tangent A straight line that touches a curve without
cutting through it at that location is called a tangent
line to the curve, and the location at which the line
and curve touch is called the point of contact. With
this said, however, this should not be taken as the pre-
cise definition of a tangent line. In general, a tangent
line is permitted to have more than one point of con-
tact, may cut through the curve at points other than
the point of contact, and may even cut through the
curve at the point of contact if this point is also an
INFLECTION POINT
. To include these possibilities,
mathematicians define the tangent line to a curve in
terms of a
LIMIT
:
The tangent line to a curve at a point Pis the
limiting position of a line through Pand
through a second point Qon the curve as that
point Qslides toward P.
But even this definition has some difficulties, for it may
be the case that the limiting position is not well-
defined. There is no unique tangent line to the corner
of a square, for example.
A curve is called smooth if every point on it has a
well-defined tangent line at that point. The theory of
DIFFERENTIAL CALCULUS
is used to find the
SLOPE
of a
tangent line to a curve.
If a curve crosses itself, then special names are
given to the point of intersection according to the
behavior of the tangent lines at that point. A point of
self-intersection is called a node (or a crunode) if there
are two distinct tangent lines at that point, and a tac-
point if the two tangent lines coincide.
If two different branches of a curve touch at a
point and have a common tangent line at that point,
then that point is called a cusp. A single cusp has two
branches meeting and terminating at a point, and a
double cusp has two branches that each continue
through the point of contact. A cusp is classified as
being of the “first kind” if the branches of the curve
belong to opposite sides of the tangent line, or of the
“second kind” if they lie on the same side (at least near
the point of contact). If one branch of a double cusp
has a point of inflection at the point of contact, then
the double cusp is called an osculinflection.
A tangent plane to a surface is a plane that touches
the surface without cutting through it. This informal
definition can be adjusted and made more precise by
considering the limit position of planes passing through
three points on the surface. If a solid
SPHERE
rests on a
tabletop, for example, then the surface of the table is a
tangent plane for the solid. At the point of contact, all
points near that point lie on the same side of the tan-
gent plane. A point on a surface is called a saddle point
if points near it lie on either side of the tangent plane at
that point. (The shape of the surface at such a point
resembles a horse’s saddle.)
In geometry, any two figures are said to be tangent
at a point Pif they touch, but do not cut through each
other, at P. In
TRIGONOMETRY
, the tangent of an angle
A, denoted tan(A),is the ratio .
See also
NORMAL TO A SURFACE
;
PARABOLOID
.
sin(A)
––––
cos(A)
492
T