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
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.
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In geometry, a cissoid (() is a plane curve generated from two given curves C_1, C_2 and a point O (the pole). Let L be a variable line passing through O and intersecting C_1 at P_1 and C_2 at P_2. Let P be the point on L so that (There are actually two such points but P is chosen so that P is in the same direction from O as P_2 is from P_1.) Then the locus of such points P is defined to be the cissoid of the curves C_1, C_2 relative to O. Slightly different but essentially equivalent definitions are used by different authors.
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 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).