In geometry, the tangent line to a plane curve at a given
point is the straight line that "just touches" the curve at that
point. Informally, it is a line through a pair of infinitely close points on
the curve. More precisely, a straight line is said to be a tangent of a curve y
= f(x) at a point x = c on the curve if the line passes through the point (c,
f(c)) on the curve and has slope f'(c) where f' is the derivative of f. A
similar definition applies to space curves and curves in n-dimensional
Euclidean space.
As it passes through the point where the tangent line and the
curve meet, called the point of tangency, the tangent line is "going in
the same direction" as the curve, and is thus the best straight-line
approximation to the curve at that point. Similarly, the tangent plane to a
surface at a given point is the plane that "just touches" the surface
at that point. The concept of a tangent is one of the most fundamental notions
in differential geometry and has been extensively generalized; see Tangent
space.
