Elements of descriptive geometry, with applications to isometric projection and othe forms of one-plane projection; a text-book for colleges and ingineering schools by O. E. Randall.

170

DESCRIPTIVE GEOMETRY

CASE 2. To draw a rectilinear tangent to the curve of intersection.
Analysis. See Problem 249, Case 2.
Construction. See Fig. 155. Let it be required to draw a rectilinear tangent to the curve at the point R.
Draw the plane Z tangent to the cone along the element B-R-P.
The intersection of the planes Z and S is the required tangent.
This intersection pierces H at y,, pierces V at c', pierces the
plane of the base at I, and must also pass through R. Therefore
Y-R-C-I is the tangent sought.
CASE 3. To find the true size of the intersection.
Analysis and Construction. See Fig. 155. Revolve the plane S
about S-s, into H. After revolution any point, as N, will fall at
nH, and other points may be found in the same way.*
The true size of the curve of intersection is represented by
nH —H-XH-O HkH-rH.
CASE 4. To develop the surface.
Analysis and Construction. Take the plane Z, Fig. 155, as the
plane of development. Starting with the element of tangency,
B-P, in this plane, roll the cone along this plane toward V, bringing the successive elements into contact with the plane Z.
Since the cone is one of revolution and the plane of the base is
perpendicular to the axis, the curve of the circular base will roll
out into the arc of a circle whose radius is equal to the slant
height of the cone.
The character of the development is shown in Fig. 156, where
B represents the vertex of the cone. The arc P-D-M-Q-A-L-P',
with center at B, with radius equal to the slant height of the cone,
and equal in length to the rectification of the curve of the circular
base of the cone, represents the development of the base.
To represent the elements in development, make P-D, P-M, P-Q,
etc., Fig. 156, such that the rectification of their arcs shall be equal
to the rectification of the corresponding arcs of the same name upon
the base of the original cone, and connect these points with B.
To develop the curve of intersection, lay off from B on the elements now in development, Fig. 156, the distances B-R, B-K, B-O,
* In the revolution of these points, distances from the axis S-s, have all been
diminished by a constant quantity.