# Difference between revisions of "Pascal limaçon"

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− | A plane algebraic curve of order 4; a [[ | + | A plane algebraic curve of order 4; a [[conchoid]] of a circle of diameter $a$ (see Fig.). |

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p071770a.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p071770a.gif" /> | ||

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

− | 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 [[ | + | 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 |

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

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

====References==== | ====References==== | ||

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====Comments==== | ====Comments==== | ||

− | + | Étienne Pascal (1588–1651) was the father of [[Pascal, Blaise|Blaise Pascal]]. | |

====References==== | ====References==== | ||

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<TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Berger, "Geometry" , '''1–2''' , Springer (1987) (Translated from French)</TD></TR> | <TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Berger, "Geometry" , '''1–2''' , Springer (1987) (Translated from French)</TD></TR> | ||

<TR><TD valign="top">[a2]</TD> <TD valign="top"> F. Gomes Teixeira, "Traité des courbes" , '''1–3''' , Chelsea, reprint (1971)</TD></TR> | <TR><TD valign="top">[a2]</TD> <TD valign="top"> F. Gomes Teixeira, "Traité des courbes" , '''1–3''' , Chelsea, reprint (1971)</TD></TR> | ||

− | <TR><TD valign="top">[a3]</TD> <TD valign="top"> J.D. Lawrence, "A catalog of special plane curves" , Dover | + | <TR><TD valign="top">[a3]</TD> <TD valign="top"> J.D. Lawrence, "A catalog of special plane curves" , Dover (1972) pp. 113–118; ISBN 0-486-60288-5 {{ZBL|0257.50002}}</TD></TR> |

</table> | </table> | ||

[[Category:Geometry]] | [[Category:Geometry]] |

## Latest revision as of 19:24, 13 December 2017

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.

The limaçon is named after Étienne 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

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

#### 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 (1972) pp. 113–118; ISBN 0-486-60288-5 Zbl 0257.50002 |

**How to Cite This Entry:**

Pascal limaçon.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Pascal_lima%C3%A7on&oldid=42506