Tangent

A tangent (from the Latin: tangere, touch ') is in the geometry of a line that touches a given curve at a certain point. For example, the rail for the wheel is a tangent, as the contact point of the wheel, a point of contact of the two geometric objects, line and circle is. Tangent curve and have the same direction at the contact point. The tangent at this point, the best linear approximation function for the curve.

Particularly simple are the conditions during the circuit all the lines can be distinguished with respect to a circle in secants, tangents and passers-by - depending on whether they have in common with the circle two points, one or no point. The circle tangent meets the circle so in exactly one point. There it is perpendicular to the associated with this contact point radius.

In the general case, the tangent is perpendicular to the point of contact to the associated radius of the circle of curvature, provided that it exists. But you can have in common with the output curve even more points. Is another point (the output curve or other curve ) is also the contact point, then one speaks of a Bitangente.

Tangent in the Analysis

Is given the curve of the graph of a real function, the tangent at the point the straight line there is the same slope as the curve. The slope of the tangent is therefore equal to the first derivative of at the point. The equation of the tangent line is thus:

(see also: point slope form).

The tangent corresponds to the best linear approximation to the function at the point:

Differential Geometry

A curve is given by a defined on the real interval [a, b] function. Is ( with ) a point on the curve, it is called the first derivative of at the point (ie ) a tangent vector. A curve tangent at this point is a straight line through the point which has the same direction as the tangent vector.

Requirements

A tangent can only exist as a rule, if the underlying function (or the underlying functions ) differentiable at this point is / are. Comparisons to but also the following section.

A simple counter-example:

The absolute value function is not differentiable at the point. The corresponding function graph has so here it is meaningless to speak at this point a "kink " of the tangent.

But there may be a right-sided and / or a left derivative at a kink; so there can be a law - tangent and / or a left - tangent.

If a function at a point of its domain, not differentiable, the value of the derivative function strives for absolute value, however, tends to infinity, so does the function graph at this point a vertical tangent ( a line parallel to the y- axis as a tangent ). An example is the function that is defined for all real numbers, but at the position is not differentiated. There exists a vertical tangent.

Synthetic and finite geometry

In the synthetic geometry and finite geometry, the term " tangent " for suitable quantities solely in terms of incidence, are thus defined without differentiability:

The third case is for the real Euclidean plane, if you see them as affine segment of the real projective plane with the standard scalar product, which is equivalent to the fact that the gradient of the functional equation that defines the quadric, in the point of touching the line crosses the quadric, a normal vector of this line is. In this respect one can be compared to the real, defined by deriving a generalized, " algebraic " tangent term also form through formal gradient calculation.

See also the illustration in the introduction comparisons: The radius is marked with a right angle symbol of the circle also represents the direction of a normal vector of the drawn tangent and ( from the center oriented toward the touch point) represents the direction of the gradient equation of the circle in the point of contact