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