  Dupin cyclides are interesting algebraic surfaces used in geometric design
and architecture to join canal surfaces smoothly and to construct model
surfaces. Dupin cyclides are special cases of Darboux cyclides, which in turn
are rather general surfaces in $\mathbb R^3$ of degree 3 or 4. This article
derives the algebraic conditions for recognition of Dupin cyclides among the
general implicit form of Darboux cyclides. We aim at practicable sets of
algebraic equations on the coefficients of the implicit equation, each such set
defining a complete intersection (of codimension 4) locally. Additionally, the
article classifies all real surfaces and lower dimensional degenerations
defined by the implicit equation for Dupin cyclides.
