added math minimal distance example

documents/math-minimal-distance-to-cubic-function/Makefile

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+SOURCE = math-minimal-distance-to-cubic-function
+make:
+	pdflatex $(SOURCE).tex -output-format=pdf
+	make clean
+
+clean:
+	rm -rf  $(TARGET) *.class *.html *.log *.aux *.out

documents/math-minimal-distance-to-cubic-function/calcMinDist.py

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+#!/usr/bin/env python
+
+import numpy
+
+class Point:
+    def __init__(self, x, y):
+        self.x = x
+        self.y = y
+
+def euclideanDist(p1, p2):
+    from math import sqrt
+    return sqrt((p1.x-p2.x)**2 + (p1.y-p2.y)**2)
+
+def getMinDist(y, precision=0.001, startX=0, endX=3):
+    """Get x of point on (x,x^2) that has minimal distance to given Point."""
+    minDist = -1
+    p1 = Point(0, y)
+    for x in numpy.arange(startX, endX, precision):
+        p2 = Point(x, x**2)
+        dist = euclideanDist(p1, p2)
+        if minDist == -1 or dist < minDist:
+            minDist = dist
+            #print(dist)
+        else:
+            if abs(i-minDist) <0.005:
+                print("x: %s" % str(x))
+            break
+    return minDist
+
+for i in numpy.arange(0, 3, 0.01):
+    minDist = getMinDist(i)
+    if abs(i-minDist) < 0.005:
+        print(i, minDist)
+
+#print(getMinDist(5, 0.00001, 2, 3))

documents/math-minimal-distance-to-cubic-function/math-minimal-distance-to-cubic-function.tex

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+\documentclass[a4paper]{scrartcl}
+\usepackage{amssymb, amsmath} % needed for math
+\usepackage[utf8]{inputenc} % this is needed for umlauts
+\usepackage[english]{babel} % this is needed for umlauts
+\usepackage[T1]{fontenc}    % this is needed for correct output of umlauts in pdf
+\usepackage[margin=2.5cm]{geometry} %layout
+\usepackage{hyperref}   % links im text
+\usepackage{braket}         % needed for \Set
+\usepackage{parskip}
+\usepackage[colorinlistoftodos]{todonotes}
+\usepackage{pgfplots}
+\pgfplotsset{compat=1.7,compat/path replacement=1.5.1}
+\usepackage{tikz}
+
+\title{Minimal distance to a cubic function}
+\author{Martin Thoma}
+
+\hypersetup{ 
+  pdfauthor   = {Martin Thoma}, 
+  pdfkeywords = {}, 
+  pdftitle    = {Minimal Distance} 
+}
+
+\def\mdr{\ensuremath{\mathbb{R}}}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Begin document                                                    %
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{document}
+\maketitle
+\begin{abstract}
+In this paper I want to discuss how to find all points on a a cubic 
+function with minimal distance to a given point.
+\end{abstract}
+
+\section{Description of the Problem}
+Let $f: \mdr \rightarrow \mdr$ be a polynomial function and $P \in \mdr^2$
+be a point. Let $d: \mdr^2 \times \mdr^2 \rightarrow \mdr_0^+$
+be the euklidean distance of two points:
+\[d \left ((x_1, y_1), (x_2, y_2) \right) := \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\]
+
+Now there is finite set of points $x_1, \dots, x_n$ such that 
+\[\forall \tilde x \in \mathbb{R} \setminus \{x_1, \dots, x_n\}: d(P, (x_1, f(x_1))) = \dots = d(P, (x_n, f(x_n))) < d(P, (\tilde x, f(\tilde x)))\]
+
+\section{Minimal distance to a constant function}
+Let $f(x) = c$ with $c \in \mdr$ be a function. 
+
+\todo[inline]{add image}
+
+Then $(x_P,f(x_P))$ has
+minimal distance to $P$. Every other point has higher distance.
+
+\section{Minimal distance to a linear function}
+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}
+
+Now you can drop a perpendicular through $P$ on $f(x)$. The slope $f_\bot$
+of the perpendicular is $- \frac{1}{m}$. Then:
+
+\begin{align}
+                 f_\bot(x) &= - \frac{1}{m} \cdot x + t_\bot\\
+    \Rightarrow        y_P &= - \frac{1}{m} \cdot x_P + t_\bot\\
+    \Leftrightarrow t_\bot &= y_P + \frac{1}{m} \cdot x_P\\
+    f(x) &= f_\bot(x)\\
+    \Leftrightarrow m \cdot x + t &= - \frac{1}{m} \cdot x + \left(y_P + \frac{1}{m} \cdot x_P \right)\\
+    \Leftrightarrow \left (m + \frac{1}{m} \right ) \cdot x &= y_P + \frac{1}{m} \cdot x_P - t\\
+    \Leftrightarrow x &= \frac{m}{m^2+1} \left ( y_P + \frac{1}{m} \cdot x_P - t \right )
+\end{align}
+
+There is only one point with minimal distance.
+
+\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 
+$b, c \in \mdr$ be a function.
+
+\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=-3,     % start the diagram at this x-coordinate
+        xmax= 3,    % end   the diagram at this x-coordinate
+        ymin=-0.25,     % start the diagram at this y-coordinate
+        ymax= 9,   % 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=-3:3, thick,samples=50, red] {0.5*x*x}; 
+      \addplot[domain=-3:3, thick,samples=50, green] {x*x}; 
+      \addplot[domain=-3:3, thick,samples=50, blue] {x*x + x};
+      \addplot[domain=-3:3, thick,samples=50, orange] {x*x + 2*x};
+      \addplot[domain=-3:3, thick,samples=50, black] {-x*x + 6};
+      \addlegendentry{$f_1(x)=\frac{1}{2}x^2$}
+      \addlegendentry{$f_2(x)=x^2$}
+      \addlegendentry{$f_3(x)=x^2+x$}
+      \addlegendentry{$f_4(x)=x^2+2x$}
+      \addlegendentry{$f_5(x)=-x^2+6$}
+    \end{axis} 
+\end{tikzpicture}
+    \caption{Quadratic functions}
+\end{figure}
+
+\subsection{Number of points with minimal distance}
+It is obvious that a quadratic function can have two points with 
+minimal distance. 
+
+For example, let $f(x) = x^2$ and $P = (0,5)$. Then $P_{f,1} \approx (2.179, 2.179^2)$
+has minimal distance to $P$, but also $P_{f,2}\approx (-2.179, 2.179^2)$.
+
+Obviously, there cannot be more than three points with minimal distance.
+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}
+    \caption{3 points with minimal distance?}
+    \todo[inline]{Is this possible?}
+\end{figure}
+
+\subsection{Calculate points with minimal distance}
+\todo[inline]{Write this}
+
+\section{Minimal distance to a cubic function}
+\subsection{Number of points with minimal distance}
+\todo[inline]{Write this}
+
+\subsection{Calculate points with minimal distance}
+\todo[inline]{Write this}
+\end{document}