# 笛卡儿叶形线

a=1

$x^3 + y^3 - 3 a x y = 0. \,$

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

## 曲线的特征

### 切线的方程

$\frac{dy}{dx} = \frac{a y - x^2}{y^2 - a x}.$

$y - y_1 = \frac{a y_1 - x_1^2}{y_1^2 - a x_1}(x - x_1).$

### 水平和竖直切线

$a y - x^2 = 0$时，笛卡儿叶形线的切线是水平的。所以：

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

$y^2 - a x = 0$时，笛卡儿叶形线的切线是竖直的。所以：

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

### 渐近线

$x + y + a = 0.$

## 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.
• 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.