geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, 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 In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segmen ...

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 ma ...

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 In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

of ''f''. A similar definition applies to

space 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 ...

s and curves in ''n''-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

. 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 that best approximates the original function at the given point. Similarly, the tangent plane to a

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

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 Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...

, "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 ...

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 (c. 287 – c. 212 BC) found the tangent to an Archimedean spiral by considering the path of a point moving along the curve. In the 1630s

Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is ...

developed the technique of

adequality Adequality is a technique developed by Pierre de Fermat in his treatise ''Methodus ad disquirendam maximam et minimam''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 the ...

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 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 Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...

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 mathem ...

. 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 ...

s from having any tangent. It has been dismissed and the modern definitions are equivalent to those of

, 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 line Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to: * a secant line, in geometry * the secant variety, in algebraic geometry * secant (trigonometry) (Latin: secans), the multiplicative inverse (or recipr ...

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 ''

''.

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 ...

parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...

hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...

s 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 A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

and not intersecting it otherwise—where the tangent line does not exist for the reasons explained above. In

convex geometry In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of num ...

, 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 Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

'',

René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...

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 In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...

of the

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 ...

\frac.

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: :

y-f(a) = k(x-a).\,

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. 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

of the function ''f'' at ''x'' = ''a'', denoted ''f'' ′(''a''). Using derivatives, the equation of the tangent line can be stated as follows: :

y=f(a)+f'(a)(x-a).\,

Calculus provides rules for computing the derivatives of functions that are given by formulas, such as the

power function Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...

trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...

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, ...

logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...

, 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, 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

contrapositive In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition. The contrapositive of a statem ...

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

\frac,

so by the point–slope formula the equation of the tangent line at (''X'', ''Y'') is :

y-Y=\frac(X) \cdot (x-X)

where (''x'', ''y'') are the coordinates of any point on the tangent line, and where the derivative is evaluated at

x=X

.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 In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalized version of the familiar arithmetic technique called long division. It can be done easily by hand, beca ...

to divide

f \, (x)

(x-X)^2

; if the remainder is denoted by

g(x)

, then the equation of the tangent line is given by :

y=g(x).

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 In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit func ...

, giving :

\frac=-\frac.

The equation of the tangent line at a point (''X'',''Y'') such that ''f''(''X'',''Y'') = 0 is then :

\frac(X,Y) \cdot (x-X)+\frac(X,Y) \cdot (y-Y)=0.

This equation remains true if

\frac(X,Y) = 0

but

\frac(X,Y) \neq 0

(in this case the slope of the tangent is infinite). If

\frac(X,Y) =  \frac(X,Y)  =0,

the tangent line is not defined and the point (''X'',''Y'') is said to be

singular Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular homology * SINGULAR, an open source Computer Algebra System (CAS) * Singular or sounder, a group of boar, ...

. 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 coordinate In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. ...

s. 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

\frac \cdot X +\frac \cdot Y+\frac \cdot Z=ng(X, Y, Z)=0.

It follows that the homogeneous equation of the tangent line is :

\frac(X,Y,Z) \cdot x+\frac(X,Y,Z) \cdot y+\frac(X,Y,Z) \cdot z=0.

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 :

f=u_n+u_+\dots+u_1+u_0\,

where each ''u''_''r'' is the sum of all terms of degree ''r''. The homogeneous equation of the curve is then :

g=u_n+u_z+\dots+u_1 z^+u_0 z^n=0.\,

Applying the equation above and setting ''z''=1 produces :

\frac(X,Y) \cdot x + \frac(X,Y) \cdot y + \frac(X,Y,1) =0

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 A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

by :

x=x(t),\quad y=y(t)

then the slope of the tangent is :

\frac=\frac

giving the equation for the tangent line at

\, t=T, \, X=x(T), \, Y=y(T)

asEdwards Art. 196 :

\frac(T) \cdot (y-Y)=\frac(T) \cdot (x-X).

\frac(T)= \frac(T) =0,

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 :

-\frac

and it follows that the equation of the normal line at (X, Y) is :

(x-X)+\frac(y-Y)=0.

Similarly, if the equation of the curve has the form ''f''(''x'', ''y'') = 0 then the equation of the normal line is given byEdwards Art. 194 :

\frac(x-X)-\frac(y-Y)=0.

If the curve is given parametrically by :

x=x(t),\quad y=y(t)

then the equation of the normal line is :

\frac(x-X)+\frac(y-Y)=0.

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 ''transl ...

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 epitro ...

shown to the right is :

(x^2+y^2-2ax)^2=a^2(x^2+y^2).\,

Expanding this and eliminating all but terms of degree 2 gives :

a^2(3x^2-y^2)=0\,

which, when factored, becomes :

y=\pm\sqrtx.

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 const ...

, 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 ''r_i'' and centers at (''x_i'', ''y_i''), for ''i'' = 1, 2 are said to be tangent to each other if :

\left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2=\left(r_1\pm r_2\right)^2.\,

* 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. :

\left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2=\left(r_1 + r_2\right)^2.\,

* Two circles are internally tangent if the

between their centres is equal to the difference between their radii. :

\left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2=\left(r_1 - r_2\right)^2.\,

Tangent plane to a surface

The tangent plane to a

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 In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...

at each point of a ''k''-dimensional manifold in the ''n''-dimensional

References

Sources

External links

* *
Tangent to a circle
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