摆线

    graph LR
    start[开始] --> input[输入A,B,C]
    input --> conditionA{A是否大于B}
    conditionA -- YES --> conditionC{A是否大于C}
    conditionA -- NO --> conditionB{B是否大于C}
    conditionC -- YES --> printA[输出A]
    conditionC -- NO --> printC[输出C]
    conditionB -- YES --> printB[输出B]
    conditionB -- NO --> printC[输出C]
    printA --> stop[结束]
    printC --> stop
    printB --> stop

参数方程

\[\begin{cases} x=r(t-\sin t) \\ y=r(1-\cos t) \\ \end{cases}\]

一般方程

消去$ \cos t $：

\[\cos t=1-\frac{y}{r}\]

消去t

\[t=\arccos(1-\frac{y}{r})\]

消去$ \sin t $：

\[\begin{align} \sin t &=t-\frac{x}{r} \\ &=\arccos(1-\frac{y}{r})-\frac{x}{r} \end{align} \tag{3}\]

则有

\[\begin{align} \sin^{2}t+\cos^{2}t &=(\arccos(1-\frac{y}{r})-\frac{x}{r})^{2}+(1-\frac{y}{r})^{2} \\ &=1 \end{align}\]

即

\[(r\arccos(1-\frac{y}{r})-x)^{2}+y(y-2r)=0\]

微分方程

\[(\frac{\mathrm{d} y}{\mathrm{d} x})^{2} =\frac{2r}{y}-1\]

推导过程

\[\begin{align} (\frac{\mathrm{d} y}{\mathrm{d} x})^{2} &=(\frac{\frac{\mathrm{d}y}{\mathrm{d}t}}{\frac{\mathrm{d}x}{\mathrm{d}t}})^{2} \\ &=(\frac{r\sin t}{r(1-\cos t)})^{2} \\ &=\frac{\sin^{2} t}{(1+\cos t)^{2}} \\ &=\frac{1-\cos t}{1+\cos t} \\ &=\frac{2}{1-\cos t}-1 \\ &=\frac{2r}{y}-1 \end{align}\]