Namespaces
Variants
Actions

Difference between revisions of "Tacnode"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
 +
{{TEX|done}}
 
''point of osculation, osculation point, double cusp''
 
''point of osculation, osculation point, double cusp''
  
The third in the series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130010/t1300102.png" />-curve singularities. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130010/t1300103.png" /> is a tacnode of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130010/t1300104.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130010/t1300105.png" />.
+
The third in the series of $A_k$-curve singularities. The point $(0,0)$ is a tacnode of the curve $X^4-Y^2=0$ in $\mathbf R^2$.
  
The first of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130010/t1300106.png" />-curve singularities are: an ordinary double point, also called a node or crunode; the cusp, or spinode; the tacnode; and the ramphoid cusp.
+
The first of the $A_k$-curve singularities are: an ordinary double point, also called a node or crunode; the cusp, or spinode; the tacnode; and the ramphoid cusp.
  
They are exemplified by the curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130010/t1300107.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130010/t1300108.png" />.
+
They are exemplified by the curves $X^{k+1}-Y^2=0$ for $k=1,2,3,4$.
  
The terms "crunode" and "spinode" are seldom used nowadays (2000).
+
The terms "crunode" and "spinode" are seldom used nowadays (2000).
  
 
See also [[Node|Node]]; [[Cusp(2)|Cusp]].
 
See also [[Node|Node]]; [[Cusp(2)|Cusp]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Dimca,   "Topics on real and complex singularities" , Vieweg (1987) pp. 175</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R.J. Walker,   "Algebraic curves" , Princeton Univ. Press (1950) (Reprint: Dover 1962)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> Ph. Griffiths,   J. Harris,   "Principles of algebraic geometry" , Wiley (1978) pp. 293; 507</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> S.S. Abhyankar,   "Algebraic geometry for scientists and engineers" , Amer. Math. Soc. (1990) pp. 3; 60</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Dimca, "Topics on real and complex singularities" , Vieweg (1987) pp. 175 {{MR|1013785}} {{ZBL|0628.14001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R.J. Walker, "Algebraic curves" , Princeton Univ. Press (1950) (Reprint: Dover 1962) {{MR|0033083}} {{ZBL|0039.37701}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> Ph. Griffiths, J. Harris, "Principles of algebraic geometry" , Wiley (1978) pp. 293; 507 {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> S.S. Abhyankar, "Algebraic geometry for scientists and engineers" , Amer. Math. Soc. (1990) pp. 3; 60 {{MR|1075991}} {{ZBL|0709.14001}} {{ZBL|0721.14001}} </TD></TR></table>

Latest revision as of 13:06, 23 July 2014

point of osculation, osculation point, double cusp

The third in the series of $A_k$-curve singularities. The point $(0,0)$ is a tacnode of the curve $X^4-Y^2=0$ in $\mathbf R^2$.

The first of the $A_k$-curve singularities are: an ordinary double point, also called a node or crunode; the cusp, or spinode; the tacnode; and the ramphoid cusp.

They are exemplified by the curves $X^{k+1}-Y^2=0$ for $k=1,2,3,4$.

The terms "crunode" and "spinode" are seldom used nowadays (2000).

See also Node; Cusp.

References

[a1] A. Dimca, "Topics on real and complex singularities" , Vieweg (1987) pp. 175 MR1013785 Zbl 0628.14001
[a2] R.J. Walker, "Algebraic curves" , Princeton Univ. Press (1950) (Reprint: Dover 1962) MR0033083 Zbl 0039.37701
[a3] Ph. Griffiths, J. Harris, "Principles of algebraic geometry" , Wiley (1978) pp. 293; 507 MR0507725 Zbl 0408.14001
[a4] S.S. Abhyankar, "Algebraic geometry for scientists and engineers" , Amer. Math. Soc. (1990) pp. 3; 60 MR1075991 Zbl 0709.14001 Zbl 0721.14001
How to Cite This Entry:
Tacnode. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tacnode&oldid=14597
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article