right|thumb|300px|the conchoid of de sluze for several values of a the conchoid(s) of de sluze is a family of plane curves studied in 1662 by renÃ© franÃ§ois walter, baron de sluze.. the curves are defined by the polar equation :r=\sec\theta+a\cos\theta \,. in cartesian coordinates, the curves satisfy the implicit equation :(x-1)(x^2+y^2)=ax^2 \, 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&ne;0). the point most distant from the asymptote is (1+a,0). (0,0) is a crunode for a<&minus;1. the area between the curve and the asymptote is, for a \ge -1, :|a|(1+a/4)\pi \, while for a < -1, the area is :\left(1-\frac a2\right)\sqrt{-(a+1)}-a\left(2+\frac a2\right)\arcsin\frac1{\sqrt{-a}}. if a<-1, the curve will have a loop. the area of the loop is :\left(2+\frac a2\right)a\arccos\frac1{\sqrt{-a}} + \left(1-\frac a2\right)\sqrt{-(a+1)}. four of the family have names of their own: :a=0, line (asymptote to the rest of the family) :a=&minus;1, cissoid of diocles :a=&minus;2, right strophoid :a=&minus;4, trisectrix of maclaurin ==references== 
Algebraic_curves Curves