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Difference between revisions of "Cusp(2)"

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''cuspidal point''
 
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====Comments====
 
====Comments====
In the above the word  "branch"  is used in a naive and non-technical sense as follows. View a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027420/c0274201.png" /> as the image of a finite or infinite interval in Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027420/c0274202.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027420/c0274203.png" /> be a single-valued analytic function defined on some interval. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027420/c0274204.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027420/c0274205.png" />) defines a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027420/c0274206.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027420/c0274207.png" />, one speaks of a branch of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027420/c0274208.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027420/c0274209.png" /> is taken for convenience. There is a second more technical (and more precise) notion of a branch in algebraic and analytic geometry which defines the branches at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027420/c02742010.png" /> as the points above <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027420/c02742011.png" /> on the normalization of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027420/c02742012.png" /> (cf. [[Normal scheme|Normal scheme]]). Using this concept a cusp is a singular point of a curve which has only one branch at this point.
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In the above the word  "branch"  is used in a naive and non-technical sense as follows. View a curve $C$ as the image of a finite or infinite interval in Euclidean space $E^n$. Let $\phi$ be a single-valued analytic function defined on some interval. If $x=\phi(y)$ (or $y=\phi(x)$) defines a subset $C_0$ of $C$, one speaks of a branch of $C$. Here $n=2$ is taken for convenience. There is a second more technical (and more precise) notion of a branch in algebraic and analytic geometry which defines the branches at a point $x\in C$ as the points above $x$ on the normalization of the curve $C$ (cf. [[Normal scheme|Normal scheme]]). Using this concept a cusp is a singular point of a curve which has only one branch at this point.
  
A curve with a cusp of the first kind (Fig.a) is, e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027420/c02742013.png" />, and one with a cusp of the second kind (Fig.b) — e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027420/c02742014.png" />.
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A curve with a cusp of the first kind (Fig.a) is, e.g., $X^4+X^2Y^2+2X^2Y-XY^2+Y^2=0$, and one with a cusp of the second kind (Fig.b) — e.g. $Y^2=X^3$.
  
 
The word  "cusp"  is also used in the theory of modular forms (see [[Fuchsian group|Fuchsian group]]; [[Modular form|Modular form]]).
 
The word  "cusp"  is also used in the theory of modular forms (see [[Fuchsian group|Fuchsian group]]; [[Modular form|Modular form]]).

Revision as of 16:57, 11 April 2014

cuspidal point

A singular point of a curve, the two branches of which have a common semi-tangent there. In the case of a plane curve one distinguishes cusps of the first and the second kind. In the former case the curve lies on one side of the tangent cone (Fig.a); in the second, on different sides (Fig.b).

Figure: c027420a

Figure: c027420b


Comments

In the above the word "branch" is used in a naive and non-technical sense as follows. View a curve $C$ as the image of a finite or infinite interval in Euclidean space $E^n$. Let $\phi$ be a single-valued analytic function defined on some interval. If $x=\phi(y)$ (or $y=\phi(x)$) defines a subset $C_0$ of $C$, one speaks of a branch of $C$. Here $n=2$ is taken for convenience. There is a second more technical (and more precise) notion of a branch in algebraic and analytic geometry which defines the branches at a point $x\in C$ as the points above $x$ on the normalization of the curve $C$ (cf. Normal scheme). Using this concept a cusp is a singular point of a curve which has only one branch at this point.

A curve with a cusp of the first kind (Fig.a) is, e.g., $X^4+X^2Y^2+2X^2Y-XY^2+Y^2=0$, and one with a cusp of the second kind (Fig.b) — e.g. $Y^2=X^3$.

The word "cusp" is also used in the theory of modular forms (see Fuchsian group; Modular form).

References

[a1] R.J. Walker, "Algebraic curves" , Springer (1978)
How to Cite This Entry:
Cusp(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cusp(2)&oldid=31520
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article