Tangent

For the tangent function see trigonometric functions. For other uses, see tangent (disambiguation).

Tangent to a curve

Tangent plane to a sphere

In geometry, the tangent line (or simply the tangent) to a curve at a given point is the straight line that "just touches" the curve at that point (in the sense explained more precisely below). As it passes through the point where the tangent line and the curve meet, or the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point. The same definition applies to space curves and curves in n-dimensional Euclidean space.

Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space.

The word "tangent" comes from the Latin tangere, meaning "to touch".

Tangent line to a curve

A tangent, a chord, and a secant to a circle

The intuitive notion that a tangent line "just 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.

In most cases, the tangent to a curve does not cross the curve at the point of tangency (though it may, when continued, cross the curve at other places away from the point of tangent) This is true, for example, of all tangents to a circle or a parabola. However, at exceptional points called inflection points, the tangent line does cross the curve at the point of tangency. An example is the point (0,0) on the graph of the cubic parabola y = x³.

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 the triangle—where the tangent line does not exist for the reasons explained above. In convex geometry, such lines are called supporting lines.

At each point, the line is always tangent to the curve. Its slope is the derivative; the positivity, negativity and zeroes of the derivative are marked by green, red and black respectively.

Analytical approach

The geometric 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 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, 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

${\frac {f(a+h)-f(a)}{h}}.$

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 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 derivative 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, 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 is too badly behaved to admit 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. The tangent line to this curve at the origin is vertical.

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 (although in a certain sense, there are two half-tangents, corresponding to two possible directions of approaching the origin).

Equations

When the curve is given by y = f(x) then the slope of the tangent is ${\frac {dy}{dx}}$ so, by the Point–slope formula the equation of the tangent line at (x, y) is

$Y-y={\frac {dy}{dx}}(X-x)$

where (X, Y) are the coordinates of a point on the line.

When the equation of the curve is given in the from f(x, y) = 0 then the value of the slope can be found by implicit differentiation, giving

${\frac {dy}{dx}}=-{\frac {\frac {\partial f}{\partial x}}{\frac {\partial f}{\partial y}}}.$

The equation of the tangent line is then

${\frac {\partial f}{\partial x}}(X-x)+{\frac {\partial f}{\partial y}}(Y-y)=0.$

For algebraic curves, 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

${\frac {\partial g}{\partial x}}x+{\frac {\partial g}{\partial y}}y+{\frac {\partial g}{\partial z}}z=ng(x,y,z)=0.$

It follows that the homogeneous equation of the tangent line is

${\frac {\partial g}{\partial x}}X+{\frac {\partial g}{\partial y}}Y+{\frac {\partial g}{\partial z}}Z=0.$

The equation of the tangent line in Cartesian coordinates can be found by setting Z=1 in this equation.

To apply this to algebraic curves, write f(x, y) as

$f=u_{n}+u_{n-1}+\dots +u_{1}+u_{0}\,$

where each u_r is the sum of all terms of degree r. The homogeneous equation of the curve is then

$g=u_{n}+u_{n-1}z+\dots +u_{1}z^{n-1}+u_{0}z^{n}=0.\,$

Applying the equation above and setting Z=1 produces

${\frac {\partial f}{\partial x}}X+{\frac {\partial f}{\partial y}}Y+u_{n-1}+\dots +(n-1)u_{1}+nu_{0}=0$

as the equation of the tangent line. 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

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

then the slope of the tangent is

${\frac {dy}{dx}}={\frac {\frac {dy}{dt}}{\frac {dx}{dt}}}$

giving the equation for the tangent line:

${\frac {dx}{dt}}(Y-y)={\frac {dy}{dt}}(X-x)$

Normal line to a curve

The line perpendicular to the tangent line to a curve at point 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

$-{\frac {1}{\frac {dy}{dx}}}$

and it follows that the equation of the normal line is

$(X-x)+{\frac {dy}{dx}}(Y-y)=0.$

Similarly, if the equation of the curve has the form f(x, y) = 0 then the equation of the tangent line is given by

${\frac {\partial f}{\partial y}}(X-x)-{\frac {\partial f}{\partial x}}(Y-y)=0.$

If the curve is given parametrically by

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

then the slope of the normal is

${\frac {dx}{dt}}(X-x)+{\frac {dy}{dt}}(Y-y)=0.$

Angle between curves

See also: Angle § 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.

Multiple tangents at the origin

The limaçon trisectrix, a curve having two tangents at the origin.

The formulas above fail when the point is a singular point. In this case there may be two or more branches of the curve which 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, 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

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

Expanding this and eliminating all but terms of degree 2 gives

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

which, when factored, becomes

$y=\pm {\sqrt {3}}x.$

So these are the equations of the two tangent lines through the origin.

Tangent circles

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_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 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 distance between their centres is equal to the difference between their radii.

$\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}=\left(r_{1}-r_{2}\right)^{2}.\,$

Surfaces and higher-dimensional manifolds

Main article: Tangent space

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. 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
• Surface normal
• Osculating circle
• Subtangent
• Tangent cone
• Tangential angle
• Tangential component
• Tangent lines to circles
• Perpendicular
• Supporting line

References

• J. Edwards (1892). Differential Calculus. London: MacMillan and Co. pp. 143 ff.

External links

• Weisstein, Eric W. "Tangent Line". MathWorld.
• Tangent to a circle With interactive animation
• Tangent and first derivative - An interactive simulation
• The Tangent Parabola by John H. Mathews