Pascal limaçon

A plane algebraic curve of order 4; a conchoid of a circle of diameter $a$ (see Fig.).

Figure: p071770a

The equation in rectangular coordinates is

$$(x^2+y^2-ax)^2=l^2(x^2+y^2);$$

in polar coordinates it is

$$\rho=a\cos\phi+l.$$

The coordinate origin is a double point, which is an isolated point for $a<l$, a node for $a>l$, and a cusp for $a=l$ (in this case Pascal's limaçon is a cardioid). The arc length can be expressed by an elliptic integral of the second kind. The area bounded by Pascal's limaçon is

$$S=\frac{\pi a^2}{2}+\pi l^2;$$

for $a>l$ the area of the inner leaf must be counted double in calculating according to this formula. The Pascal limaçon is a special case of a Descartes oval, it is an epitrochoid (see Trochoid).

The limaçon is named after E. Pascal, who first treated it in the first half of the 17th century.

References

[1]	A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)

Comments

E. Pascal is the father of B. Pascal, the famous one.

References

[a1]	M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French)
[a2]	F. Gomes Teixeira, "Traité des courbes" , 1–3 , Chelsea, reprint (1971)
[a3]	J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972) pp. 113–118