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''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|Node]]; [[Cusp(2)|Cusp]].

====References====
<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>