星形线

星形线（astroid）或稱為四尖瓣線（tetracuspid），是一個有四個尖點的內擺線，也屬於超橢圓的一種。所有星形线皆可以依以下的方程式比例縮放而得^[1]：

x^{2/3}+y^{2/3}=a^{2/3}.\,

其英文名稱得名自希臘文的「星星」，星形线幾乎和橢圓的渐屈线相同。

若讓一個半徑為1/4的圓在一個半徑為1的圓內部，延著圓的圓周旋轉，小圓圓周上的任一點形成的軌跡即為星形线。星形线的參數方程為^[1]：

x=a\cos ^{3}\theta ,\qquad y=a\sin ^{3}\theta .

星形线是一個幾何虧格（英语：geometric genus）為0代數曲線的實數軌跡，其方程式如下：

(x^{2}+y^{2}-1)^{3}+27x^{2}y^{2}=0.\,

因此星形线為六次曲線，在實數平面上有四個尖瓣的奇点，分別是星形线的四個頂點，在無限遠處還有二個複數的尖瓣的奇點，四個重根的複數奇點，因此星形线共有十個奇点。

星形线的對偶曲線（英语：Dual curve）是十字架形曲線（英语：cruciform curve），其方程式為 $\textstyle x^{2}y^{2}=x^{2}+y^{2}.$ 。星形線的渐屈线為另一個二倍大的渐屈线。

一個半徑為 $a$ 之圓的內擺線構成的星形线，其面積為 ${\frac {3}{8}}\pi a^{2}$ ，周長為6a。

參考資料

^ ^1.0 ^1.1 清华大学数学科学系. 微积分. 北京: 清华大学出版社有限公司. 2003: 47–48 [2012-12-21]. ISBN 7302067856. （原始内容存档于2019-05-28）.

J. Dennis Lawrence. A catalog of special plane curves. Dover Publications. 1972: 4–5,34–35,173–174. ISBN 0-486-60288-5.
Wells D. The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. 1991: 10–11. ISBN 0-14-011813-6.
R.C. Yates. Astroid. A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards. 1952: 1 ff.

外部連結

维基共享资源上的相关多媒体资源：星形线

埃里克·韦斯坦因. Astroid. MathWorld.
"Astroid" at The MacTutor History of Mathematics archive （页面存档备份，存于互联网档案馆）
"Astroïde" at Encyclopédie des Formes Mathématiques Remarquables （页面存档备份，存于互联网档案馆） (in French)
Article on 2dcurves.com
Visual Dictionary Of Special Plane Curves, Xah Lee （页面存档备份，存于互联网档案馆）
Bars of an Astroid （页面存档备份，存于互联网档案馆） by Sándor Kabai, The Wolfram Demonstrations Project.

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