CONCHOID (kou'koid) (Gk, so-ixeetaie, kon chorides, mussel-shaped, from Konh, konelle, shell + citlog, form) OF NIC'OME'DES. A 'shell-shaped' curve invented by Nicomedes (n.c. 180). It is related to the problems of tri secting an angle (see TRISECTION PROBLEsI), of constructing two geometric means between two given straight lines, and of duplicating the cube. The curve may he constructed by drawing a straight line' LM for the directrix, and through any point P as the pole drawing, a pencil of lines cutting LM in 11„ The conchoid is the locus of points found by laying ofT a constant length each way from R,. 11„,. . .. on these rays. This constant length is called the modulus. The curve differs in general shape according as the modulus is equal to, greater than, or less than the distance of the fixed point from the fixed straight line. The figure shows the forms of the curve in the last two eases. The loop occurs when the modulus is greater than the perpendion lar distance of P and LAI. When the modulus
equals this distance. P is a cusp on the curve. The directrix LSl is an asymptote to the two brandies of the curve. If tlw foot of the per pendicular from the pole to the directrix be taken as the origin, and the distance be called b, and the modulus (t, the equation of the conehoid is (y = 0. Its order is the fourth, and its class the sixth unless P is a cusp, in which ease its class is the fifth. (See CURVES.) P is, in general, a double point, and the curve meets its asymptote at four consecu tive points at infinity. The curve may easily be described mechanically, and is frequently used in architecture as a bounding line of the vertical section of columns. Consult: Sundara Row, Geometric Paper Folding (Chicago, 1901) ; Klein, Fortriige fiber ausgewiihlte Fragen der Elementargeotnetrir ( Leipzig, 1895). translated as Fam-ous Problems of Geometry (Boston, 1897).