Pascal limaçon

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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


in polar coordinates it is


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.

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


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


Étienne Pascal (1588–1651) was the father of Blaise Pascal.


[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 (1972) pp. 113–118; ISBN 0-486-60288-5 Zbl 0257.50002
How to Cite This Entry:
Pascal limaçon. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article