# 笛卡儿叶形线

${\displaystyle x^{3}+y^{3}-3axy=0.\,}$

${\displaystyle r={\frac {3a\sin \theta \cos \theta }{\sin ^{3}\theta +\cos ^{3}\theta }}.}$

## 曲线的特征

### 切线的方程

${\displaystyle {\frac {dy}{dx}}={\frac {ay-x^{2}}{y^{2}-ax}}.}$

${\displaystyle y-y_{1}={\frac {ay_{1}-x_{1}^{2}}{y_{1}^{2}-ax_{1}}}(x-x_{1}).}$

### 水平和竖直切线

${\displaystyle ay-x^{2}=0}$时，笛卡儿叶形线的切线是水平的。所以：

${\displaystyle x=a{\sqrt[{3}]{2}}.}$

${\displaystyle y^{2}-ax=0}$时，笛卡儿叶形线的切线是竖直的。所以：

${\displaystyle y=a{\sqrt[{3}]{2}}.}$

### 渐近线

${\displaystyle x+y+a=0.}$

## 参考文献

• Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 218, 1987.
• Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 77-82, 1997.
• Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 106-109, 1972.
• MacTutor History of Mathematics Archive. "Folium of Descartes." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Foliumd.html页面存档备份，存于互联网档案馆）.
• Stroeker, R. J. "Brocard Points, Circulant Matrices, and Descartes' Folium." Math. Mag. 61, 172-187, 1988.
• Yates, R. C. "Folium of Descartes." In A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 98-99, 1952.