Touch (math)

The touch is a concept from the mathematical branch of differential geometry . Two geometric objects such as function graphs , curves or curved surfaces touch each other at a common point if the tangents of the two objects coincide at this point. This point is called the point of contact. The tangents can be determined with the help of differential calculus .

Generally speaking, there is a -th order contact ${\displaystyle n}$ at a common point if all derivatives up to the -th order match at this point. ${\displaystyle n}$

Touching two functions

Let be two functions defined on the interval that are differentiable at an inner point of the interval . Then the functions touch each other and exactly at the point when ${\displaystyle f,g\colon I\to \mathbb {R} }$ ${\displaystyle I\subset \mathbb {R} }$ ${\displaystyle a}$ ${\displaystyle I}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle a}$

${\displaystyle f(a)=g(a)\qquad {\text{und}}\qquad f'(a)=g'(a)}$

applies.

Touching two curves

The concept of contact between two differentiable functions can easily be transferred to two curves with a differentiable path .

Let and be two curves with a differentiable path, where is an interval. Does a point exist with ${\displaystyle \gamma \colon I\to \mathbb {R} ^{n}}$ ${\displaystyle {\bar {\gamma }}\colon I\to \mathbb {R} ^{n}}$ ${\displaystyle I\subset \mathbb {R} }$ ${\displaystyle a\in I}$

${\displaystyle \gamma (a)={\bar {\gamma }}(a)\qquad {\text{und}}\qquad \gamma '(a)={\bar {\gamma }}'(a)}$

then the point of contact of the two curves is called and . ${\displaystyle a}$ ${\displaystyle \gamma (a)}$ ${\displaystyle {\bar {\gamma }}}$

Correspondingly, a point of contact of the -th order of two curves is called with at least a -fold differentiable path if all derivatives of the two curves coincide at the point . ${\displaystyle a\in I\subset \mathbb {R} }$ ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle a}$ ${\displaystyle k}$

At every point on a curve in which the tangent does not touch the curve in a higher order, there is a clearly defined circle which touches the curve in this point in a higher order. It is called the circle of curvature or oscillation circle. For example, the unit circle around the origin is the oscillation circle of the cosine function in the point . ${\displaystyle (0,1)}$

See also

• Contact between circles and triangles: inscribed circle , angled circle
• Implementation in technology: cogwheel , toothing law

Individual evidence

1. ↑ Touching two functions . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .
2. ↑ Heinrich Brauner : Differential Geometry . Vieweg, Braunschweig 1981, ISBN 3-528-03809-8 , p. 81.