Tangent (geometry)

Tangent comes from Latin tangere , touch: in Geometry, the tangent with a Courbe in one of its points is a line which “touches” the curve with more close in the vicinity of this point. The curve and its tangent then form a null angle in this point.

The concept of tangent makes it possible to carry out approximations: for the resolution of certain problems which require to know the behavior of the curve in the vicinity of a point, one can compare this one to his tangent. This explains the relationship between the concept of tangent and the differential Calculus.

To be satisfied sometimes as one makes it define the tangent as a line which “touches the curve without crossing it” would be incorrect, since

nothing prevents the curve from a little further recrossing one of its tangents (the concept of tangent at the point M describes well the situation only in one small vicinity of the point M ).
there are exceptional situations where the tangent in M precisely crosses the curve at the point M . It is said whereas there is a inflection in M .

The counterpart of the concept of tangent for the Surface S is that of tangent plan. It can be defined by considering the whole of the curves plotted on surface and passing by a given point, and by considering the whole of the tangents obtained. One can then generalize with objects of size larger than 2.

Geometrical definition of the tangent

One starts by defining the Droite secant between two points M and NR of the curve: it is the line which connects them. The tangent in M can then be defined as the limiting position of the secant when the point NR tends towards M .

To be made perfectly rigorous, this definition requires to introduce concepts of Topologie allowing the calculation of such a Limite. It is however very picturesque.

Example: tangent with the circle

In each one of its points the Cercle admits a tangent. The tangent in M is the line passing by M and perpendicular to the ray resulting from M .

The tangents with the circle of center O and ray R are the lines located at the Distance R of the point O . In fact also the lines cut the circle in exactly a point, but it is about a property particular to the circle.

Angle between two curves

Is two curves C and It passing by the same point M ; it is supposed that they have both of the tangents in this point.

the Angle between the tangents is called angle of the two curves in M
in the particular case where this angle is right, it is said that the two curves are orthogonal in M .
in the particular case where this angle is flat, the tangents are identical and it is said that both curved are tangent in M .

Calculations of tangent

Tangent with a graph of numerical function

Here F is a function defined on an interval of the form $] has \ alpha, a+ \ alpha \ alpha >0$ , with actual values. One is concerned with know if the graph, of equation '' y=f (X) '', admits a tangent at the point '' has '' coordinates '' (has, F (a)) ''.

The secant between the points of X-coordinate has and a+h is the line passing by has and of slope $p (H) = \ frac {F (a+h) - F (a)} {H}$ , which is a rate of variation of F . There are three possibilities

if p (H) admits a finished limit p when H tends towards 0, there is a tangent: the line passing by has and of slope p . The function is then derivable and the slope of the tangent is the value of the Dérivée.
: The equation of this tangent is then $y=f \, '(A) (x-a) +f (a)$

if p (H) admits an infinite limit, there is a tangent: line of equation x=a .
: That occurs for example for the function $f (X) = \ sqrt$
if not, not of tangent.

Tangent with a parameterized arc

This time F is a function defined on an interval of the form $] has \ alpha, a+ \ alpha \ alpha >0$ with values in a vector space '' E '' of finished size. One makes the study in the vicinity of the point of parameter '' has ''.

A first condition to be able to speak about secant is that in the vicinity of has, the curve does not pass that once by the point has . In this case, one can again calculate the slope of the secant and seek if it has a limit.

In any case, the concept of tangent does not depend on the selected Paramétrage. To prove it, it is simply a question of applying a theorem of composition of limits.

Bond with differential calculus

If F admits a vector derived not no one at the point has , it is said that has is a regular point and there is a tangent, directed by the vector f' (A) .

If F admits a succession of derivative null in then a first nonnull derivative has while going to the order p

f' (a)=f

(a)= \ dowries = f^ {(p-1)}(a)=0 \ qquad f^ {(p)}(A) \ neq 0

then there is a tangent, directed by the first nonnull derivative. In such a point one says that there is a contact order p between the curve and its tangent (whereas in a regular point the contact is only of order 1).

Attention: the French tradition is to use the “regular” word for two distinct concepts, the regularity of F like function or that of the arc. It is possible to parameterize a Carré way $\ mathcal C^ \ infty$ , which shows well that the regularity within the meaning of the functions does not give necessarily the existence of tangents. Simply, for such a parameter setting, at the tops, all the derivative will be null.

Semi-tangents

For a more precise study, one can introduce semi-tangents on the right and to on the left define the behavior for the values of the parameter strictly higher or strictly lower than has. The extra information contained in a semi-tangent is the direction of the movement.

It is said that there is a semi-tangent on the right when the following limit exists

T_d = \ lim \ limits_ {H \ to 0^+} \ frac {F (a+h) - F (a)} {\|F (a+h) - F (a) \|}

The semi-tangent is then the half-line of origin this vector

T_d

It is said that there is a semi-tangent on the left when the following limit exists (attention with the order)

T_g = \ lim \ limits_ {H \ to 0^-} \ frac {F (A) - F (a-h)}{\|F (A) - F (a-h) \|}

The semi-tangent is then the half-line of origin this vector

T_g

If there are semi-tangents, the following vocabulary is used:

not angular when the semi-tangents form a nonflat angle

The graph of the function absolute value gives an example of angular point

not of graining when the semi-tangents are opposite: there is then a tangent, but the curve makes a kind of half-turn, from where the name.

In the case of a deltoïde, one sees three points of reflection.

tangent without graining for the most frequent case: the semi-tangents are equal.

Curve in polar coordinates

If the arc admits for parameter the polar angle $\ rho (\ theta)$ , the derived vector admits for expression in the mobile base $\ rho' (\ theta) U (\ theta) + \ rho (\ theta) v (\ theta)$ .

all the points other than the origin are regular, and thus have a tangent
if the arc passes by the origin in $\ theta_0$ , then the secant $M_0M (\ theta)$ is not other than the line of angle $\ theta$ . There is thus always a tangent: line of angle $\ theta_0$ .

In any rigor, so that the secants exist there is necessary to add the condition which the arc passes only once by the origin for $\ theta$ rather small.

Tangent for an implicit curve

One considers a curve of Cartesian equation F (X, there) =C in the Euclidean plan, for a function F of class $\ mathcal C^1$ on open of the plan.

The Théorème of the implicit functions makes it possible to be brought back to an arc parameterized and of to determine existence and possible equation of the tangent to this curve in a given point. Precisely, a point M= (X, there) pertaining to the curve is known as regular when the Gradient of F is nonnull in this point. And in this case, the tangent is Orthogonal E with the vector gradient.

Position compared to the tangent

Convexity

The graph of a derivable numerical function is convex if and only if the curve is always above its tangents. It is concave if and only if the curve is below its tangents.

In the cases that one meets in practice, the curve is alternatively concave or convex on various intervals separated by points from inflection (for which the tangent crosses the curve).

One can extend to the arcs parameterized by seeking the points of inflection and the direction in which the concavity of the curve is turned. A tool for the knowledge is the calculation of the sign of the Courbure.

For example the convex concept of closed Courbe is defined, i.e. which is always located on a side of its tangents. For such a curve, the curve does not change a sign.

Use of differential calculus for the remarkable points

One places oneself in the plan, and one will proceed to the thorough study of an arc F in the vicinity of one of his points has . It is supposed that the first nonnull derivative which is that of order p , that the first derived not colinéaire with $f^ {\, (p)}(a)$ is that of order Q . There is then a Repère judicious to carry out the study: $(F (a), f^ {\, (p)}(A), f^ {\, (Q)}(A))$

In this reference mark, the arc takes the form (X (T), Y (T)) . One carries out then the Développement limited of the functions X and Y

X (T) = \ frac1 {p!} t^p + O (t^p) \ qquad Y (T) = \ frac1 {Q!} t^q + O (t^q)

Known facts are found when T tends towards 0 or towards X: X and Y tends towards 0 (continuity of the curve), the slope Y/X tends towards 0 (the tangent is given by the first basic vector). But moreover there is the sign of X and Y for T rather small. The sign of X says to us if we are ahead or behind (compared to the direction of $f^ {\, (p)}(a)$ ). The sign of Y indicates to us if we are above or below the tangent.

for p odd, Q even: X changes sign, not Y , one advances while remaining above the tangent. It is an ordinary point.

for p odd, Q odd: X and Y changes sign, one advances while crossing the tangent. It is a point of inflection.

for p even, Q odd: X does not change a sign, but Y if, one makes half-turn but while passing on other side of the tangent. It is a point of reflection of first species (case of the deltoïde above).

for p even, Q even: X and Y has an unchanged sign, one sets out again in opposite direction, while remaining same side of the tangent. It is a point of reflection of second species.

Extension to surfaces and beyond

That is to say M a point of a surface S . One considers the whole of all the curves plotted on S and passing by M and having a tangent in M . If the meeting of all the tangents thus obtained form a plan, it is called tangent Plan on the surface.

One proceeds in the same way for curved subspaces of larger size of E : the Subvariety S.

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