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.

CLASSIFICATION OF SURFACES

87

line somewhat removed from the first. If this secant plane be
revolved about the element of tangency as an axis toward the
position of the tangent plane, the element of tangency will remain
stationary and the second line will gradually approach the position
of the first; and when the second line takes the position consecutive to the element of tangency, or practically the position of the
element of tangency itself, the secant plane will take the position
of the tangent plane.
If a plane is tangent to a single curved surface at a given point,
it will be tangent to the surface all along the rectilinear element
containing the point of tangency. For if through any point of
this element a cutting plane be passed oblique to the elements, it
will cut from the consecutive element (which is also a line of the
tangent plane) a point consecutive to the assumed point.
These two consecutive points lie in the tangent plane and at the
same time lie on the line cut from the surface by the oblique plane.
A straight line through these two points is tangent to the line cut
from the surface at the assumed point, and lies in the tangent
plane. The tangent plane is then tangent to the surface at this
point, since it contains two rectilinear tangents to the surface at
this point.
If then a plane is tangent to a single curved surface, any plane
passed oblique to the element of tangency will cut from the tangent plane a straight line which will be tangent to the line cut
from the surface at the point where the element of tangency intersects the oblique plane.
Two surfaces are tangent to each other when they are both
tangent to the same surface at a common point, or when planes
passed through their point of contact cut from the two surfaces
lines which are tangent to each other at the point of contact.
227. Normals to Surfaces. A straight line is normal to a surface
at a given point when it is perpendicular to the plane which is
tangent to the surface at that point.
A plane is normal to a surface at a given point when it contains
the rectilinear normal to the surface at that point.
Evidently there can be but one rectilinear normal to a surface
at a given point, but an infinite number of plane normals.