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Definition 2025

The tacnode (at the origin) of the curve

(x^{2}+y^{2}-3x)^{2}-4x^{2}(2-x)=0

tacnode ‎(plural tacnodes)

(geometry) A point of a curve where two or more osculating circles to the curve at that point are tangent, so that two branches of the curve have ordinary tangency at the double point.
- 2012, Joe Harris, Algebraic Geometry: A First Course, page 262:
  A tacnode is defined to be a singularity equivalent to the origin in the curve $y^{2}-x^{2}=0$ , that is, the union of two smooth arcs meeting with contact of order two. Similarly, an oscnode is a singularity equivalent to $y^{2}-x^{6}=0$ , i.e., consisting of two smooth branched with contact of order three, and so on.