In
geometry, the tangent line (or simply tangent) to a plane
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
at a given
point is the
straight line that "just touches" the curve at that point.
Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
defined it as the 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 at a point if the line passes through the point on the curve and has slope , 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.
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
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, ...
that best approximates the original function at the given point.
Similarly, the tangent plane to a
surface at a given point is the
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
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; .
The word "tangent" comes from the
Latin , "to touch".
History
Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of ge ...
makes several references to the tangent ( ''ephaptoménē'') to a circle in book III of the ''
Elements'' (c. 300 BC). In
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''.
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists i ...
(c. 287 – c. 212 BC) found the tangent to an
Archimedean spiral
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a cons ...
by considering the path of a point moving along the curve.
In the 1630s
Fermat developed the technique of
adequality Adequality is a technique developed by Pierre de Fermat in his treatise ''Methodus ad disquirendam maximam et minimam'' to calculate tangents and other problems in analysis and used this to calculate tangents to the parabola. The technique of adequality is similar to taking the difference between
and
and dividing by a power of
. Independently
Descartes used his
method of normals In calculus, the method of normals was a technique invented by Descartes for finding normal and tangent lines to curves. It represented one of the earliest methods for constructing tangents to curves. The method hinges on the observation that th ...
based on the observation that the radius of a circle is always normal to the circle itself.
These methods led to the development of
differential calculus in the 17th century. Many people contributed.
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.
René-François de Sluse
René-François Walter de Sluse (; also Renatius Franciscus Slusius or Walther de Sluze; 2 July 1622 – 19 March 1685) was a Walloon mathematician and churchman, who served as the canon of Liège and abbot of Amay.
Biography
He was born in Vis� ...
and
Johannes Hudde
Johannes (van Waveren) Hudde (23 April 1628 – 15 April 1704) was a burgomaster (mayor) of Amsterdam between 1672 – 1703, a mathematician and governor of the Dutch East India Company.
As a "burgemeester" of Amsterdam he ordered that t ...
found algebraic algorithms for finding tangents. Further developments included those of
John Wallis and
Isaac Barrow, leading to the theory of
Isaac Newton and
Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mat ...
.
An 1828 definition of a tangent was "a right line which touches a curve, but which when produced, does not cut it". This old definition prevents
inflection point
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case o ...
s from having any tangent. It has been dismissed and the modern definitions are equivalent to those of
Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
, who defined the tangent line as the line through a pair of
infinitely close points on the curve.
Tangent line to a plane curve
The intuitive notion that a tangent line "touches" a curve can be made more explicit by considering the sequence of straight lines (
secant lines) 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
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case o ...
''.
Circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
s,
parabolas,
hyperbolas and
ellipses do not have any inflection point, but more complicated curves do have, like the graph of a
cubic function
In mathematics, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d
where the coefficients , , , and are complex numbers, and the variable takes real values, and a\neq 0. In other words, it is both a polynomial function of degree ...
, which has exactly one inflection point, or a sinusoid, which has two inflection points per each
period
Period may refer to:
Common uses
* Era, a length or span of time
* Full stop (or period), a punctuation mark
Arts, entertainment, and media
* Period (music), a concept in musical composition
* Periodic sentence (or rhetorical period), a concept ...
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 lines.
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
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
in the 17th century. In the second book of his ''
Geometry'',
René Descartes 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".
Intuitive description
Suppose that a curve is given as the graph of a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
, ''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
In single-variable calculus, the difference quotient is usually the name for the expression
: \frac
which when taken to the limit as ''h'' approaches 0 gives the derivative of the function ''f''. The name of the expression stems from the fact ...
:
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:
:
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
Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
in the 19th century and is based on the notion of
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
. 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 of the function ''f'' at ''x'' = ''a'', denoted ''f'' ′(''a''). Using derivatives, the equation of the tangent line can be stated as follows:
:
Calculus provides rules for computing the derivatives of functions that are given by formulas, such as the
power function,
trigonometric functions,
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
,
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''
1/3 illustrates the first possibility: here the difference quotient at ''a'' = 0 is equal to ''h''
1/3/''h'' = ''h''
−2/3, which becomes very large as ''h'' approaches 0. This curve has a tangent line at the origin that is vertical.
The graph ''y'' = ''x''
2/3 illustrates another possibility: this graph has a ''
cusp
A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth.
Cusp or CUSP may also refer to:
Mathematics
* Cusp (singularity), a singular point of a curve
* Cusp catastrophe, a branch of bifurc ...
'' 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
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
, as a ''double tangent''.
The graph ''y'' = , ''x'', of the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
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
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
so by the
point–slope formula the equation of the tangent line at (''X'', ''Y'') is
:
where (''x'', ''y'') are the coordinates of any point on the tangent line, and where the derivative is evaluated at
.
[Edwards Art. 191]
When the curve is given by ''y'' = ''f''(''x''), the tangent line's equation can also be found by using
polynomial division to divide
by
; if the remainder is denoted by
, then the equation of the tangent line is given by
:
When the equation of the curve is given in the form ''f''(''x'', ''y'') = 0 then the value of the slope can be found by
implicit differentiation, giving
:
The equation of the tangent line at a point (''X'',''Y'') such that ''f''(''X'',''Y'') = 0 is then
[
:
This equation remains true if but (in this case the slope of the tangent is infinite). If the tangent line is not defined and the point (''X'',''Y'') is said to be singular.
For ]algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
s, computations may be simplified somewhat by converting to homogeneous coordinates. Specifically, let the homogeneous equation of the curve be ''g''(''x'', ''y'', ''z'') = 0 where ''g'' is a homogeneous function of degree ''n''. Then, if (''X'', ''Y'', ''Z'') lies on the curve, Euler's theorem implies
It follows that the homogeneous equation of the tangent line is
:
The equation of the tangent line in Cartesian coordinates can be found by setting ''z''=1 in this equation.[Edwards Art. 192]
To apply this to algebraic curves, write ''f''(''x'', ''y'') as
:
where each ''u''''r'' is the sum of all terms of degree ''r''. The homogeneous equation of the curve is then
:
Applying the equation above and setting ''z''=1 produces
:
as the equation of the tangent line.[Edwards Art. 193] The equation in this form is often simpler to use in practice since no further simplification is needed after it is applied.
If the curve is given parametrically by
:
then the slope of the tangent is
:
giving the equation for the tangent line at as[Edwards Art. 196]
:
If 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
:
and it follows that the equation of the normal line at (X, Y) is
:
Similarly, if the equation of the curve has the form ''f''(''x'', ''y'') = 0 then the equation of the normal line is given by[Edwards Art. 194]
:
If the curve is given parametrically by
:
then the equation of the normal line is
:
Angle 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.[Edwards Art. 195]
Multiple tangents at a point
The formulas above fail when the point is a 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 translating
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
the curve) this gives a method for finding the tangent lines at any singular point.
For example, the equation of the limaçon trisectrix
In geometry, a limaçon trisectrix is the name for the quartic plane curve that is a trisectrix that is specified as a limaçon. The shape of the limaçon trisectrix can be specified by other curves particularly as a rose, conchoid or ep ...
shown to the right is
:
Expanding this and eliminating all but terms of degree 2 gives
:
which, when factored, becomes
:
So these are the equations of the two tangent lines through the origin.[Edwards Art. 197]
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 derivatives 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.
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'' 2/3, 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 line to a space curve
Tangent circles
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
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
, with radii
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
of ''ri'' and centers at (''xi'', ''yi''), for ''i'' = 1, 2 are said to be tangent to each other if
:
* Two circles are externally tangent if the distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
between their centres is equal to the sum of their radii.
:
* Two circles are internally tangent if the distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
between their centres is equal to the difference between their radii.
:
Tangent plane to a surface
The tangent plane to a 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''.
Higher-dimensional manifolds
More generally, there is a ''k''-dimensional tangent space at each point of a ''k''-dimensional manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
in the ''n''-dimensional Euclidean space.
See also
* Newton's method
* Normal (geometry)
* Osculating circle
* Osculating curve
In differential geometry, an osculating curve is a plane curve from a given family that has the highest possible order of contact with another curve.
That is, if ''F'' is a family of smooth curves, ''C'' is a smooth curve (not in general belonging ...
* Perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
* Subtangent
* Supporting line In geometry, a supporting line ''L'' of a curve ''C'' in the plane is a line that contains a point of ''C'', but does not separate any two points of ''C''."The geometry of geodesics", Herbert Busemannp. 158/ref> In other words, ''C'' lies completely ...
* Tangent cone
* Tangential angle
In geometry, the tangential angle of a curve in the Cartesian plane, at a specific point, is the angle between the tangent line to the curve at the given point and the -axis. (Some authors define the angle as the deviation from the direction of ...
* Tangential component
* Tangent lines to circles
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is 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. More ...
* Tangent vector
* Multiplicity (mathematics)#Behavior of a polynomial function near a multiple root
* Algebraic curve#Tangent at a point
References
Sources
*
External links
*
*
Tangent to a circle
With interactive animation
— An interactive simulation
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Differential geometry
Differential topology
Analytic geometry
Elementary geometry