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(Redirected from Conchoid of de sluze)
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:

References

  1. ^ Smith, David Eugene (1958), History of Mathematics, Volume 2, Courier Dover Publications, p. 327, ISBN  9780486204307.
  2. ^ "Conchoid of de Sluze by J. Dziok et al.on Computers and Mathematics with Applications 61 (2011) 2605–2613" (PDF).
From Wikipedia, the free encyclopedia
(Redirected from Conchoid of de sluze)
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:

References

  1. ^ Smith, David Eugene (1958), History of Mathematics, Volume 2, Courier Dover Publications, p. 327, ISBN  9780486204307.
  2. ^ "Conchoid of de Sluze by J. Dziok et al.on Computers and Mathematics with Applications 61 (2011) 2605–2613" (PDF).

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