Conchoid of de Sluze
From Wikipedia, the free encyclopedia
The conchoid(s) of de Sluze is a family of plane curves studied in 1662 by René François Walter, baron de Sluze.
In polar coordinates
- r = secθ + acosθ
and in implicit Cartesian coordinates
- (x − 1)(x2 + y2) = ax2
(except that for a=0 the implicit form has an acnode (0,0) not present in polar form).
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
- | a | (1 + a / 4)π for a≥−1
- for a<−1
The area of the loop is
- for a<−1
Four of the family have names of their own:
- a=0, line (asymptote to rest of family)
- a=−1, cissoid of Diocles (clue to geometric construction)
- a=−2, right strophoid
- a=−4, trisectrix of Maclaurin
[edit] See also
[edit] External links
- Eric W. Weisstein, Conchoid of de Sluze at MathWorld.
- 2D Curves
- Famous Curves (includes a scaling factor)