Conchoid from de Sluze

Conchoid by de Sluze for various ${\displaystyle a}$

The de Sluze conchoid is a family of flat curves that was examined by René François Walther de Sluze in 1662 . In polar coordinates it is expressed as follows:

${\displaystyle r=\sec \theta +a\cos \theta }$

The secant is the reciprocal function of the cosine .

The following applies to Cartesian coordinates : ${\displaystyle (x,y)}$

${\displaystyle (x-1)(x^{2}+y^{2})=ax^{2}}$

However, the Cartesian form has for a solution point that does not exist in the polar coordinate form. ${\displaystyle a=0}$ ${\displaystyle (0,0)}$

These expressions have an asymptote (for ). The point furthest from the asymptote a is . In curves intersect for yourself. ${\displaystyle x=1}$ ${\displaystyle a\neq 0}$ ${\displaystyle (1+a,0)}$ ${\displaystyle (0,0)}$ ${\displaystyle a<-1}$

The area between the curve and the asymptote is calculated as follows:

${\displaystyle |a|(1+a/4)\pi \,}$ For ${\displaystyle a\geq -1}$

${\displaystyle \left(1-{\frac {a}{2}}\right){\sqrt {-(a+1)}}-a\left(2+{\frac {a}{2}}\right)\arcsin {\frac {1}{\sqrt {-a}}}}$ For ${\displaystyle a<-1}$

The area of ​​the loop is

${\displaystyle \left(2+{\frac {a}{2}}\right)a\arccos {\frac {1}{\sqrt {-a}}}+\left(1-{\frac {a}{2}}\right){\sqrt {-(a+1)}}}$ For ${\displaystyle a<-1}$

Four curves of the family have special names:

• ${\displaystyle a=0}$, Straight line (asymptote for the other curves of the family)
• ${\displaystyle a=-1}$, Cissoids
• ${\displaystyle a=-2}$, right strophoid
• ${\displaystyle a=-4}$, Trisffektix from Maclaurin

Web links

• Eric W. Weisstein : Conchoid of de Sluze . In: MathWorld (English).
• John J. O'Connor, Edmund F. Robertson :  Conchoid of de Sluze. In: MacTutor History of Mathematics archive .