The Conchoid of de Sluze for varying values of a

The conchoid(s) of de Sluze is a family of plane curves studied in 1662 by René François Walter, baron de Sluze. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... Events February 1 - The Chinese pirate Koxinga seizes the island of Taiwan after a nine-month siege. ... René François Walther de Sluze (Slusius), canon of Liège, (July 7, 1622 - March 19, 1685) was a mathematician. ...

In polar coordinates This article describes some of the common coordinate systems that appear in elementary mathematics. ...

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). An acnode is an isolated point not on an curve, but whose coordinates satisfy the equation of the curve. ...

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. An asymptote is a straight line or curve which a curve approaches as one moves along the curve. ... A crunode is a point where a curve intersects itself so that both branches of the curve have distinct tangent lines. ...

The area between the curve and the asymptote is

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

A representation of one line Three lines — the red and blue lines have same slope, while the red and green ones have same y-intercept. ... The cissoid of Diocles is an unbounded plane curve with a single cusp, which is symmetric about the line of tangency of the cusp, and whose pair of symmetrical branches both approach the same asymptote (but in opposite directions) as a point moving along the cissoid moves farther away from... This article or section is in need of attention from an expert on the subject. ... This article may be confusing for some readers, and should be edited to enhance clarity. ...

See also

• Conchoid (mathematics)

A conchoid is a curve derived from a fixed point O, another curve, and a length d. ...

External links

• Eric W. Weisstein, Conchoid of de Sluze at MathWorld.
• 2D Curves
• Famous Curves (includes a scaling factor)