# Tangent

## In mathematics, straight line touching a plane curve without crossing it / From Wikipedia, the free encyclopedia

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In geometry, the **tangent line** (or simply **tangent**) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve.^{[1]}^{[2]} More precisely, a straight line is tangent to the curve *y* = *f*(*x*) at a point *x* = *c* 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.

The point where the tangent line and the curve meet or intersect is called the * point of tangency*. The tangent line is said to be "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point.
The tangent line to a point on a differentiable curve can also be thought of as a

*tangent line approximation*, the graph of the affine function that best approximates the original function at the given point.

^{[3]}

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.

The word "tangent" comes from the Latin *tangere*, "to touch".