包络线

设曲线族的每条曲线${\displaystyle C_{s}}$ 为${\displaystyle t\mapsto (x(s,t),y(s,t))}$ 。

设存在包络线。由于包络线的每点都与曲线族的其中一条曲线的其中一点相切，对于任意的${\displaystyle s}$ ，设${\displaystyle (x(s,h(s)),y(s,h(s)))}$ 表示${\displaystyle C_{s}}$ 和包络线相切的那点。由此式可见，${\displaystyle s}$ 是包络线的变数。要求出包络线，就即要求出${\displaystyle h(s)}$ 。

在${\displaystyle C_{s}}$ 的切向量为${\displaystyle <{\frac {\partial x}{\partial t}},{\frac {\partial y}{\partial t}}>}$ ，其中${\displaystyle t=h(s)}$ 。

在E的切向量为${\displaystyle <{\frac {dx}{ds}},{\frac {dy}{ds}}>}$ 。因为${\displaystyle x}$ 是${\displaystyle s}$ 和${\displaystyle t}$ 的函数，而此处${\displaystyle t=h(s)}$ ，局部求导有：

${\displaystyle {\frac {dx}{ds}}={\frac {\partial x}{\partial h}}{\frac {dh}{ds}}+{\frac {\partial x}{\partial s}}{\frac {ds}{ds}}={\frac {\partial x}{\partial h}}h'(s)+{\frac {\partial x}{\partial s}}}$ 

类似地得 ${\displaystyle {\frac {dy}{ds}}={\frac {\partial y}{\partial h}}h'(s)+{\frac {\partial y}{\partial s}}}$ 。

因为${\displaystyle E}$ 和${\displaystyle C_{s}}$ 在该点相切，因此其切向量应平行，故有

${\displaystyle {\frac {\partial x}{\partial t}}=\lambda ({\frac {\partial x}{\partial h}}h'(s)+{\frac {\partial x}{\partial s}})}$ 

${\displaystyle {\frac {\partial y}{\partial t}}=\lambda ({\frac {\partial y}{\partial h}}h'(s)+{\frac {\partial y}{\partial s}})}$ 

其中${\displaystyle \lambda \neq 0}$ 。可用此两式消去${\displaystyle h'(s)}$ 。整理后得： ${\displaystyle {\frac {\partial y}{\partial h}}{\frac {\partial x}{\partial s}}={\frac {\partial y}{\partial s}}{\frac {\partial x}{\partial h}}}$ 

• https://web.archive.org/web/20070621035057/http://www.math.neu.edu/~bridger/Envelope/envelope.htm

• 克莱罗方程

• Special Plane Curves: Envelope （页面存档备份，存于互联网档案馆）
• Envelopes of Lines and Circles （页面存档备份，存于互联网档案馆）