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192 PROBLEMS IN GEOMETRICAL DRAWING.
Involute.
An involute is a curved line which may be assumed to be
generated in the following manner : Suppose a string be placed
around a cylinder from a to b, in the
fig. 28 c, direction 01 the arrow (see Fig. 28),
and having a pencil attached at b ;
keep the string tight and move the
pencil toward c, and the involute,
b c, is generated.
To draw an involute.
Solution
:
From the point b, (see Fig. 28)
set off any number of radial lines at
equal distances, as 1, 2, 3,"
4, 5. From
points of intersection draw the tangents (perpendicular to the
radial lines). Set off on the first tangent the length of the arc 1
to b ; on the second tangent the arc 2 to b, etc. This will give
the points through which to draw the involute.
To draw a spiral from a given
point, c.
Solution
:
Draw the line a b through the
point c. Set off the centers r and
S, one-fourth as far from c as the
distance is to be between two lines
in the spiral. Using r as center,
describe the arc from c to 1, and
using S as center, describe the arc
from 1 to 2 ; using r as center, de-
scribe the arc from 2 to 3, etc.
If a cone (see
Conical Sections.
Fig. 30), is cut by a plane on the line a b,
which is parallel to the center line, the
section will be a hyperbola.
If cut by a plane on the line c d, which
is parallel to the side, the section will be a
parabola.
If cut by a plane on the linej^, which
is parallel to the base line, the section will
be a circle.
If cut by a line, e f, which is
neither parallel to the side, the center-
line nor the base, the section will be an
ellipse.
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