C. Auguste Dupin prefect "G", who appears in all three stories seeking his counsel. In "The Murders in the Rue Morgue", Dupin investigates the murder of a mother and daughter in Paris. He investigates another murder in "The Mystery of Marie Rogêt". This story was based on the true story of Mary Rogers, a saleswoman at a cigar store in Manhattan whose body was found floating in the Hudson River in 1841. Dupin's final appearance, in "The Purloined Letter", features an investigation of a letter stolen from the French queen. Poe called this story "perhaps, the best of my tales of ratiocination". Throughout
Dupin cyclide of Dupin cyclides is invariant under the larger group of all Lie sphere transformations; any two Dupin cyclides are Lie-equivalent. They form (in some sense) the simplest class of Lie-invariant surfaces after the spheres, and are therefore particularly significant in Lie sphere geometry. The definition also means that a Dupin cyclide is the envelope of the one-parameter family of spheres tangent to three given mutually tangent spheres. It follows that it is tangent to infinitely many Soddy's hexlet configurations of spheres. A further special property of a cyclide is: An elliptic cyclide can be represented parametrically by the following formulas