Conchoid

Conchoid (Gr. κόγχη, shell, and εἶδος, form), a plane curve invented by the Greek mathematician Nicomedes, who devised a mechanical construction for it and applied it to the problem of the duplication of the cube, the construction of two mean proportionals between two given quantities, and possibly to the trisection of an angle as in the 8th lemma of Archimedes. Proclus grants Nicomedes the credit of this last application, but it is disputed by Pappus, who claims that his own discovery was original. The conchoid has been employed by later mathematicians, notably Sir Isaac Newton, in the construction of various cubic curves.  
  
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The conchoid is generated as follows:—Let O be a fixed point and BC a fixed straight line; draw any line through O intersecting BC in P and take on the line PO two points X, X′, such that PX = PX′ = a constant quantity. Then the locus of X and X′ is the conchoid. The conchoid is also the locus of any point on a rod which is constrained to move so that it always passes through a fixed point, while a fixed point on the rod travels along a straight line. To obtain the equation to the curve, draw AO perpendicular to BC, and let AO = a; let the constant quantity PX = PX′ = b. Then taking O as pole and a line through O parallel to BC as the initial line, the polar equation is r = a cosec θ ± b, the upper sign referring to the branch more distant from O. The cartesian equation with A as origin and BC as axis of x is x²y² = (a + y)² (b² - y²). Both branches belong to the same curve and are included in this equation. Three forms of the curve have to be distinguished according to the ratio of a to b. If a be less than b, there will be a node at O and a loop below the initial point (curve 1 in the figure); if a equals b there will be a cusp at O (curve 2); if a be greater than b the curve will not pass through O, but from the cartesian equation it is obvious that O is a conjugate point (curve 3). The curve is symmetrical about the axis of y and has the axis of x for its asymptote.

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