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.

SURFACES OF DOUBLE CURVATURE

147

340. Problem 215. Given an ellipsoid of revolution with the long
axis parallel to G-L; required to assume a point upon the surface
and to draw a plane tangent to the surface at this point.
341. Problem 216. To draw a plane tangent to any surface of
revolution.
Analysis. The directions given in connection with the solution
of the foregoing problems will be sufficient for the solution of all
problems relative to tangency of planes to surfaces of revolution.
342. Problem 217. To draw a plane containing a given straight
line and tangent to a given sphere.
Analysis 1. After the required plane is constructed, an auxiliary
plane through the center of the sphere and perpendicular to the
given line will cut from the sphere a great circle, from the line a
point, and from the required plane a straight line passing through
the point and tangent to the great circle. Therefore, through the
center of the sphere and perpendicular to the given line, pass a
plane cutting the sphere in a great circle and cutting the line in
a point. Through this point and tangent to the circle draw a
straight line. This line together with the given line will determine
the required plane.
Analysis 2. If through any point of the given line a series of
rectilinear tangents to the sphere be drawn, they will form the
elements of a cone whose axis will contain the point on the line
and the center of the sphere. The surface of this cone will be
tangent to the sphere in the circumference of a small circle whose
plane will be perpendicular to the axis of the cone.
A plane tangent to the cone must also be tangent to the sphere.
Therefore a plane through the given line and tangent to the cone
will be the required plane.
Analysis 3. If from each of two points on the given line a series
of rectilinear tangents to the sphere be drawn, they will form the
elements of two cones whose axes will contain the points on the
line and the center of the sphere, and whose surfaces will be tangent
to the sphere in the circumferences of small circles, which may be
taken as the bases of the cones. The planes of these small circles
will be perpendicular to the axes of their respective cones, and will
intersect in a straight line which will be a chord of the sphere.