tangent
A tangent (from latin : tangere touch ') is in the geometry of a straight line having a given curve at a certain point contacts . For example, the rail for the wheel is a tangent, since the point of contact of the wheel is a point of contact between the two geometric objects, straight line and circle. Tangent and curve have the same direction at the point of contact. The tangent is the best linear approximation function for the curve at this point .
The relationships are particularly simple with a circle: All straight lines can be differentiated into secants , tangents and passers-by - depending on whether they have two points in common with the circle, one point or no point at all. The circle tangent therefore meets the circle at exactly one point. It stands there perpendicular to the contact radius belonging to this point .
In the general case, too, the tangent is perpendicular to the radius of the circle of curvature belonging to the point of contact , if this exists. However, it can have other points in common with the initial curve. If another point (of the output curve or another curve) is also the point of contact, one speaks of a bitangent .
Tangent in calculus
If the given curve is the graph of a real function , then the tangent at the point is the straight line that has the same slope as the curve there. The slope of the tangent is thus equal to the first derivative of at the site : . The equation of the tangent is thus:
(see also: Point slope form ).
The tangent corresponds to the best linear approximation for the function at the point :
- For
Differential geometry
Let a curve im be given by a function defined on the real interval . If (with ) a curve point, the first derivative of at this point is called a tangential vector . A curve tangent at this point is a straight line through the point that has the same direction as the tangent vector.
requirements
As a rule, a tangent can only exist if the underlying function (or the underlying functions) is / are differentiable at this point . But also compare the following section .
A simple counterexample:
The amount function can not be differentiated at this point . The associated function graph has a "kink" at this point, so it makes no sense to speak of the tangent here .
At a kink, however, there may be a right-hand and / or a left-hand discharge; so there can be a right tangent and / or a left tangent.
If a function is not differentiable at one point in its domain of definition, but the value of the derivative function tends towards infinity in terms of absolute value , the function graph has a perpendicular tangent at this point (a parallel to the y-axis as a tangent). An example of this is the function that is defined for all real numbers, but is not differentiable at that point . There is a vertical tangent there.
Synthetic and finite geometry
In synthetic geometry and finite geometry , the term "tangent" for suitable sets can be defined solely with terms of incidence, i.e. without any preconditions for differentiability:
- For a quadratic set , in a projective plane , a tangent is a straight line that has exactly one point in common with this set or is entirely contained in it.
- With this definition, there is exactly one tangent in each point of the oval , especially for an oval in a projective plane. No straight line has more than two points in common with the oval.
- In analytical terms , this means for a projective quadric over a Papposian projective plane , whichsatisfiesthe Fano axiom , the most important special case of a quadric set: A projective straight line is a tangent of the quadric if and only if the coefficient vector of the straight line satisfies the homogeneous quadratic equation that the Quadric (defined as a set of points).
The third case is for the real Euclidean plane , if it is viewed as an affine section of the real projective plane with the standard scalar product , tantamount to the fact that the gradient of the functional equation that defines the quadric is at the point where the straight line forms the quadric is a normal vector of this straight line. In this respect, an “algebraic” tangent term, which is defined by derivation, and the real one, can also be formed by formal gradient calculation.
Compare also the figure in the introduction: The radius of the circle marked with the right angle symbol simultaneously represents the direction of a normal vector of the drawn tangent and (oriented from the center to the point of contact) the direction of the gradient of the circular equation at its point of contact.
See also
Web links
Individual evidence
- ^ Albrecht Beutelspacher , Ute Rosenbaum: Projective geometry . From the basics to the applications (= Vieweg Studium: advanced course in mathematics ). 2nd, revised and expanded edition. Vieweg, Wiesbaden 2004, ISBN 3-528-17241-X , 4 square quantities ( table of contents [accessed on July 31, 2013]).