipse, after being found, would have to
be transferred to the end of B. But since centre line G G is obviously
at the same angle to A A that A A is to G G, we may start from the
centre line of the body whose elliptical appearance is to be drawn, and
draw a centre line A A at the same angle to G G as the end of B is
supposed to be viewed from. This is done in Figure 231 _a_, in which the
end face of B is to be drawn viewed from a point on the line G G, but at
an angle of 45 degrees; hence line A A is drawn at an angle of 45
degrees to centre line G G, and centre line E is drawn from the centre
of the end of B at a right angle to G G, and from where it cuts A A, as
at F, a side view of B is drawn, or a single line of a length equal to
the diameter of B may be drawn at a right angle to A A and equidistant
on each side of F. A line, D D, at a right angle to A A, and at any
convenient distance above F, is then drawn, and from its intersection
with A A as a centre, a circle C equal to the diameter of B is drawn;
one-half of the circumference of C is divided off into any number of
equal divisions as by arcs _a_, _b_, _c_, _d_, _e_, _f_. From these
points of division, lines _g_, _h_, _i_, _j_, _k_, _l_ are drawn, and
also lines _m_, _n_, _o_, _p_, _q_, _r_. From the intersection of these
last lines with the face in the side view, lines _s_, _t_, _u_, _t_,
_w_, _x_, _y_, _z_ are drawn, and from point F line E is drawn. Now it
is clear that the width of the end face of the cylinder will appear the
same from any point of view it may be looked at, hence the sides H H are
made to equal the diameter of the cylinder B and marked up to centre
line E.
[Illustration: Fig. 231 _a_.]
[Illustration: Fig. 232.]
It is obvious also that the lines _s_, _z_, drawn from the extremes of
the face to be projected will define the width of the ellipse, hence we
have four of the points (marked respectively 1, 2, 3, 4) in the ellipse.
To obtain the remaining points, lines _t_, _u_, _v_, _w_, _x_, _y_
(which start from the point on the face F where the lines _m_, _n_, _o_,
_p_, _q_, _r_, respectively meet it) are drawn across the face of B as
shown. The compasses are then set to the radius _g_; that is, from
centre line D to division _a_ on the circle, and this radius is
transferred to the face to be projected the compass-point being rested
at the intersection of centre line G and line _t_, and two arcs as 5 and
6 drawn, giving two more points in the curve of the ellipse. The
compasses are then set to the length of line _h_ (that is, from centre
line D to point of division _b_), and this distance is transferred,
setting the compasses on centre line G where it is intersected by line
_u_, and arcs 7, 8 are marked, giving two more points in the ellipse. In
like manner points 9 and 10 are obtained from the length of line _i_, 11
and 12 from that of _j_; points 13 and 14 from the length of _k_, and 15
and 16 from _l_, and the ellipse may be drawn in from these points.
It may be pointed out, however, that since points 5 and 6 are the same
distance from G that points 15 and 16 are, and since points 7 and 8 are
the same distance from G that points 13 and 14 are, while points 9 and
10 are the same distance from G that 11 and 12 are, the lines, _j_, _k_,
_l_ are unnecessary, since _l_ and _g_ are of equal length, as are also
_h_ and _k_ and _i_ and _j_. In Figure 232 the cylinders are line shaded
to make them show plainer to the eye, and but three lines (_a_, _b_,
_c_) are used to get the radius wherefrom to mark the arcs where the
points in the ellipse shall fall; thus, radius _a_ gives points 1, 2, 3
and 4; radius _b_ gives points 5, 6, 7 and 8, and radius _c_ gives 9,
10, 11 and 12, the extreme diameter being obtained from lines S, Z, and
H, H.