Given: A right circular cone, green in the diagram. The next example shows, that the embedding of a surface into a threefold orthogonal system is not unique.Įxamples Right circular cone Special examples are systems of confocal conic sections.ĭupin's theorem is a tool for determining the curvature lines of a surface by intersection with suitable surfaces (see examples), without time-consuming calculation of derivatives and principal curvatures. The idea of threefold orthogonal systems can be seen as a generalization of orthogonal trajectories. Hence, only the curvature lines of the cylinder are of interest: A horizontal plane intersects a cylinder at a circle and a vertical plane has lines with the cylinder in common.

A plane has no curvature lines, because any normal curvature is zero. The set of curvature lines of a right circular cylinder consists of the set of circles (maximal curvature) and the lines (minimal curvature).

But this example is of no interest, because a plane has no curvature lines.Ī simple example with at least one pencil of curved surfaces: 1) all right circular cylinders with the z-axis as axis, 2) all planes, which contain the z-axis, 3) all horizontal planes (see diagram).Ī curvature line is a curve on a surface, which has at any point the direction of a principal curvature (maximal or minimal curvature). The most simple example of a threefold orthogonal system consists of the coordinate planes and their parallels. The intersection curve of any pair of surfaces of different pencils of a threefold orthogonal system is a curvature line.Ī threefold orthogonal system of surfaces consists of three pencils of surfaces such that any pair of surfaces out of different pencils intersect orthogonally.In differential geometry Dupin's theorem, named after the French mathematician Charles Dupin, is the statement: C.Two planes (purple, blue) as members of a threefold orthogonal system intersect a cylinder at curvature lines (blue circle, purble line).Auguste Dupin by writer Gerard Dole might actually have been adventures of Charles Dupin himself - or perhaps that they were, in fact, one and the same individual. A more interesting idea is Vincent Mollet's cited theory that some of the cases ascribed to C. Charles's wife is identified as Amelie de Lagardère, a relative of the Delgardies, and they are credited with a daughter named Augustine who is said to have bore Nero Wolfe as son to Mycroft Holmes, a proposition that few Wold Newton enthusiasts would likely support. According to this theory, Charles is also the ancestor of a notorious gentleman thief, as well as father to a brilliant detective. The French Wold Newton Universe website also refers to Vincent Mollet's suggestion that Gerard Dole's Dupin stories were actually, at least partly, adventures of his father - or that both men were actually one and the same.Īccording to the scholars behind the French Wold Newton Universe website, the surname 'Dupin' is actually Poe's name for the 'Lupin' family, and that Charles Dupin's mother had had an affair with her husband's (supposed) coachman Louis Lupin. In the French Wold Newton Universe, it is argued that Dupin's real name is Lupin and that he married Antoinette Corday by whom he had a son Theophraste Lupin, the petty thief outclassed by his own offspring, Arsène Lupin. In Tarzan Alive, Philip José Farmer mentions that Dupin's father, Charles Dupin, was one of twin sons to Honoré Delagardie and his wife, who had been present at the Wold Newton Meteorite strike. Dupin is also one of many literary characters to appear in The League of Extraordinary Gentleman by Alan Moore and Kevin O'Neill. Subsequent pastiches have been penned by other authors, notably Michael Harrison in The Exploits of the Chevalier Dupin (1968) and its expanded edition The Murders in the Rue Royale. Nevertheless, on occasion Dupin gives the benefit of his intellect and wide reading to the solving of crimes in Paris in the 1840s, sometimes at the request of the authorities, in a series of stories by the American author Edgar Allan Poe: Rather, he spends his days sitting in gloom with a close friend reading through copious volumes of forgotten lore, through the courtesies of his family's creditors. The Chevalier Dupin is generally considered the first character in detective fiction, although he is not by profession a detective.

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