Family of algebraic curves of the form r = sec(θ) + a*cos(θ)
The Conchoid of de Sluze for several values of a
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.[1][2]
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). (0,0) is a crunode for a < −1.
The area between the curve and the asymptote is, for a ≥ −1,
while for a < −1, the area is
If a < −1, the curve will have a loop. The area of the loop is
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
^Smith, David Eugene (1958), History of Mathematics, Volume 2, Courier Dover Publications, p. 327, ISBN 9780486204307.
^"Conchoid of de Sluze by J. Dziok et al.on Computers and Mathematics with Applications 61 (2011) 2605–2613" (PDF).
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