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{{About||the tangent function|Tangent (trigonometry)|other uses|Tangent (disambiguation)}}
[[Image:Tangent to a curve.svg|220px|right|thumb|Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot.]]
[[Image:Image Tangent-plane.svg|220px|right|thumb|Tangent plane to a sphere]]

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. [[Leibniz]] defined it as the line through a pair of [[infinitesimal|infinitely close]] points on the curve.<ref>Leibniz, G., "[[Nova Methodus pro Maximis et Minimis]]", ''[[Acta Eruditorum]]'', Oct. 1684.</ref> More precisely, a straight line is said to be a tangent of a curve {{nowrap|''y'' {{=}} ''f''(''x'')}} at a point {{nowrap|''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 (topology)|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]].

The word "tangent" comes from the [[Latin]] {{lang|la|[[wikt:en:tangere#Latin|tangere]]}}, "to touch".

==History==

[[Euclid]] makes several references to the tangent ({{lang|grc|ἐφαπτομένη}} ''ephaptoménē'') to a circle in book III of the ''[[Euclid's Elements|Elements]]''  (c. 300 BC).<ref>{{cite web|last1=Euclid|title=Euclid's Elements|url=http://aleph0.clarku.edu/~djoyce/elements/bookIII/bookIII.html|accessdate=1 June 2015}}</ref> In [[Apollonius of Perga|Apollonius]] work ''Conics'' (c. 225 BC) he defines a tangent as being ''a line such that no other straight line could
fall between it and the curve''.<ref name="Shenk">{{cite web|last1=Shenk|first1=Al|title=e-CALCULUS Section  2.8|url=http://math.ucsd.edu/~ashenk/Section2_8.pdf|pages=2.8|accessdate=1 June 2015}}</ref>

[[Archimedes]] (c.  287 – c.  212 BC) found the tangent to an [[Archimedean spiral]] by considering the path of a point moving along the curve.<ref name="Shenk"/>

In the 1630s [[Fermat]] developed the technique of [[adequality]] to calculate tangents and other problems in analysis and used this to calculate tangents to the parabola. The technique of adeqality is similar to taking the difference between <math>f(x+h)</math> and <math>f(x)</math> and dividing by a power of <math>h</math>. Independently [[Descartes]] used his [[method of normals]] based on the observation that the radius of a circle is always normal to the circle itself.<ref>{{cite book|last=Katz|first=Victor J.|year=2008|title=A History of Mathematics|edition=3rd|publisher=Addison Wesley|isbn=978-0321387004|page=510}}</ref>

These methods led to the development of [[differential calculus]] in the [[17th century]]. Many people contributed. [[Gilles de Roberval|Roberval]] discovered a general method of drawing tangents, by considering a curve as described by a moving point whose motion is the resultant of several simpler motions.<ref>{{cite journal|last=Wolfson|first=Paul R.|year=2001|title=The Crooked Made Straight: Roberval and Newton on Tangents| journal=The American Mathematical Monthly | volume=108 | number=3 | pages=206–216 | doi=10.2307/2695381}}</ref>
[[René-François de Sluse]] and [[Johannes Hudde]] found algebraic algorithms for finding tangents.<ref>{{cite book|last=Katz|first=Victor J.|year=2008|title=A History of Mathematics|edition=3rd|publisher=Addison Wesley|isbn=978-0321387004|pages=512–514}}</ref> Further developments included those of [[John Wallis]] and [[Isaac Barrow]], leading to the theory of [[Isaac Newton]] and [[Gottfried Leibniz]].
 
An 1828 definition of a tangent was "a right line which touches a curve, but which when produced, does not cut it".<ref>Noah Webster, ''American Dictionary of the English Language'' (New York: S. Converse, 1828), vol. 2, p. 733, [https://archive.org/stream/americandictiona02websrich#page/n733/mode/2up]</ref> This old definition prevents [[inflection point]]s from having any tangent. It has been dismissed and the modern definitions are equivalent to those of [[Gottfried Wilhelm Leibniz|Leibniz]] who defined the tangent line as the line through a pair of [[infinitesimal|infinitely close]] points on the curve.

==Tangent line to a curve==
[[Image:CIRCLE LINES.svg|thumb|220px|A tangent, a [[chord (geometry)|chord]], and a [[secant line|secant]] to a circle]]
The intuitive notion that a tangent line "touches" a curve can be made more explicit by considering the sequence of straight lines ([[secant line]]s) passing through two points, ''A'' and ''B'', those that lie on the function curve. The tangent at ''A'' is the limit when point ''B'' approximates or tends to ''A''. The existence and uniqueness of the tangent line depends on a certain type of mathematical smoothness, known as "differentiability." For example, if two circular arcs meet at a sharp point (a vertex) then there is no uniquely defined tangent at the vertex because the limit of the progression of secant lines depends on the direction in which "point ''B''" approaches the vertex.

At most points, the tangent touches the curve without crossing it (though it may, when continued, cross the curve at other places away from the point of tangent). A point where the tangent (at this point) crosses the curve is called an ''[[inflection point]]''. [[Circle]]s, [[parabola]]s, [[hyperbola]]s and [[ellipse]]s do not have any inflection point, but more complicated curves do have, like the graph of a [[cubic function]], which has exactly one inflection point, or a sinusoid, which has two inflection points per each [[Periodic function|period]] of the [[sine]].

Conversely, it may happen that the curve lies entirely on one side of a straight line passing through a point on it, and yet this straight line is not a tangent line. This is the case, for example, for a line passing through the vertex of a [[triangle]] and not intersecting it otherwise—where the tangent line does not exist for the reasons explained above. In [[convex geometry]], such lines are called [[supporting hyperplane|supporting lines]].

[[Image:Graph of sliding derivative line.gif|left|thumb|400px|At each point, the moving line is always tangent to the [[curve]]. Its slope is the [[derivative]]; green marks positive derivative, red marks negative derivative and black marks zero derivative. The point (x,y) = (0,1) where the tangent intersects the curve, is not a [[Maxima and minima|max]], or a min, but is a [[point of inflection]].]]

===Analytical approach===
The geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or the '''tangent line problem,''' was one of the central questions leading to the development of [[calculus]] in the 17th century. In the second book of his ''[[La Geometrie|Geometry]]'', [[René Descartes]]<ref>{{cite book |publisher=Courier [http://store.doverpublications.com/0486600688.html Dover] |isbn=0-486-60068-8 |pages=95 |last=Descartes |first=René |title=The geometry of René Descartes |year=1954 |url-access=registration |url=https://archive.org/details/geometryofrenede00rend }}</ref> [[s:fr:Page:Descartes La Géométrie.djvu/52|said]] of the problem of constructing the tangent to a curve, "And I dare say that this is not only the most useful and most general problem in geometry that I know, but even that I have ever desired to know".<ref>{{cite journal |author=R. E. Langer |date=October 1937 |title=Rene Descartes |journal=[[American Mathematical Monthly]] |volume=44 |issue=8 |pages=495–512 |publisher=Mathematical Association of America |doi=10.2307/2301226 |jstor=2301226}}</ref>

====Intuitive description====
Suppose that a curve is given as the graph of a [[function (mathematics)|function]], ''y'' = ''f''(''x''). To find the tangent line at the point ''p'' = (''a'', ''f''(''a'')), consider another nearby point ''q'' = (''a'' + ''h'', ''f''(''a'' + ''h'')) on the curve. The [[slope]] of the [[secant line]] passing through ''p'' and ''q'' is equal to the [[difference quotient]]

: <math>\frac{f(a+h)-f(a)}{h}.</math>

As the point ''q'' approaches ''p'', which corresponds to making ''h'' smaller and smaller, the difference quotient should approach a certain limiting value ''k'', which is the slope of the tangent line at the point ''p''. If ''k'' is known, the equation of the tangent line can be found in the point-slope form:

: <math> y-f(a) = k(x-a).\,</math>

====More rigorous description====
To make the preceding reasoning rigorous, one has to explain what is meant by the difference quotient approaching a certain limiting value ''k''. The precise mathematical formulation was given by [[Augustin-Louis Cauchy|Cauchy]] in the 19th century and is based on the notion of [[limit of a function|limit]]. Suppose that the graph does not have a break or a sharp edge at ''p'' and it is neither plumb nor too wiggly near ''p''. Then there is a unique value of ''k'' such that, as ''h'' approaches 0, the difference quotient gets closer and closer to ''k'', and the distance between them becomes negligible compared with the size of ''h'', if ''h'' is small enough. This leads to the definition of the slope of the tangent line to the graph as the limit of the difference quotients for the function ''f''. This limit is the [[Derivative#Definition via difference quotients|derivative]] of the function ''f'' at ''x'' = ''a'', denoted ''f''&nbsp;′(''a''). Using derivatives, the equation of the tangent line can be stated as follows:

: <math> y=f(a)+f'(a)(x-a).\,</math>

Calculus provides rules for computing the derivatives of functions that are given by formulas, such as the [[power function]], [[trigonometric functions]], [[exponential function]], [[logarithm]], and their  various combinations. Thus, equations of the tangents to graphs of all these functions, as well as many others, can be found by the methods of calculus.

====How the method can fail====
Calculus also demonstrates that there are functions and points on their graphs for which the limit determining the slope of the tangent line does not exist. For these points the function ''f'' is ''non-differentiable''. There are two possible reasons for the method of finding the tangents based on the limits and derivatives to fail: either the geometric tangent exists, but it is a vertical line, which cannot be given in the point-slope form since it does not have a slope, or the graph exhibits one of three behaviors that precludes a geometric tangent.

The graph ''y'' = ''x''<sup>1/3</sup> illustrates the first possibility:  here the difference quotient at ''a'' = 0 is equal to ''h''<sup>1/3</sup>/''h'' = ''h''<sup>−2/3</sup>, which becomes very large as ''h'' approaches 0. This curve has a tangent line at the origin that is vertical.

The graph ''y'' = ''x''<sup>2/3</sup> illustrates another possibility: this graph has a ''[[Cusp (singularity)|cusp]]'' at the origin. This means that, when ''h'' approaches 0, the difference quotient at ''a'' = 0 approaches plus or minus infinity depending on the sign of ''x''. Thus both branches of the curve are near to the half vertical line for which ''y''=0, but none is near to the negative part of this line. Basically, there is no tangent at the origin in this case, but in some context one may consider this line as a tangent, and even, in [[algebraic geometry]], as a ''double tangent''.

The graph ''y'' = |''x''| of the [[absolute value]] function consists of two straight lines with different slopes joined at the origin. As a point ''q'' approaches the origin from the right, the secant line always has slope 1. As a point ''q'' approaches the origin from the left, the secant line always has slope −1. Therefore, there is no unique tangent to the graph at the origin. Having two different (but finite) slopes is called a ''corner''.

Finally, since differentiability implies continuity, the [[Contraposition|contrapositive]] states ''discontinuity'' implies non-differentiability. Any such jump or point discontinuity will have no tangent line. This includes cases where one slope approaches positive infinity while the other approaches negative infinity, leading to an infinite jump discontinuity

===Equations===
When the curve is given by ''y'' = ''f''(''x'') then the slope of the tangent is <math>\frac{dy}{dx},</math>
so by the [[Linear equation#Point–slope form|point–slope formula]] the equation of the tangent line at (''X'',&nbsp;''Y'') is
:<math>y-Y=\frac{dy}{dx}(X) \cdot (x-X)</math>
where (''x'',&nbsp;''y'') are the coordinates of any point on the tangent line, and where the derivative is evaluated at <math>x=X</math>.<ref name=E191>Edwards Art. 191</ref>

When the curve is given by ''y'' = ''f''(''x''), the tangent line's equation can also be found<ref>Strickland-Constable, Charles, "A simple method for finding tangents to polynomial graphs", ''[[Mathematical Gazette]]'', November 2005, 466–467.</ref> by using [[polynomial division]] to divide <math>f \, (x)</math> by <math>(x-X)^2</math>; if the remainder is denoted by <math>g(x)</math>, then the equation of the tangent line is given by

:<math>y=g(x).</math>

When the equation of the curve is given in the form  ''f''(''x'',&nbsp;''y'') = 0 then the value of the slope can be found by [[Implicit and explicit functions#Implicit differentiation|implicit differentiation]], giving
:<math>\frac{dy}{dx}=-\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}}.</math>
The equation of the tangent line at a point (''X'',''Y'') such that ''f''(''X'',''Y'') = 0 is then<ref name=E191/>
:<math>\frac{\partial f}{\partial x}(X,Y) \cdot (x-X)+\frac{\partial f}{\partial y}(X,Y) \cdot (y-Y)=0.</math>
This equation remains true if <math>\frac{\partial f}{\partial y}(X,Y) = 0</math> but <math>\frac{\partial f}{\partial x}(X,Y) \neq 0</math> (in this case the slope of the tangent is infinite). If <math>\frac{\partial f}{\partial y}(X,Y) =  \frac{\partial f}{\partial x}(X,Y)  =0, </math> the tangent line is not defined and the point (''X'',''Y'') is said [[singular point of a curve|singular]].

{{clear}}
For [[algebraic curve]]s, computations may be simplified somewhat by converting to [[homogeneous coordinate]]s. Specifically, let the homogeneous equation of the curve be ''g''(''x'',&nbsp;''y'',&nbsp;''z'') = 0 where ''g'' is a homogeneous function of degree ''n''. Then, if (''X'',&nbsp;''Y'',&nbsp;''Z'') lies on the curve, [[Homogeneous function#Positive homogeneity|Euler's theorem]] implies
<math display=block>\frac{\partial g}{\partial x} \cdot X +\frac{\partial g}{\partial y} \cdot Y+\frac{\partial g}{\partial z} \cdot Z=ng(X, Y, Z)=0.</math>
It follows that the homogeneous equation of the tangent line is
:<math>\frac{\partial g}{\partial x}(X,Y,Z) \cdot x+\frac{\partial g}{\partial y}(X,Y,Z) \cdot y+\frac{\partial g}{\partial z}(X,Y,Z) \cdot z=0.</math>
The equation of the tangent line in Cartesian coordinates can be found by setting ''z''=1 in this equation.<ref name=E192>Edwards Art. 192</ref>

To apply this to algebraic curves, write ''f''(''x'',&nbsp;''y'') as
:<math>f=u_n+u_{n-1}+\dots+u_1+u_0\,</math>
where each ''u''<sub>''r''</sub> is the sum of all terms of degree ''r''. The homogeneous equation of the curve is then
:<math>g=u_n+u_{n-1}z+\dots+u_1 z^{n-1}+u_0 z^n=0.\,</math>
Applying the equation above and setting ''z''=1 produces
:<math>\frac{\partial f}{\partial x}(X,Y) \cdot x + \frac{\partial f}{\partial y}(X,Y) \cdot y + \frac{\partial g}{\partial z}(X,Y,1) =0</math>
as the equation of the tangent line.<ref name=E193>Edwards Art. 193</ref> The equation in this form is often simpler to use in practice since no further simplification is needed after it is applied.<ref name=E192 />

If the curve is given [[Parametric equation|parametrically]] by
:<math>x=x(t),\quad y=y(t)</math>
then the slope of the tangent is
:<math>\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}</math>
giving the equation for the tangent line at <math>\, t=T, \, X=x(T), \, Y=y(T)</math> as<ref name=E196>Edwards Art. 196</ref>
:<math>\frac{dx}{dt}(T) \cdot (y-Y)=\frac{dy}{dt}(T) \cdot (x-X).</math>
If <math>\frac{dx}{dt}(T)= \frac{dy}{dt}(T) =0, </math> the tangent line is not defined. However, it may occur that the tangent line exists and may be computed from an implicit equation of the curve.

===Normal line to a curve===
The line perpendicular to the tangent line to a curve at the point of tangency is called the ''normal line'' to the curve at that point. The slopes of perpendicular lines have product −1, so if the equation of the curve is ''y'' = ''f''(''x'') then slope of the normal line is
:<math>-\frac{1}{\frac{dy}{dx}}</math>
and it follows that the equation of the normal line at (X, Y) is
:<math>(x-X)+\frac{dy}{dx}(y-Y)=0.</math>
Similarly, if the equation of the curve has the form ''f''(''x'',&nbsp;''y'') = 0 then the equation of the normal line is given by<ref name=E194>Edwards Art. 194</ref>
:<math>\frac{\partial f}{\partial y}(x-X)-\frac{\partial f}{\partial x}(y-Y)=0.</math>

If the curve is given parametrically by
:<math>x=x(t),\quad y=y(t)</math>
then the equation of the normal line is<ref name=E196 />
:<math>\frac{dx}{dt}(x-X)+\frac{dy}{dt}(y-Y)=0.</math>

===Angle between curves===
{{see also|Angle#Angles between curves}}
The angle between two curves at a point where they intersect is defined as the angle between their tangent lines at that point. More specifically, two curves are said to be tangent at a point if they have the same tangent at a point, and orthogonal if their tangent lines are orthogonal.<ref name=E195>Edwards Art. 195</ref>

===Multiple tangents at a point===
[[Image:LimaçonTrisectrix.svg|right|thumb|300px|The limaçon trisectrix: a curve with two tangents at the origin.]]
The formulas above fail when the point is a [[Singular point of a curve|singular point]]. In this case there may be two or more branches of the curve that pass through the point, each branch having its own tangent line. When the point is the origin, the equations of these lines can be found for algebraic curves by factoring the equation formed by eliminating all but the lowest degree terms from the original equation. Since any point can be made the origin by a change of variables (or by [[Translation (geometry)|translating]] the curve) this gives a method for finding the tangent lines at any singular point.

For example, the equation of the [[limaçon trisectrix]] shown to the right is
:<math>(x^2+y^2-2ax)^2=a^2(x^2+y^2).\,</math>
Expanding this and eliminating all but terms of degree 2 gives
:<math>a^2(3x^2-y^2)=0\,</math>
which, when factored, becomes
:<math>y=\pm\sqrt{3}x.</math>
So these are the equations of the two tangent lines through the origin.<ref name=E197>Edwards Art. 197</ref>

When the curve is not self-crossing, the tangent at a reference point may still not be uniquely defined because the curve is not differentiable at that point although it is differentiable elsewhere. In this case the [[left and right derivative]]s are defined as the limits of the derivative as the point at which it is evaluated approaches the reference point from respectively the left (lower values) or the right (higher values). For example, the curve ''y'' = |''x'' | is not differentiable at ''x'' = 0: its left and right derivatives have respective slopes –1 and 1; the tangents at that point with those slopes are called the left and right tangents.<ref>Thomas, George B. Jr., and Finney, Ross L. (1979), ''Calculus and Analytic Geometry'', Addison Wesley Publ. Co.: p. 140.</ref>

Sometimes the slopes of the left and right tangent lines are equal, so the tangent lines coincide. This is true, for example, for the curve ''y'' = ''x'' <sup>2/3</sup>, for which both the left and right derivatives at ''x'' = 0 are infinite; both the left and right tangent lines have equation ''x'' = 0.

==Tangent circles==
{{Main|Tangent circles}}

[[File:Tangent circles.svg|thumb|200px|Two pairs of tangent circles. Above internally and below externally tangent]]Two circles of non-equal radius, both in the same plane, are said to be tangent to each other if they meet at only one point. Equivalently, two [[circles]], with [[radii]] of ''r<sub>i</sub>'' and centers at (''x<sub>i</sub>'', ''y<sub>i</sub>''), for ''i''&nbsp;=&nbsp;1,&nbsp;2 are said to be tangent to each other if

:<math>\left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2=\left(r_1\pm r_2\right)^2.\,</math>
* Two circles are '''externally tangent'''  if the [[Distance#Geometry|distance]] between their centres is equal to the sum of their radii.
:<math>\left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2=\left(r_1 + r_2\right)^2.\,</math>
* Two circles are '''internally tangent'''  if the [[Distance#Geometry|distance]] between their centres is equal to the difference between their radii.<ref>[http://homepage.eircom.net/~phabfys/circles.html Circles For Leaving Certificate Honours Mathematics by Thomas O’Sullivan 1997]</ref>
:<math>\left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2=\left(r_1 - r_2\right)^2.\,</math>

==Surfaces and higher-dimensional manifolds==
{{Main|Tangent space}}

The ''tangent plane'' to a [[Surface (topology)|surface]] at a given point ''p'' is defined in an analogous way to the tangent line in the case of curves. It is the best approximation of the surface by a plane at ''p'', and can be obtained as the limiting position of the planes passing through 3 distinct points on the surface close to ''p'' as these points converge to ''p''. More generally, there is a ''k''-dimensional [[tangent space]] at each point of a ''k''-dimensional [[manifold]] in the ''n''-dimensional [[Euclidean space]].

==See also==
* [[Newton's method]]
* [[Normal (geometry)]]
* [[Osculating circle]]
* [[Osculating curve]]
* [[Perpendicular]]
* [[Subtangent]]
* [[Supporting line]]
* [[Tangent cone]]
* [[Tangential angle]]
* [[Tangential and normal components|Tangential component]]
* [[Tangent lines to circles]]
* [[Multiplicity (mathematics)#Behavior of a polynomial function near a multiple root]]
* [[Algebraic curve#Tangent at a point]]

==References==
{{Reflist}}

==Sources==
* {{cite book|author=J. Edwards|title=Differential Calculus|publisher=MacMillan and Co.|location=London|pages=143 ff.|year=1892|url=https://books.google.com/books?id=unltAAAAMAAJ&pg=PA143#v=onepage&q&f=false}}

==External links==
{{commons category|Tangency}}
{{Collier's Poster|Tangent}}
* {{springer|title=Tangent line|id=p/t092170}}
* {{MathWorld|title=Tangent Line|urlname=TangentLine}}
* [http://www.mathopenref.com/tangent.html Tangent to a circle] With interactive animation
* [http://www.vias.org/simulations/simusoft_difftangent.html Tangent and first derivative] — An interactive simulation

[[Category:Differential geometry]]
[[Category:Differential topology]]
[[Category:Analytic geometry]]
[[Category:Elementary geometry]]