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In geometry, a cissoid (() is a plane curve generated from two given curves C_1, C_2 and a point O (the pole). Let L be a variable line passing through O and intersecting C_1 at P_1 and C_2 at P_2. Let P be the point on L so that (There are actually two such points but P is chosen so that P is in the same direction from O as P_2 is from P_1.) Then the locus of such points P is defined to be the cissoid of the curves C_1, C_2 relative to O. Slightly different but essentially equivalent definitions are used by different authors. For example, P may be defined to be the point so that This is equivalent to the other definition if C_1 is replaced by its reflection through O. Or P may be defined as the midpoint of P_1 and P_2; this produces the curve generated by the previous curve scaled by a factor of 1/2. If C_1 and C_2 are given in polar coordinates by and respectively, then the equation describes the cissoid of C_1 and C_2 relative to the origin. However, because a point may be represented in multiple ways in polar coordinates, there may be other branches of the cissoid which have a different equation. Specifically, C_1 is also given by So the cissoid is actually the union of the curves given by the equations It can be determined on an individual basis depending on the periods of f_1 and f_2, which of these equations can be eliminated due to duplication. For example, let C_1 and C_2 both be the ellipse The first branch of the cissoid is given by which is simply the origin. The ellipse is also given by so a second branch of the cissoid is given by which is an oval shaped curve. If each C_1 and C_2 are given by the parametric equations and then the cissoid relative to the origin is given by When C_1 is a circle with center O then the cissoid is conchoid of C_2. When C_1 and C_2 are parallel lines then the cissoid is a third line parallel to the given lines. Let C_1 and C_2 be two non-parallel lines and let O be the origin.

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Concepts associés (2)

Conchoid of de Sluze

In algebraic geometry, the conchoids of de Sluze are a family of plane curves studied in 1662 by Walloon mathematician René François Walter, baron de Sluze. The curves are defined by the polar equation In cartesian coordinates, the curves satisfy the implicit equation except that for a = 0 the implicit form has an acnode (0,0) not present in polar form. They are rational, circular, cubic plane curves. These expressions have an asymptote x = 1 (for a ≠ 0). The point most distant from the asymptote is (1 + a, 0).

Cissoïde

En géométrie, la cissoïde, ou courbe cissoïdale ou cissoïdale, de deux courbes (C) et (C) par rapport à un point fixe O est le lieu géométrique des points P tels que : où P1 est un point de (C) et P un point de (C), P et P étant alignés avec O. La cissoïde peut aussi être vue comme la courbe médiane de pôle O des courbes C' et C', images de C et C par une homothétie de centre O et de rapport 2. Elle est parfois définie comme l'ensemble des points P tels que : où P est un point de (C) et P2 un point de (C), P et P étant alignés avec O.