Editing Analytic geometry (section)

==Spherical and nonlinear planes and their tangents==
Tangent is the linear approximation of a spherical or other curved or twisted line of a function.

===Tangent lines and planes===
{{main|Tangent}}
In [[geometry]], the '''tangent line''' (or simply '''tangent''') to a plane [[curve]] at a given [[Point (geometry)|point]] is the [[straight line]] that "just touches" the curve at that point. Informally, it is a line through a pair of [[infinitesimal|infinitely close]] points on the curve. More precisely, a straight line is said to be a tangent of a curve {{nowrap|1=''y'' = ''f''(''x'')}} at a point {{nowrap|1=''x'' = ''c''}} on the curve if the line passes through the point {{nowrap|(''c'', ''f''(''c''))}} on the curve and has slope {{nowrap|''f''{{'}}(''c'')}} where ''f''{{'}} is the [[derivative]] of ''f''. A similar definition applies to [[space curve]]s 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 (mathematics)|surface]] at a given point is the [[Plane (mathematics)|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]].