From f65ad3de8836d2db95fc5b3209fe3ce2db13e809 Mon Sep 17 00:00:00 2001 From: Mark Date: Sat, 4 Jan 2025 11:02:26 -0800 Subject: [PATCH] Added geometric optimization --- Advanced/Geometric Optimization/main.tex | 251 +++++++++++++++++++++++ 1 file changed, 251 insertions(+) create mode 100755 Advanced/Geometric Optimization/main.tex diff --git a/Advanced/Geometric Optimization/main.tex b/Advanced/Geometric Optimization/main.tex new file mode 100755 index 0000000..e2b6d43 --- /dev/null +++ b/Advanced/Geometric Optimization/main.tex @@ -0,0 +1,251 @@ +% use [nosolutions] flag to hide solutions. +% use [solutions] flag to show solutions. +\documentclass[ + solutions +]{../../resources/ormc_handout} +\usepackage{../../resources/macros} + +\uptitlel{Advanced 2} +\uptitler{\smallurl{}} +\title{Geometric Optimization} +\subtitle{ + Prepared by Mark on \today \par + Based on a handout by Nakul \& Andreas +} + +\begin{document} + + \maketitle + + \section{Optimization} + + \problem{} + Let $A$ and $B$ be two points on the same side of a given line $\ell$. \par + Find a point $C$ on $\ell$ so that $|AC| + |BC|$ is minimized. + + + \begin{center} + \begin{tikzpicture}[scale = 2] + \draw[-] (-2,0) -- (3,0); + + \fill[fill=black] (-0.6, 1) circle (0.03) node[below] {$A$}; + \fill[fill=black] (1.6, 0.75) circle (0.03) node[below] {$B$}; + \fill[fill=black] (0.5, 0) circle (0.03) node[below] {$C$}; + + \draw[-] (-0.6, 1) -- (0.5, 0) -- (1.6, 0.75); + \end{tikzpicture} + \end{center} + + \vfill + \pagebreak + + + \definition{} + An \textit{ellipse} with foci $A$, $B$ and radius $r$ is the set of all points $C$ where $|AB| + |BC| = r$. + + \problem{} + Consider a reflective ellipse with foci $A$ and $B$. \par + Find all points $X$ on the ellipse where $A$ can aim a laser at so that the beam reaches $B$. \par + \hint{use \ref{simtri}} + + \begin{center} + \begin{tikzpicture}[ + dot/.style={draw, fill, circle, inner sep=1.2}, + scale = 0.75 + ] + \def\a{5} % large half axis + \def\b{3} % small half axis + + \draw (0,0) ellipse ({\a} and {\b}); + + % Foci + \node[dot,label={above right:$A$}] (A) at ({-sqrt(\a*\a-\b*\b)},0) {}; + \node[dot,label={above:$B$}] (B) at ({+sqrt(\a*\a-\b*\b)},0) {}; + + % Node on ellipse + \def\angle{150} + \node[dot,label={\angle:$X$}] (X) at (\angle:{\a} and {\b}) {}; + \draw (A) -- (X) -- (B); + \end{tikzpicture} + \end{center} + + \vfill + \pagebreak + + + + + \problem{} + Let $C$ be a point in the interior of a given angle. Find points A and B on the sides + of the angle such that the perimeter of the triangle ABC is a minimum. + \vfill + + \problem{} + In a convex quadrilateral ABCD, find the point T for which the sum of the distances to + the vertices is minimal. + \vfill + \pagebreak + + + \problem{} + A road needs to be constructed from town A to town B, crossing a river, over which a + perpendicular bridge is to be constructed. + Where should the bridge be placed to minimize $|AR_1| + |R_1R_2| + |R_2B|$? + + \begin{center} + \begin{tikzpicture}[scale = 1.5] + \draw[-] (-5, 0.5) -- (5, 0.5); + \draw[-] (-5, -0.5) -- (5, -0.5); + + \fill[fill=black] (-3, -3) circle (0.06) node[below] {$A$}; + \fill[fill=black] (0.5, -0.5) circle (0.06) node[below] {$R_1$}; + \fill[fill=black] (0.5, 0.5) circle (0.06) node[below right] {$R_2$}; + \fill[fill=black] (3, 1) circle (0.06) node[below] {$B$}; + + \draw[-] (-3, -3) -- (0.5, -0.5) -- (0.5, 0.5) -- (3,1); + \end{tikzpicture} + \end{center} + + \pagebreak + + + + + + \problem{} + Consider an equilateral triangle triangle with vertices labeled $A$, $B$, and $C$. \par + Let P be a point inside this triangle. Place $D$, $E$, and $F$ so that $PD$, $PE$, and $PF$ + are the perpendiculars from $P$ to the sides of the triangle. \par + + Find all points $P$ where $|PD| + |PE| + |PF|$ is minimized. + + \begin{center} + \begin{tikzpicture}[scale = 4] + \draw[-] + (-1, 0) + -- (1, 0) + -- (0, 1.47) + -- cycle + ; + + \fill[fill=black] (0, 1.47) circle (0.03) node[above] (A) {$A$}; + \fill[fill=black] (1, 0) circle (0.03) node[below right] {$B$}; + \fill[fill=black] (-1, 0) circle (0.03) node[below left] {$C$}; + + \fill[fill=black] (0.39, 0.9) circle (0.03) node[above right] {$D$}; + \fill[fill=black] (-0.3, 0) circle (0.03) node[below right] {$E$}; + \fill[fill=black] (-0.555, 0.65) circle (0.03) node[above left] {$F$}; + + \fill[fill=black] (-0.3, 0.5) circle (0.03) node[below right] {$P$}; + + + \draw[-] (-0.3, 0.5) -- (0.39, 0.9); + \draw[-] (-0.3, 0.5) -- (-0.3, 0); + \draw[-] (-0.3, 0.5) -- (-0.555, 0.65); + + \draw[-] (-0.2, 0) -- (-0.2, 0.1) -- (-0.3, 0.1); + + + \end{tikzpicture} + \end{center} + + \vfill + \pagebreak + + + + + \problem{} + With the same setup as \ref{equi}, find all points $P$ where $|PA| + |PB| + |PC|$ is minimized. + \vfill + + + \problem{} + Solve \ref{equi2} for a triangle that isn't equilateral. + \vfill + \pagebreak + + + + \problem{} + Draw a circle, then draw two distinct tangents $\ell_1$ and $\ell_2$ that intersect at point $A$. \par + Let $P$ be a point on the circle between the tangents, and $BC$ be the tangent at that point. + Describe how $P$ shoud be selected in order to minimize the perimeter of triangle $ABC$. + + \begin{center} + \begin{tikzpicture}[scale = 2] + \draw (0, 0) circle (1); + + \fill[fill=black] (-4, -1) circle (0.04) node[below] (A) {$A$}; + \draw[-] (-4, -1) -- (2, -1); + \draw[-] (-4, -1) -- (1.2, 1.78); + \draw[-] (-1, -1) -- (-1, 0.6); + + \fill[fill=black] (-1, 0.6) circle (0.04) node[above left] {$B$}; + \fill[fill=black] (-1, 0) circle (0.04) node[right] {$P$}; + \fill[fill=black] (-1, -1) circle (0.04) node[below] {$C$}; + \end{tikzpicture} + \end{center} + + \vspace{2cm} + + + \problem{} + Now, assume that $\ell_1$ and $\ell_2$ intersect at $A$, and pick a point $P$ between them. \par + Find $BC$ through $P$ so that the perimeter of $ABC$ is minimized. + + \vfill + \pagebreak + + + \section{Bonus Problems} + + \problem{} + Given a cube $A_1B_1C_1D_1A_2B_2C_2D_2$ with side length $l$, \par + find the angle and distance between lines $A_1B_2$ and $A_2C_1$. + + \begin{solution} + Triangle $A_1B_2D_2$ is equilateral. \par + Also, point $A_2$ is equidistant from each of this triangle's vertices. \par + Therefore, its projection onto the plane formed by $A_1$, $B_2$, and $D_2$ is the center of the triangle. \par + + \vspace{2mm} + + Similarly, $C_1$ is mapped to the center of $A_1B_2D_2$. \par + Therefore, lines $A_1B_2$ and $A_2C_1$ are perpendicular and the distance between them is + equal to the distance from the center of triangle $A_1B_2D_2$ to its side. + + \vspace{2mm} + + Since all the sides of this triangle have length $l\sqrt{2}$, the distance in question is $\frac{a}{\sqrt{6}}$. + + \end{solution} + + \vfill + + \problem{} + Consider a cube $A_1B_1C_1D_1A_2B_2C_2D_2$, and let $K$, $L$, \par + and $M$ be midpoints of the edges $A_2D_2$, $A_1B_1$, and $C_1C_2$. \par + Show that the triangle formed by $KLM$ is equilateral, and that its center is the center of the cube. + + \begin{solution} + Let $O$ be the center of the cube. Then, $|OK| = |C_1D_2|$, $|2OL| = |D_2A_1|$, and $2|OM| = |A_1C_1|$. \par + Since triangle $C_1D_2A_1$is equilateral, triangle $KLM$ is equilateral and has $O$ as its center. + \end{solution} + + \vfill + + \pagebreak + + + \problem{} + Consider all $n$-gons with a certain perimeter. + Show that the $n$-gon with maximal area has equal sides + \vfill + + \problem{} + Consider all $n$-gons with a certain perimeter. + Show that the $n$-gon with maximal area has equal angles + \vfill + +\end{document} \ No newline at end of file