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Martin Thoma 2013-12-16 10:51:15 +01:00
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documents/math-minimal-distance-to-cubic-function/linear-functions.tex
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@ -109,8 +109,16 @@ $a \leq b$, $m \neq 0$ be a linear function.
\end{figure}
The point with minimum distance can be found by:
\[\underset{x\in\mdr}{\arg \min d_{P,f}(x)} = \begin{cases}
\[\underset{x\in[a,b]}{\arg \min d_{P,f}(x)} = \begin{cases}
S_1(f, P) &\text{if } S_1(f, P) \cap [a,b] \neq \emptyset\\
\Set{a} &\text{if } S_1(f, P) \ni x < a\\
\Set{b} &\text{if } S_1(f, P) \ni x > b
\end{cases}\]
Because:
\begin{align}
\underset{x\in[a,b]}{\arg \min d_{P,f}(x)} &= \underset{x\in[a,b]}{\arg \min d_{P,f}(x)^2}\\
&=\underset{x\in[a,b]}{\arg \min} x^2 - 2x_P x + (x_P^2 + y_P^2 - 2 y_P c + c^2)\\
&=\underset{x\in[a,b]}{\arg \min} x^2 - 2 x_P x + x_P^2\\
&=\underset{x\in[a,b]}{\arg \min} (x-x_P)^2\\
\end{align}