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added some ideas for the case of intervalls [a,b] of R

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Martin Thoma 2013-12-12 23:05:04 +01:00
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documents/math-minimal-distance-to-cubic-function/constant-functions.tex
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@ -1,12 +1,13 @@
\chapter{Constant functions}
\section{Defined on $\mdr$}
Let $f(x) = c$ with $c \in \mdr$ be a constant function.
Let $f:\mdr \rightarrow \mdr$, $f(x) := c$ with $c \in \mdr$ be a constant function.
\begin{figure}[htp]
\centering
\begin{tikzpicture}
\begin{axis}[
legend pos=north west,
legend cell align=left,
axis x line=middle,
axis y line=middle,
grid = major,
@ -45,4 +46,62 @@ Then $(x_P,f(x_P))$ has
minimal distance to $P$. Every other point has higher distance.
See Figure~\ref{fig:constant-min-distance}.
\section{Defined on a closed interval of $\mdr$}
This means:
\[S_0(f, P) = \Set{x_P} \text{ with } P = (x_P, y_P)\]
\clearpage
\section{Defined on a closed interval $[a,b] \subseteq \mdr$}
Let $f:[a,b] \rightarrow \mdr$, $f(x) := c$ with $a,b,c \in \mdr$ and
$a \leq b$ be a constant function.
\begin{figure}[htp]
\centering
\begin{tikzpicture}
\begin{axis}[
legend pos=north west,
legend cell align=left,
axis x line=middle,
axis y line=middle,
grid = major,
width=0.8\linewidth,
height=8cm,
grid style={dashed, gray!30},
xmin=-5, % start the diagram at this x-coordinate
xmax= 5, % end the diagram at this x-coordinate
ymin= 0, % start the diagram at this y-coordinate
ymax= 3, % end the diagram at this y-coordinate
axis background/.style={fill=white},
xlabel=$x$,
ylabel=$y$,
tick align=outside,
minor tick num=-3,
enlargelimits=true,
tension=0.08]
\addplot[domain=-5:-2, thick,samples=50, red] {1};
\addplot[domain=-1:3, thick,samples=50, green] {1.5};
\addplot[domain=3:5, thick,samples=50, blue, densely dotted] {3};
\addplot[black, mark = *, nodes near coords=$P$,every node near coord/.style={anchor=225}] coordinates {(2, 2)};
\addplot[blue, mark = *, nodes near coords=$P_{h,\text{min}}$,every node near coord/.style={anchor=225}] coordinates {(3, 3)};
\addplot[green, mark = x, nodes near coords=$P_{g,\text{min}}$,every node near coord/.style={anchor=120}] coordinates {(2, 1.5)};
\addplot[red, mark = *, nodes near coords=$P_{f,\text{min}}$,every node near coord/.style={anchor=225}] coordinates {(-2, 1)};
\draw[thick, dashed] (axis cs:2,1.5) -- (axis cs:2,2);
\draw[thick, dashed] (axis cs:2,2) -- (axis cs:-2,1);
\draw[thick, dashed] (axis cs:2,2) -- (axis cs:3,3);
\addlegendentry{$f(x)=1, D = [-5,-2]$}
\addlegendentry{$g(x)=1.5, D = [-1,3]$}
\addlegendentry{$h(x)=3, D = [3,5]$}
\end{axis}
\end{tikzpicture}
\caption{Three constant functions and their points with minimal distance}
\label{fig:constant-min-distance-closed-intervall}
\end{figure}
The point with minimum distance can be found by:
\[\underset{x\in\mdr}{\arg \min d_{P,f}(x)} = \begin{cases}
S_0(f,P) &\text{if } S_0(f,P) \cap [a,b] \neq \emptyset \\
\Set{a} &\text{if } S_0(f,P) \ni x_P < a\\
\Set{b} &\text{if } S_0(f,P) \ni x_P > b
\end{cases}\]