Asymptotic point

Graph of (with enlargement). Asymptotic point at (0 | 0) ${\displaystyle y=x\cdot \sin \left({\frac {1}{x}}\right)}$

In analysis and geometry , an asymptotic point of a curve is - broadly speaking - an asymptotic line that has degenerated into a point.

In this sense, an asymptotic point can be understood as an accumulation point of curve points that is not itself a curve point.

y = (x² − 1) / (x − 1); continuously eliminable definition gap at x = 1

A definition gap that can be continuously corrected is also an asymptotic point.
Often, however, the term is restricted to points that the curve circles an infinite number of times without finally reaching them. This is not possible with function graphs; Spirals are typical for this type of asymptotic point .

The logarithmic spiral has one asymptotic point , the clothoid two.

A precise definition of this concept of asymptotic point is not very easy. It amounts to the fact that the tangent angle of the curve tends towards the limit value ± ± as it approaches the accumulation point .

In this form, the term is also used in spherical geometry .

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  Logarithmic spiral

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  Clothoid

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  Loxodrome

Individual evidence

1. ^ Lexicon of Mathematics. Heidelberg, Berlin (spectrum) 2002, article "Asymptote"
2. ↑ Naas, Schmid: Mathematical dictionary. Berlin, Stuttgart (Teubner) 1965³