C. Auguste Dupin of life into it?" Many tropes that would later become commonplace in detective fiction first appeared in Poe's stories: the eccentric but brilliant detective, the bumbling constabulary, the first-person narration by a close personal friend. Dupin also initiates the storytelling device where the detective announces his solution and then explains the reasoning leading up to it. Like Sherlock Holmes, Dupin uses his considerable deductive prowess and observation to solve crimes. Poe also portrays the police in an unsympathetic manner as a sort of foil to the detective. The character helped established the genre of detective fiction, distinct from mystery fiction,
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