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.

116

DESCRIPTIVE GEOMETRY

All meridian lines of the same surface of revolution are the same.
If a plane is tangent to a surface of revolution at a given point,
it will be perpendicular to the meridian plane of the surface passing
through this point. For if through the point of tangency we pass
a plane perpendicular to the axis, it will cut from the surface the
circumference of a circle, from the tangent plane a straight line tangent to the circle at the point of tangency, and from the meridian
plane a straight line which is the radius of the circle at the point
of tangency, and therefore perpendicular to the rectilinear tangent.
Through the point of tangency and parallel to the axis draw a
straight line. This line is in the meridian plane, is perpendicular
to the plane passed perpendicular to the axis, and is therefore
perpendicular to the rectilinear tangent.
Since then the tangent plane has a straight line (the rectilinear
tangent) perpendicular to two straight lines (the radius and the line
parallel to the axis) of the meridian plane, it must be perpendicular
l'C" N< ~ to the meridian plane.
s,, -( l \- S 286. Representation of Surfaces
of Revolution. Surfaces of revo_____ _ \ ~ lution are usually represented by:a!a-m -L assuming their axes perpendicular
p,,, to H, although this is not neces/I I \ sary. The intersection of the sure' I. ' face with H, or the horizontal pro< y'"I pa jection of some important section
8I |~ (of the surface made by a plane perd' |P' g, pendicular to the axis, is taken for
the horizontal projection of the surb face, and the vertical projection of
FIG. 117 the meridian curve whose plane
is parallel to V is taken as the vertical projection of the surface.
287. To assume a Point upon any Surface of Revolution.
Analysis and Construction. Let the surface of revolution be
represented as in Fig. 117, where A-B represents the axis, where
the circle e,-f, represents the horizontal projection of the largest
horizontal circle of the surface, and where d'-e'-f'-g' represents