In mathematics, straight line touching a plane curve without crossing it
For the tangent function, see Tangent (trigonometry). For other uses, see Tangent (disambiguation).
Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot.Tangent plane to a sphere
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".
^In "Nova Methodus pro Maximis et Minimis" (Acta Eruditorum, Oct. 1684), Leibniz appears to have a notion of tangent lines readily from the start, but later states: "modo teneatur in genere, tangentem invenire esse rectam ducere, quae duo curvae puncta distantiam infinite parvam habentia jungat, seu latus productum polygoni infinitanguli, quod nobis curvae aequivalet", ie. defines the method for drawing tangents through points infinitely close to each other.
^Thomas L. Hankins (1985). Science and the Enlightenment. Cambridge University Press. p. 23. ISBN 9780521286190.
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