From 102fe0447b50221e92a7bd74ca2b9a9158e1f1e5 Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Thu, 2 Jan 2014 16:16:13 +0100 Subject: [PATCH] =?UTF-8?q?=C3=84nderungen=20der=20Zugfahrt=20eingearbeite?= =?UTF-8?q?t.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../constant-functions.tex | 23 ++++---- .../cubic-functions.tex | 51 ++++++++++-------- .../introduction.tex | 5 +- .../linear-functions.tex | 44 +++++++++++++-- ...ath-minimal-distance-to-cubic-function.pdf | Bin 454923 -> 451332 bytes .../problem-description.tex | 17 +++--- .../quadratic-case-2.1.tex | 27 ++++------ .../quadratic-case-2.2.tex | 8 +-- .../quadratic-case-2.3.tex | 2 +- .../quadratic-functions.tex | 31 +++++------ 10 files changed, 129 insertions(+), 79 deletions(-) diff --git a/documents/math-minimal-distance-to-cubic-function/constant-functions.tex b/documents/math-minimal-distance-to-cubic-function/constant-functions.tex index 023e069..c714bb3 100644 --- a/documents/math-minimal-distance-to-cubic-function/constant-functions.tex +++ b/documents/math-minimal-distance-to-cubic-function/constant-functions.tex @@ -1,8 +1,13 @@ \chapter{Constant functions} \section{Defined on $\mdr$} +\begin{lemma} Let $f:\mdr \rightarrow \mdr$, $f(x) := c$ with $c \in \mdr$ be a constant function. -The situation can be seen in Figure~\ref{fig:constant-min-distance}. +Then $(x_P, f(x_P))$ is the only point on the graph of $f$ with +minimal distance to $P$. +\end{lemma} + +The situation can be seen in Figure~\ref{fig:constant-min-distance}. \begin{figure}[htp] \centering \begin{tikzpicture} @@ -43,20 +48,19 @@ The situation can be seen in Figure~\ref{fig:constant-min-distance}. \label{fig:constant-min-distance} \end{figure} +\begin{proof} The point $(x, f(x))$ with minimal distance can be calculated directly: \begin{align} - d_{P,f}(x) &= \sqrt{(x_P - x)^2 + (y_P - f(x))^2}\\ - &= \sqrt{(x_P^2 - 2x_P x + x^2) + (y_P^2 - 2 y_P c + c^2)} \\ - &= \sqrt{x^2 - 2 x_P x + (x_P^2 + y_P^2 - 2 y_P c + c^2)}\label{eq:constant-function-distance}\\ + d_{P,f}(x) &= \sqrt{(x - x_P)^2 + (f(x) - y_P)^2}\\ + &= \sqrt{(x^2 - 2x_P x + x_P^2) + (c^2 - 2 c y_P + y_P^2)} \\ + &= \sqrt{x^2 - 2 x_P x + (x_P^2 + c^2 - 2 c y_P + y_P^2)}\label{eq:constant-function-distance}\\ \xRightarrow{\text{Theorem}~\ref{thm:fermats-theorem}} 0 &\stackrel{!}{=} (d_{P,f}(x)^2)'\\ &= 2x - 2x_P\\ \Leftrightarrow x &\stackrel{!}{=} x_P \end{align} -Then $(x_P,f(x_P))$ has -minimal distance to $P$. Every other point has higher distance. -See Figure~\ref{fig:constant-min-distance} to see that intuition -yields to the same results. +So $(x_P,f(x_P))$ is the only point with minimal distance to $P$. $\qed$ +\end{proof} This result means: @@ -124,8 +128,7 @@ given by: \begin{proof} \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} \big (x^2 - 2x_P x + x_P^2 + \overbrace{(y_P^2 - 2 y_P c + c^2)}^{\text{constant}} \big )\\ - &=\underset{x\in[a,b]}{\arg \min} (x^2 - 2 x_P x + x_P^2)\\ + &=\underset{x\in[a,b]}{\arg \min} \big ((x-x_P)^2 + \overbrace{(y_P^2 - 2 y_P c + c^2)}^{\text{constant}} \big )\\ &=\underset{x\in[a,b]}{\arg \min} (x-x_P)^2 \end{align} diff --git a/documents/math-minimal-distance-to-cubic-function/cubic-functions.tex b/documents/math-minimal-distance-to-cubic-function/cubic-functions.tex index e374f17..5a448d8 100644 --- a/documents/math-minimal-distance-to-cubic-function/cubic-functions.tex +++ b/documents/math-minimal-distance-to-cubic-function/cubic-functions.tex @@ -1,8 +1,8 @@ \chapter{Cubic functions} \section{Defined on $\mdr$} -Let $f(x) = a \cdot x^3 + b \cdot x^2 + c \cdot x + d$ be a cubic function -with $a \in \mdr \setminus \Set{0}$ and -$b, c, d \in \mdr$ be a function. +Let $f:\mdr \rightarrow \mdr, f(x) = a \cdot x^3 + b \cdot x^2 + c \cdot x + d$ +be a cubic function with $a \in \mdr \setminus \Set{0}$ and +$b, c, d \in \mdr$. \begin{figure}[htp] \centering @@ -50,8 +50,8 @@ $b, c, d \in \mdr$ be a function. %\todo[inline]{Write this} \subsection{Calculate points with minimal distance} -\begin{theorem} - There cannot be an algebraic solution to the problem of finding +\begin{theorem}\label{thm:no-finite-solution} + There cannot be a finite, closed form solution to the problem of finding a closest point $(x, f(x))$ to a given point $P$ when $f$ is a polynomial function of degree $3$ or higher. \end{theorem} @@ -71,29 +71,29 @@ $b, c, d \in \mdr$ be a function. General algebraic equations of degree 5 don't have a solution formula.\footnote{TODO: Quelle} Although here seems to be more structure, the resulting algebraic - equation can be almost any polynomial of degree 5:\footnote{Thanks to Peter Košinár on \href{http://math.stackexchange.com/a/584814/6876}{math.stackexchange.com} for this one} + equation can be almost any polynomial of degree 5:\footnote{Thanks to Peter Košinár on \href{http://math.stackexchange.com/a/584814/6876}{math.stackexchange.com} for the idea.} \begin{align} 0 &\stackrel{!}{=} f'(x) \cdot \left (f(x) - y_p \right ) + (x - x_p)\\ - &= \underbrace{3 a^2}_{= \tilde{a}} x^5 + \underbrace{5ab}_{\tilde{b}}x^4 + \underbrace{2(2ac + b^2 )}_{=: \tilde{c}}x^3 &+& \underbrace{3(ad+bc-ay_p)}_{\tilde{d}} x^2 \\ - & &+& \underbrace{(2 b d+c^2+1-2 b y_p)}_{=: \tilde{e}}x+\underbrace{c d-c y_p-x_p}_{=: \tilde{f}}\\ + &= \underbrace{3 a^2}_{= \tilde{a}} x^5 + \underbrace{5ab}_{= \tilde{b}}x^4 + \underbrace{2(2ac + b^2 )}_{= \tilde{c}}x^3 &+& \underbrace{3(ad+bc-ay_p)}_{= \tilde{d}} x^2 \\ + & &+& \underbrace{(2 b d+c^2+1-2 b y_p)}_{= \tilde{e}}x+\underbrace{c d-c y_p-x_p}_{= \tilde{f}}\\ 0 &\stackrel{!}{=} \tilde{a}x^5 + \tilde{b}x^4 + \tilde{c}x^3 + \tilde{d}x^2 + \tilde{e}x + \tilde{f} \end{align} \begin{enumerate} - \item For any coefficient $\tilde{a} \in \mdr_{> 0}$ of $x^5$ we can choose $a$ such that we get $\tilde{a}$. - \item For any coefficient $\tilde{b} \in \mdr \setminus \Set{0}$ of $x^4$ we can choose $b$ such that we get $\tilde{b}$. - \item With $c$, we can get any value of $\tilde{c} \in \mdr$. - \item With $d$, we can get any value of $\tilde{d} \in \mdr$. - \item With $y_p$, we can get any value of $\tilde{e} \in \mdr$. - \item With $x_p$, we can get any value of $\tilde{f} \in \mdr$. + \item For any coefficient $\tilde{a} \in \mdr_{> 0}$ of $x^5$ we can choose $a := \frac{1}{3} \sqrt{\tilde{a}}$ such that we get $\tilde{a}$. + \item For any coefficient $\tilde{b} \in \mdr \setminus \Set{0}$ of $x^4$ we can choose $b := \frac{1}{5a} \cdot \tilde{b}$ such that we get $\tilde{b}$. + \item With $c := -2b^2 + \frac{1}{4a} \tilde{c}$, we can get any value of $\tilde{c} \in \mdr$. + \item With $d := -bc + a y_p + \frac{1}{a} \tilde{d}$, we can get any value of $\tilde{d} \in \mdr$. + \item With $y_p := \frac{1}{2b}(2bd + c^2)\cdot \tilde{e}$, we can get any value of $\tilde{e} \in \mdr$. + \item With $x_p := cd - c y_P+\tilde{f}$, we can get any value of $\tilde{f} \in \mdr$. \end{enumerate} The first restriction guaratees that we have a polynomial of degree 5. The second one is necessary, to get a high range of $\tilde{e}$. - This means, that there is no solution formula for the problem of + This means that there is no finite solution formula for the problem of finding the closest points on a cubic function to a given point, because if there was one, you could use this formula for finding roots of polynomials of degree 5. $\qed$ @@ -101,9 +101,6 @@ $b, c, d \in \mdr$ be a function. \subsection{Another approach} -\todo[inline]{Currently, this is only an idea. It might be usefull -to move the cubic function $f$ such that $f$ is point symmetric -to the origin. But I'm not sure how to make use of this symmetry.} Just like we moved the function $f$ and the point to get in a nicer situation, we can apply this approach for cubic functions. @@ -144,7 +141,7 @@ nicer situation, we can apply this approach for cubic functions. \caption{Cubic functions with $b = d = 0$} \end{figure} -First, we move $f_0$ by $\frac{b}{3a}$ to the right, so +First, we move $f_0$ by $\frac{b}{3a}$ in $x$ direction, so \[f_1(x) = ax^3 + \frac{b^2 (c-1)}{3a} x + \frac{2b^3}{27 a^2} - \frac{bc}{3a} + d \;\;\;\text{ and }\;\;\;P_1 = (x_P + \frac{b}{3a}, y_P)\] @@ -161,6 +158,19 @@ because &= ax^3 + \frac{b^2}{3a}\left (1-2+c \right )x + \frac{b^3}{9a^2} \left (1-\frac{1}{3} \right )- \frac{bc}{3a} + d \end{align} +The we move it in $y$ direction by $- (\frac{2b^3}{27 a^2} - \frac{bc}{3a} + d)$: + +\[f_2(x) = ax^3 + \frac{b^2 (c-1)}{3a} x \;\;\;\text{ and }\;\;\;P_2 = (x_P + \frac{b}{3a}, y_P - (\frac{2b^3}{27 a^2} - \frac{bc}{3a} + d))\] + +Multiply everything by $\sgn(a)$: + +\[f_3(x) = \underbrace{|a|}_{=: \alpha}x^3 + \underbrace{\frac{b^2 (c-1)}{3|a|}}_{=: \beta} x \;\;\;\text{ and }\;\;\;P_2 = (x_P + \frac{b}{3a}, \sgn(a) (y_P - \frac{2b^3}{27 a^2} + \frac{bc}{3a} - d))\] + +Now the problem seems to be much simpler. The function $\alpha x^3 + \beta x$ +with $\alpha > 0$ is centrally symmetric to $(0, 0)$. + +\todo[inline]{Und weiter?} + \subsection{Number of points with minimal distance} As this leads to a polynomial of degree 5 of which we have to find roots, there cannot be more than 5 solutions. @@ -211,9 +221,6 @@ As soon as the values don't change much, you are close to a root. The problem of this approach is choosing a starting value that is close enough to the root. So we have to have a \enquote{good} initial guess. - -\subsubsection{Quadratic minimization} -\todo[inline]{TODO} \clearpage \subsubsection{Muller's method} diff --git a/documents/math-minimal-distance-to-cubic-function/introduction.tex b/documents/math-minimal-distance-to-cubic-function/introduction.tex index 3703a2c..1c86085 100644 --- a/documents/math-minimal-distance-to-cubic-function/introduction.tex +++ b/documents/math-minimal-distance-to-cubic-function/introduction.tex @@ -7,9 +7,8 @@ is by applying the \href{https://en.wikipedia.org/wiki/PID_algorithm}{PID algori This algorithm needs to know the signed current error. So you need to be able to get the minimal distance of a point (the position of the car) to a cubic spline (the prefered path) -combined with the direction (left or right). -As you need to get the signed error (and one steering direction might -be prefered), it is not only necessary to +combined with sign (which represents the steering direction). +As one steering direction might be prefered, it is not only necessary to get the minimal absolute distance, but might also help to get all points on the spline with minimal distance. diff --git a/documents/math-minimal-distance-to-cubic-function/linear-functions.tex b/documents/math-minimal-distance-to-cubic-function/linear-functions.tex index 94d8826..4fb246b 100644 --- a/documents/math-minimal-distance-to-cubic-function/linear-functions.tex +++ b/documents/math-minimal-distance-to-cubic-function/linear-functions.tex @@ -5,7 +5,8 @@ $f(x) := m \cdot x + t$ with $m \in \mdr \setminus \Set{0}$ and $t \in \mdr$ be a linear function. - Then the points $(x, f(x))$ with minimal distance are given by: + Then there is only one point $(x, f(x))$ on the graph of $f$ with + minimal distance to $P = (x_P, y_P)$. This point is given by \[x = \frac{m}{m^2+1} \left ( y_P + \frac{1}{m} \cdot x_P - t \right )\] \end{theorem} @@ -35,6 +36,8 @@ \addplot[domain=-5:5, thick,samples=50, red] {0.5*x}; \addplot[domain=-5:5, thick,samples=50, blue, dashed] {-2*x+6}; \addplot[black, mark = *, nodes near coords=$P$,every node near coord/.style={anchor=225}] coordinates {(2, 2)}; + \newcommand{\R}{0.9} + \addplot [domain=0:2*pi,samples=50, dotted]({\R*cos(deg(x))+2},{\R*sin(deg(x))+2}); \addplot[blue, nodes near coords=$f_\bot$,every node near coord/.style={anchor=225}] coordinates {(1.5, 3)}; \addplot[red, nodes near coords=$f$,every node near coord/.style={anchor=225}] coordinates {(0.9, 0.5)}; \addlegendentry{$f(x)=\frac{1}{2}x$} @@ -57,7 +60,7 @@ &= \frac{m}{m^2+1} \left ( y_P + \frac{1}{m} \cdot x_P - t \right )\label{eq:solution-linear-r} \end{align} It is obvious that a minium has to exist, the $x$ from Equation~\ref{eq:solution-linear-r} - has to be this minimum. + has to be this minimum. $\qed$ \end{proof} \clearpage @@ -113,4 +116,39 @@ The point with minimum distance can be found by: \Set{b} &\text{if } S_1(f, P) \ni x > b \end{cases}\] -\todo[inline]{argument? proof?} +If $S_1(f, P) \cap [a,b] \neq \emptyset$, then $\underset{x\in[a,b]}{\arg \min d_{P,f}(x)} = S_1(f,P) \cap [a,b]$, +because $S_1(f,P)$ gives all global minima of $f$. Those are also +minima for the intervall $[a,b]$. There are not more minima, because +$S_1$ gives all minima of $P$ to $f$. + +If $S_1(f, P) \cap [a,b] = \emptyset$, then it is not that simple. +But we can calculate the distance function: + +\begin{align} + d_{P,f}(x) &= \sqrt{(x-x_P)^2 + (f(x) - y_P)^2}\\ + &= \sqrt{(x^2 - 2x x_P + x_P^2) + (mx + (t-y_P))^2}\\ + &= \sqrt{(x^2 - 2x x_P + x_P^2) + m^2 x^2 + 2mx(t-y_P) + (t-y_P)^2}\\ + &= \sqrt{x^2(1+m^2) + x(-2 x_P + 2m(t-y_P)) + (x_P^2 + (t-y_P)^2)} +\end{align} + +This function (defined on $\mdr$) is symmetry to the axis +\begin{align} + x_S &= - \frac{-2 x_P + 2m(t-y_P)}{2(1+m^2)}\\ + &= \frac{x_P - m(t-y_P)}{1+m^2}\\ + &= \frac{m}{m^2+1} (y_P + \frac{1}{m} x_P - t) +\end{align} + +$f$ is on $(-\infty, x_S]$ strictly monotonically decreasing and +on $[x_S, + \infty)$ strictly monotonically increasing. + +Thus we can conclude: +\[\forall x,y \in \mdr: x \leq y < x_S \Rightarrow d_{P,f}(x_S) < d_{P,f}(y) \leq d_{P,f}(x)\] +\[\forall x,y \in \mdr: x_S < y \leq x \Rightarrow d_{P,f}(x_S) < d_{P,f}(y) \leq d_{P,f}(x)\] + +When $S_1(f, P) \cap [a,b] = \emptyset$, then you can have two cases: +\begin{itemize} + \item $a \leq b < x_S$: $b$ has the shortest distance in $[a,b]$ + on the graph of $f$ to $P$. + \item $x_S < a \leq b$: $a$ has the shortest distance in $[a,b]$ + on the graph of $f$ to $P$. +\end{itemize} diff --git a/documents/math-minimal-distance-to-cubic-function/math-minimal-distance-to-cubic-function.pdf b/documents/math-minimal-distance-to-cubic-function/math-minimal-distance-to-cubic-function.pdf index c36d4efb02b773bfe6f49c079fc18d2c4b55243d..3e863079df3cb1174e2bd7c882f99e626f8ee898 100644 GIT binary patch delta 236030 zcmZsiV{j%w*QR58;!JGY#uMANolLAJwr$%sCeFmRZD-I$z2jxscg`ozKsy=G4TysnBB>*P&~1$Wlj>!cCVc#-uH}qi|1t^qr~?dCk6M;2 zmx@BN^W@(i@O)sdKB9~5irDK!=z?sC^FY&u%L{d#N)!}Uk%m=)j5j#OI(ui(*S=-u>od#EMdP z3*$wxJXxTn#JhPI>0FkV&zfzMSm9wHxy{^CJ1LG!5}A$DwT=2wPZJ`+X}Z!3J~JIH ztLycAn@`Wzb;xj*9Yf1Y;I>fJ(YP(}W}rE;N+fBmsDhGn@twdGqzjL&_gCxQXMap{ 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