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documents/math-minimal-distance-to-cubic-function/math-minimal-distance-to-cubic-function.tex
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@ -45,7 +45,40 @@ Now there is finite set of points $x_1, \dots, x_n$ such that
\section{Minimal distance to a constant function}
Let $f(x) = c$ with $c \in \mdr$ be a function.
\todo[inline]{add image}
\begin{figure}[htp]
\centering
\begin{tikzpicture}
\begin{axis}[
legend pos=north west,
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=-5:5, thick,samples=50, red] {1};
\addplot[domain=-5:5, thick,samples=50, green] {2};
\addplot[domain=-5:5, thick,samples=50, blue] {3};
\addplot[black, mark = *, nodes near coords=$P$,every node near coord/.style={anchor=225}] coordinates {(2, 2)};
\draw[thick, dashed] (axis cs:2,0) -- (axis cs:2,3);
\addlegendentry{$f(x)=1$}
\addlegendentry{$g(x)=2$}
\addlegendentry{$h(x)=3$}
\end{axis}
\end{tikzpicture}
\caption{3 constant functions}
\end{figure}
Then $(x_P,f(x_P))$ has
minimal distance to $P$. Every other point has higher distance.
@ -54,7 +87,37 @@ minimal distance to $P$. Every other point has higher distance.
Let $f(x) = m \cdot x + t$ with $m \in \mdr \setminus \Set{0}$ and
$t \in \mdr$ be a function.
\todo[inline]{add image}
\begin{figure}[htp]
\centering
\begin{tikzpicture}
\begin{axis}[
legend pos=north east,
axis x line=middle,
axis y line=middle,
grid = major,
width=0.8\linewidth,
height=8cm,
grid style={dashed, gray!30},
xmin= 0, % 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=-5:5, thick,samples=50, red] {0.5*x};
\addplot[domain=-5:5, thick,samples=50, blue] {-2*x+6};
\addplot[black, mark = *, nodes near coords=$P$,every node near coord/.style={anchor=225}] coordinates {(2, 2)};
\addlegendentry{$f(x)=\frac{1}{2}x$}
\addlegendentry{$g(x)=-2x+6$}
\end{axis}
\end{tikzpicture}
\caption{The shortest distance of $P$ to $f$ can be calculated by using the perpendicular}
\end{figure}
Now you can drop a perpendicular through $P$ on $f(x)$. The slope $f_\bot$
of the perpendicular is $- \frac{1}{m}$. Then:
@ -70,6 +133,7 @@ of the perpendicular is $- \frac{1}{m}$. Then:
\end{align}
There is only one point with minimal distance.
\clearpage
\section{Minimal distance to a quadratic function}
Let $f(x) = a \cdot x^2 + b \cdot x + c$ with $a \in \mdr \setminus \Set{0}$ and
@ -126,47 +190,58 @@ But can there be three points?
\begin{figure}[htp]
\centering
\begin{tikzpicture}
\begin{axis}[
legend pos=north west,
axis x line=middle,
axis y line=middle,
grid = major,
width=0.8\linewidth,
height=8cm,
grid style={dashed, gray!30},
xmin=-0.7, % start the diagram at this x-coordinate
xmax= 0.7, % end the diagram at this x-coordinate
ymin=-0.25, % start the diagram at this y-coordinate
ymax= 0.5, % end the diagram at this y-coordinate
axis background/.style={fill=white},
xlabel=$x$,
ylabel=$y$,
%xticklabels={-2,-1.6,...,7},
%yticklabels={-8,-7,...,8},
tick align=outside,
minor tick num=-3,
enlargelimits=true,
tension=0.08]
\addplot[domain=-0.7:0.7, thick,samples=50, orange] {x*x};
\draw (axis cs:0,0.5) circle[radius=0.5];
\draw[red, thick] (axis cs:0,0.5) -- (axis cs:0.101,0.0102);
\draw[red, thick] (axis cs:0,0.5) -- (axis cs:-0.101,0.0102);
\draw[red, thick] (axis cs:0,0.5) -- (axis cs:0,0);
\addlegendentry{$f(x)=x^2$}
\end{axis}
\end{tikzpicture}
\begin{tikzpicture}
\begin{axis}[
legend pos=north west,
axis x line=middle,
axis y line=middle,
grid = major,
width=0.8\linewidth,
height=8cm,
grid style={dashed, gray!30},
xmin=-0.7, % start the diagram at this x-coordinate
xmax= 0.7, % end the diagram at this x-coordinate
ymin=-0.25, % start the diagram at this y-coordinate
ymax= 0.5, % end the diagram at this y-coordinate
axis background/.style={fill=white},
xlabel=$x$,
ylabel=$y$,
%xticklabels={-2,-1.6,...,7},
%yticklabels={-8,-7,...,8},
tick align=outside,
minor tick num=-3,
enlargelimits=true,
tension=0.08]
\addplot[domain=-0.7:0.7, thick,samples=50, orange] {x*x};
\draw (axis cs:0,0.5) circle[radius=0.5];
\draw[red, thick] (axis cs:0,0.5) -- (axis cs:0.101,0.0102);
\draw[red, thick] (axis cs:0,0.5) -- (axis cs:-0.101,0.0102);
\draw[red, thick] (axis cs:0,0.5) -- (axis cs:0,0);
\addlegendentry{$f(x)=x^2$}
\end{axis}
\end{tikzpicture}
\caption{3 points with minimal distance?}
\todo[inline]{Is this possible?}
\todo[inline]{Is this possible? http://math.stackexchange.com/q/553097/6876}
\end{figure}
\subsection{Calculate points with minimal distance}
\todo[inline]{Write this}
\section{Minimal distance to a cubic function}
Let $f(x) = a \cdot x^3 + b \cdot x^2 + c \cdot x + d$ with $a \in \mdr \setminus \Set{0}$ and
$b, c, d \in \mdr$ be a function.
\subsection{Number of points with minimal distance}
\todo[inline]{Write this}
\subsection{Special points}
\todo[inline]{Write this}
\subsection{Voronoi}
For $b^2 \geq 3ac$
\todo[inline]{Write this}
\subsection{Calculate points with minimal distance}
\todo[inline]{Write this}
\end{document}