奇怪的东西
__Floze3__
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个人记录
\begin{aligned}
f(h) &= (h - x)^2 + (h - y)^2 \\
&= 2h^2 -2(x + y)h + (x^2 + y^2)
\end{aligned}
\begin{aligned}
f(l) = (l - h) ^ 2 + f(h) + (l - k) ^ 2 + f(k)
\end{aligned}
\begin{aligned}
f(h) + (l - h)^2 &= ah^2 + bh + c + l^2 - 2hl + h^2 \\
&= (a + 1)h^2 + (b - 2l)h + (c + l^2)
\end{aligned}
当 h = \frac{2l - b}{2a + 2} 时取最小。此时
\begin{aligned}
f(h) + (l - h)^2 &= (a + 1) \times \frac{(2l - b)^2}{4(a + 1)^2} + (b - 2l) \times \frac{2l - b}{2a + 2} + (c + l^2) \\
&= \frac{4l^2 - 4lb + b^2}{4(a + 1)} - \frac{b^2 - 4bl + 4l^2}{2(a+1)} + (c + l^2) \\
&= (1 - \frac{4}{4(a + 1)})l^2 + \frac{4b}{4(a + 1)}l + c - \frac{b^2}{4(a + 1)}
\end{aligned}
由于 a = a'，可以得到：
\begin{aligned}
f(l) = (2 + \frac{8}{4(a + 1)})l^2 + \frac{4b + 4b'}{4(a + 1)}l + c + c' - \frac{b^2 + b'^2}{4(a + 1)}
\end{aligned}