Tacnode

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[edit]

[a1] |  A. Dimca, "Topics on real and complex singularities" , Vieweg (1987) pp. 175 MR1013785 Zbl 0628.14001

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[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