TheInfoList

In
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

, 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 (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

at a given
point Point or points may refer to: Places * Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...
is the
straight line 290px, A representation of one line segment. In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature In mathematics, curvature is any of several str ...

that "just touches" the curve at that point.
Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the "#1666–1676, 1666–1676" section. ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist, and diplomat. He is a promin ...

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 of a function, argument (input value). Derivatives are a fundament ...

of ''f''. A similar definition applies to
space curve In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s and curves in ''n''-dimensional
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
. 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 an affinity (from the Latin, ''affinis'', "connected with"), is a geometric transformation that preserves line (geometry), lines and parallelism (geometry), parallelism (but not necessarily Eucli ...
that best approximates the original function at the given point. Similarly, the tangent plane to a
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
at a given point is the
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
and has been extensively generalized; . The word "tangent" comes from the
Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an appa ...

, "to touch".

# History

Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

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 (; grc, ; ; ) was a Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its popula ...

(c.  287 – c.  212 BC) found the tangent to an
Archimedean spiral The Archimedean spiral (also known as the arithmetic spiral) is a spiral In , a spiral is a which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the are:
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 lawyer at the '' Parlement'' of Toulouse Toulouse ( , ; oc, Tolosa ; la, Tolosa ) is the capital of the French departments of France, department ...

developed the technique of
adequality Adequality is a technique developed by Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French people, French mathematician who is given credit for early developments that led to infinitesimal ...
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 $f\left(x+h\right)$ and $f\left(x\right)$ and dividing by a power of $h$. Independently used his
method of normalsIn calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. 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 mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (fro ...
and
Johannes Hudde Johannes (van Waveren) Hudde (23 April 1628 – 15 April 1704) was a burgomaster Manneken Pis dressed as a burgomaster from the Seven Noble Houses of Brussels. Burgomaster (alternatively spelled burgermeister, literally ''master of the cit ...
found algebraic algorithms for finding tangents. Further developments included those of
John Wallis John Wallis (; la, Wallisius; ) was an English clergyman and mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), f ...

and
Isaac Barrow Isaac Barrow (October 1630 – 4 May 1677) was an English Christian theologian and mathematician who is generally given credit for his early role in the development of infinitesimal calculus; in particular, for the discovery of the fundamental ...

, leading to the theory of
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics a ...

and
Gottfried Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the "#1666–1676, 1666–1676" section. ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist, and diplomat. He is a promin ...
. 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathem ...

s from having any tangent. It has been dismissed and the modern definitions are equivalent to those of
Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the "#1666–1676, 1666–1676" section. ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist, and diplomat. He is a promin ...

, who defined the tangent line as the line through a pair of infinitely close points on the curve.

# Tangent line to a 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 In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
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 In differential calculus In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathem ...

''.
Circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

s,
parabola In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

s,
hyperbola File:Hyperbel-def-ass-e.svg, 300px, Hyperbola (red): features In mathematics, a hyperbola () (adjective form hyperbolic, ) (plural ''hyperbolas'', or ''hyperbolae'' ()) is a type of smooth function, smooth plane curve, curve lying in a plane, defi ...

s and
ellipse In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

s do not have any inflection point, but more complicated curves do have, like the graph of a
cubic function 210px, Graph of a cubic function with 3 real roots (where the curve crosses the horizontal axis—where ). The case shown has two critical points. Here the function is . In mathematics Mathematics (from Ancient Greek, Greek: ) includes ...

, 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 * Period, a descriptor for a historical or period drama ...

of the
sine In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

. 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 In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The ...

and not intersecting it otherwise—where the tangent line does not exist for the reasons explained above. In
convex geometryIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, 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 mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

in the 17th century. In the second book of his ''
Geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...
'',
René Descartes René Descartes ( or ; ; Latinisation of names, Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, Mathematics, mathematician, and scientist who invented analytic geometry, linking the previously sep ...

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 function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
, ''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 Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', ...

of the
secant line In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
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 of a function, limit as ''h'' approaches 0 gives the derivative of the Function (mathematics), function ''f''. The ...
: $\frac.$ 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: : $y-f\left(a\right) = k\left(x-a\right).\,$

### 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 was ...

in the 19th century and is based on the notion of
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...

. 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
of the function ''f'' at ''x'' = ''a'', denoted ''f'' ′(''a''). Using derivatives, the equation of the tangent line can be stated as follows: : $y=f\left(a\right)+f\text{'}\left(a\right)\left(x-a\right).\,$ Calculus provides rules for computing the derivatives of functions that are given by formulas, such as the
power function Exponentiation is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europ ...

,
trigonometric functions In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

,
exponential function The exponential function is a mathematical function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of ...

,
logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

, 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 Cusp may refer to: Mathematics *Cusp (singularity) In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
'' 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

, as a ''double tangent''. The graph ''y'' = , ''x'', of the
absolute value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities 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 In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...
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 :$y-Y=\frac\left(X\right) \cdot \left(x-X\right)$ 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 to divide $f \, \left(x\right)$ by $\left(x-X\right)^2$; if the remainder is denoted by $g\left(x\right)$, then the equation of the tangent line is given by :$y=g\left(x\right).$ 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, giving :$\frac=-\frac.$ The equation of the tangent line at a point (''X'',''Y'') such that ''f''(''X'',''Y'') = 0 is then :$\frac\left(X,Y\right) \cdot \left(x-X\right)+\frac\left(X,Y\right) \cdot \left(y-Y\right)=0.$ This equation remains true if $\frac\left(X,Y\right) = 0$ but $\frac\left(X,Y\right) \neq 0$ (in this case the slope of the tangent is infinite). If $\frac\left(X,Y\right) = \frac\left(X,Y\right) =0,$ the tangent line is not defined and the point (''X'',''Y'') is said to be
singular Singular may refer to: * Singular, the grammatical number In linguistics, grammatical number is a grammatical category of nouns, pronouns, adjectives, and verb agreement (linguistics), agreement that expresses count distinctions (such as "one", ...
. 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 cu ...
s, computations may be simplified somewhat by converting to
homogeneous coordinate Homogeneity and heterogeneity are concepts often used in the sciences Science (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally ...
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 In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that if ''n'' and ''a'' are coprime In number theory, two integer An integer (from the Latin wikt:integer#Latin, ''integer'' mea ...
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 :$\frac\left(X,Y,Z\right) \cdot x+\frac\left(X,Y,Z\right) \cdot y+\frac\left(X,Y,Z\right) \cdot z=0.$ 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 :$\frac\left(X,Y\right) \cdot x + \frac\left(X,Y\right) \cdot y + \frac\left(X,Y,1\right) =0$ 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 by :$x=x\left(t\right),\quad y=y\left(t\right)$ then the slope of the tangent is :$\frac=\frac$ giving the equation for the tangent line at $\, t=T, \, X=x\left(T\right), \, Y=y\left(T\right)$ asEdwards Art. 196 :$\frac\left(T\right) \cdot \left(y-Y\right)=\frac\left(T\right) \cdot \left(x-X\right).$ If $\frac\left(T\right)= \frac\left(T\right) =0,$ 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 :$\left(x-X\right)+\frac\left(y-Y\right)=0.$ 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 :$\frac\left(x-X\right)-\frac\left(y-Y\right)=0.$ If the curve is given parametrically by :$x=x\left(t\right),\quad y=y\left(t\right)$ then the equation of the normal line is :$\frac\left(x-X\right)+\frac\left(y-Y\right)=0.$

## 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 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 :$\left(x^2+y^2-2ax\right)^2=a^2\left(x^2+y^2\right).\,$ Expanding this and eliminating all but terms of degree 2 gives :$a^2\left(3x^2-y^2\right)=0\,$ 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 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 A shape or figure is the form of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type. A p ...

, with radii of ''ri'' and centers at (''xi'', ''yi''), for ''i'' = 1, 2 are said to be tangent to each other if :$\left\left(x_1-x_2\right\right)^2+\left\left(y_1-y_2\right\right)^2=\left\left(r_1\pm r_2\right\right)^2.\,$ * Two circles are externally tangent if the
distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

between their centres is equal to the sum of their radii. :$\left\left(x_1-x_2\right\right)^2+\left\left(y_1-y_2\right\right)^2=\left\left(r_1 + r_2\right\right)^2.\,$ * Two circles are internally tangent if the
distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

between their centres is equal to the difference between their radii.Circles For Leaving Certificate Honours Mathematics by Thomas O’Sullivan 1997
/ref> :$\left\left(x_1-x_2\right\right)^2+\left\left(y_1-y_2\right\right)^2=\left\left(r_1 - r_2\right\right)^2.\,$

# Surfaces

The ''tangent plane'' to a
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
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 , the tangent space of a 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 manifold at a point can ...
at each point of a ''k''-dimensional
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

in the ''n''-dimensional
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
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Newton's method In numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathem ...

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Normal (geometry) In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...
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Osculating circle In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point ''p'' on the curve has been traditionally defined as the circle passing through ''p'' and a pair of additional points on the curve in ...

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Osculating curve Image:Osculating circle.svg, A curve ''C'' containing a point ''P'' where the Radius of curvature (mathematics), radius of curvature equals ''r'', together with the tangent line and the osculating circle touching ''C'' at ''P'' In differential geom ...
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Perpendicular In elementary geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...

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Supporting lineIn geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...
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Tangent cone In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...
* Tangential angle * Tangential component *
Tangent lines to circlesIn Euclidean geometry, Euclidean plane geometry, a tangent line to a circle is a line that touches the circle at exactly one point, never entering the circle's interior. Tangent lines to circles form the subject of several theorems, and play an impor ...
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Tangent vector :''For a more general — but much more technical — treatment of tangent vectors, see tangent space.'' In mathematics, a tangent vector is a Vector (geometry), vector that is tangent to a curve or Surface (mathematics), surface at a given point. T ...
* Multiplicity (mathematics)#Behavior of a polynomial function near a multiple root * Algebraic curve#Tangent at a point

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