卡西尼卵形線
喺幾何上,卡西尼卵形線(hippopede ,嚟自古希臘文ἱπποπέδη,意思係「horse fetter」)係一條平面曲線,由呢個方程定義
- ,
where it is assumed that c>0 and c>d since the remaining cases either reduce to a single point or can be put into the given form with a rotation. Hippopedes are bicircular rational algebraic curves of degree 4 and symmetric with respect to both the x and y axes. When d>0 the curve has an oval form and is often known as an oval of Booth, and when d<0 the curve resembles a sideways figure eight, or lemniscate, and is often known as a lemniscate of Booth, after James Booth (1810–1878) who studied them. Hippopedes were also investigated by Proclus (for whom they are sometimes called Hippopedes of Proclus) and Eudoxus. For d = −c, the hippopede corresponds to the lemniscate of Bernoulli.
作為環面嘅截面
[編輯]![](http://upload.wikimedia.org/wikipedia/commons/thumb/0/02/Hippopede02.svg/350px-Hippopede02.svg.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Hippopede01.svg/350px-Hippopede01.svg.png)
Hippopedes can be defined as the curve formed by the intersection of a torus and a plane, where the plane is parallel to the axis of the torus and tangent to it on the interior circle. Thus it is a spiric section which in turn is a type of toric section.
If a circle with radius a is rotated about an axis at distance b from its center, then the equation of the resulting hippopede in polar coordinates
or in Cartesian coordinates
- .
Note that when a>b the torus intersects itself, so it does not resemble the usual picture of a torus.
參考
[編輯]- Lawrence JD. (1972) Catalog of Special Plane Curves, Dover. Pp. 145–146.
- Booth J. A Treatise on Some New Geometrical Methods, Longmans, Green, Reader, and Dyer, London, Vol. I (1873) and Vol. II (1877).
- Weisstein, Eric W., "Hippopede" - MathWorld.(英文)
- "Hippopede" at 2dcurves.com
- "Courbes de Booth" at Encyclopédie des Formes Mathématiques Remarquables
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