Tacnode

A plane curve singularity where two distinct smooth branches meet with higher-order tangency (osculation), so that they share the same tangent direction and meet with intersection multiplicity greater than two; a standard local model is y^2 = x^4 or two branches with contact of order two.

A tacnode is a planar singularity where two smooth branches have a common tangent and osculate to higher order at their contact point; geometrically the branches meet with higher-order tangency rather than transversely.