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 SINGLE CURVATURE

125

upon this circumference. The rectilinear tangent to the circle at
this point is a line of the required plane and should pierce H and V
in the traces already found.
303. Problem 179. Given a cylinder whose base is a circle on V
and whose axis is oblique to H and V; required to assume a point
upon the surface and to draw a plane tangent to the surface at this
point.
304. Problem 180. Given a cylinder whose axis is oblique to H
and V, the plane of whose base is perpendicular to H but oblique to V,
and the vertical projection of whose base is a circle; required to
assume a point upon the surface and to draw a plane tangent to the
surface at this point.
305. Problem 181. Given a cylinder whose axis is [A = 0, 4, 0;
B = 0, 8, 6] and whose base on H is a circle of 3-unit radius;
required to assume a point upon the surface and to draw a plane
tangent to the surface at this point.
306. Problem 182. Given a cylinder whose axis is oblique to H and
V and whose base is a circle in a plane parallel to H and above it;
required to assume a point upon the surface and to draw a plane
tangent to the surface at this point.
307. Problem 183. To draw a plane tangent to a cylinder and
through a point without the surface.
Analysis. The required plane will contain an element of the
surface of the cylinder. Therefore a straight line through the
given point and parallel to the elements of the cylinder will be
a line of the required plane (see Section 42) and should pierce H
and V in points of the required traces.
Since the required plane will be tangent to the surface all along
the element of tangency, any plane oblique to the elements of the
cylinder will cut from the surface a curved line, from the auxiliary
line parallel to the elements a point, and from the required plane
a straight line passing through this point and tangent to the curve
at a point on the element of tangency. Therefore, if through
any point of the auxiliary line we draw a plane cutting from the
surface of the cylinder a curved line, the straight line through
the assumed point and tangent to the curve will be a line of the
required plane.