title: Conchoid of de Sluze
text: # 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
  
      $r=\sec \theta +a\cos \theta \,.$
  
  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 ≠ 0). The point most distant from the asymptote is (1 + a, 0). (0,0) is a crunode for a < −1.
  
  The area between the curve and the asymptote is, for a ≥ −1,
  
      $|a|(1+a/4)\pi \,$
  
  while for a < −1, the area is
  
      $\left(1-{\frac {a}{2}}\right){\sqrt {-(a+1)}}-a\left(2+{\frac {a}{2}}\right)\arcsin {\frac {1}{\sqrt {-a}}}.$
  
  If a < −1, the curve will have a loop. The area of the loop is
  
      $\left(2+{\frac {a}{2}}\right)a\arccos {\frac {1}{\sqrt {-a}}}+\left(1-{\frac {a}{2}}\right){\sqrt {-(a+1)}}.$
  
  Four of the family have names of their own:
  
  * a = 0, line (asymptote to the rest of the family)
  * a = −1, cissoid of Diocles
  * a = −2, right strophoid
  * a = −4, trisectrix of Maclaurin
  