#LyX 2.1 created this file. For more info see http://www.lyx.org/ \lyxformat 474 \begin_document \begin_header \textclass article \begin_preamble \usepackage{etex} \usepackage[all, knot]{xy} \xyoption{arc} \usepackage{qtree} \usepackage{tipa} \usepackage{pstricks} \usepackage{pst-plot} \usepackage{rotating} \usepackage{tikz} \usetikzlibrary{scopes} \end_preamble \use_default_options false \maintain_unincluded_children false \language english \language_package default \inputencoding auto \fontencoding global \font_roman lmodern \font_sans default \font_typewriter default \font_math auto \font_default_family default \use_non_tex_fonts false \font_sc false \font_osf false \font_sf_scale 100 \font_tt_scale 100 \graphics default \default_output_format pdf \output_sync 0 \bibtex_command default \index_command default \paperfontsize default \spacing single \use_hyperref false \papersize default \use_geometry false \use_package amsmath 1 \use_package amssymb 1 \use_package cancel 0 \use_package esint 1 \use_package mathdots 1 \use_package mathtools 0 \use_package mhchem 1 \use_package stackrel 0 \use_package stmaryrd 0 \use_package undertilde 0 \cite_engine basic \cite_engine_type default \biblio_style plain \use_bibtopic false \use_indices false \paperorientation portrait \suppress_date false \justification true \use_refstyle 0 \index Index \shortcut idx \color #008000 \end_index \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \paragraph_indentation default \quotes_language english \papercolumns 1 \papersides 1 \paperpagestyle default \tracking_changes false \output_changes false \html_math_output 0 \html_css_as_file 0 \html_be_strict true \end_header \begin_body \begin_layout Section Introduction \end_layout \begin_layout Standard In order to properly typeset the document and have instant preview working, the following packages have to be installed: \end_layout \begin_layout LyX-Code xy, qtree, tipa, pstricks, pst-plot, rotating, tikz, pgf, etex \end_layout \begin_layout Standard Due to the use of postscript specials in Section \begin_inset CommandInset ref LatexCommand ref reference "sec:PSTricks" \end_inset , only the Postscript or PDF (ps2pdf) output formats will correctly render the included images. However, the PDF (XeTeX) output seems to also work, while trying to typeset the document with \family typewriter pdflatex \family default or \family typewriter pdflualatex \family default will generate errors. \end_layout \begin_layout Section Diagrams by XY(-pic) \end_layout \begin_layout Standard Few examples of diagrams were taken from Aaron Landa's archive of xy-pic examples, see \begin_inset Flex URL status collapsed \begin_layout Plain Layout http://www.math.columbia.edu/~lauda/xy/ \end_layout \end_inset . \end_layout \begin_layout Description Braid \end_layout \begin_layout Standard \begin_inset Preview \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Plain Layout \backslash xy \end_layout \begin_layout Plain Layout (5,10)*{}; (-10,-10)*{} ** \backslash crv{(6,-2)&(-12,4)} \end_layout \begin_layout Plain Layout \backslash POS?(.25)*{ \backslash hole}="x" \backslash POS?(.45)*{ \backslash hole}="y" \backslash POS?(.6)*{ \backslash hole}="z"; \end_layout \begin_layout Plain Layout "y"+(0,-1); (2,-10)*{} ** \backslash crv{} \backslash POS?(.2)*{ \backslash hole}="M"; \end_layout \begin_layout Plain Layout (-10,10)*{}; "z" ** \backslash crv{(-9,0)}; \end_layout \begin_layout Plain Layout "z"; "M" ** \backslash crv{}; \end_layout \begin_layout Plain Layout "M"; "x" ** \backslash crv{(5,0)}; \end_layout \begin_layout Plain Layout "x"; "y" ** \backslash crv{(0,7) & (-5,6)}; \end_layout \begin_layout Plain Layout \backslash endxy \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Description Knots \end_layout \begin_layout Standard \begin_inset Preview \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Plain Layout \backslash xy \end_layout \begin_layout Plain Layout (6,9)*{}="1"; \end_layout \begin_layout Plain Layout (-8.5,-1)*{}="2"; \end_layout \begin_layout Plain Layout "1";"2" ** \backslash crv{~*=<.5pt>{.} (0,30)}?(.75)* \backslash dir{>}+(-2,2)*{z}; \end_layout \begin_layout Plain Layout (-6.5,8)*{}="1"; \end_layout \begin_layout Plain Layout (-.5,-9)*{}="2"; \end_layout \begin_layout Plain Layout "1";"2" ** \backslash crv{~*=<.5pt>{.} (-28.5,9.3)}?(.7)* \backslash dir{>}+(-2,-2)*{u}; \end_layout \begin_layout Plain Layout (-9.5,-3.35)*{}="1"; \end_layout \begin_layout Plain Layout (8.5,-3)*{}="2"; \end_layout \begin_layout Plain Layout "1";"2" ** \backslash crv{~*=<.5pt>{.} (-17.67,-24.19)}?(.7)* \backslash dir{>}+(-1,-3)*{x}; \end_layout \begin_layout Plain Layout (1,-10)*{}="1"; \end_layout \begin_layout Plain Layout (6.5,7.13)*{}="2"; \end_layout \begin_layout Plain Layout "1";"2" ** \backslash crv{~*=<.5pt>{.} (17.67,-24.19)}?(.7)* \backslash dir{>}+(3,-1)*{y}; \end_layout \begin_layout Plain Layout (11,-1)*{}="1"; \end_layout \begin_layout Plain Layout (-4,8)*{}="2"; \end_layout \begin_layout Plain Layout "1";"2" ** \backslash crv{~*=<.5pt>{.} (28.5,9.3)}?(.93)* \backslash dir{>}+(1,2)*{w}; \end_layout \begin_layout Plain Layout \backslash endxy \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Description Circles \end_layout \begin_layout Standard \begin_inset Preview \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Plain Layout \backslash xy \end_layout \begin_layout Plain Layout (0,0)* \backslash xycircle(14,8){++ \backslash dir{<}}; \end_layout \begin_layout Plain Layout (0,0)* \backslash xycircle<10pt>{{.}}; \end_layout \begin_layout Plain Layout (0,0)* \backslash xycircle(5,2){:}; \end_layout \begin_layout Plain Layout (0,15)* \backslash xycircle(10,4){--}; \end_layout \begin_layout Plain Layout \backslash endxy \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Description Cobordism \end_layout \begin_layout Standard \begin_inset Preview \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Plain Layout \backslash xy \end_layout \begin_layout Plain Layout (0,0)* \backslash ellipse(3,1){.}; \end_layout \begin_layout Plain Layout (0,0)* \backslash ellipse(3,1)__,=:a(-180){-}; \end_layout \begin_layout Plain Layout (-6,-8)* \backslash ellipse(3,1){.}; \end_layout \begin_layout Plain Layout (6,-8)* \backslash ellipse(3,1){.}; \end_layout \begin_layout Plain Layout (0,-8)* \backslash ellipse(3,1){.}; \end_layout \begin_layout Plain Layout (-6,-8)* \backslash ellipse(3,1)__,=:a(-180){-}; \end_layout \begin_layout Plain Layout (6,-8)* \backslash ellipse(3,1)__,=:a(-180){-}; \end_layout \begin_layout Plain Layout (0,-8)* \backslash ellipse(3,1)__,=:a(180){-}; \end_layout \begin_layout Plain Layout (-3,6)* \backslash ellipse(3,1){-}; \end_layout \begin_layout Plain Layout (3,6)* \backslash ellipse(3,1){-}; \end_layout \begin_layout Plain Layout (-3,12)*{}="1"; \end_layout \begin_layout Plain Layout (3,12)*{}="2"; \end_layout \begin_layout Plain Layout (-9,12)*{}="A2"; \end_layout \begin_layout Plain Layout (9,12)*{}="B2"; \end_layout \begin_layout Plain Layout "1";"2" ** \backslash crv{(-3,7) & (3,7)}; \end_layout \begin_layout Plain Layout (-3,0)*{}="A"; \end_layout \begin_layout Plain Layout (3,0)*{}="B"; \end_layout \begin_layout Plain Layout (-3,1)*{}="A1"; \end_layout \begin_layout Plain Layout (3,1)*{}="B1"; \end_layout \begin_layout Plain Layout "A";"A1" ** \backslash dir{-}; \end_layout \begin_layout Plain Layout "B";"B1" ** \backslash dir{-}; \end_layout \begin_layout Plain Layout "B2";"B1" ** \backslash crv{(8,7) & (3,5)}; \end_layout \begin_layout Plain Layout "A2";"A1" ** \backslash crv{(-8,7) & (-3,5)}; \end_layout \begin_layout Plain Layout %REFLECT \end_layout \begin_layout Plain Layout (3,-16)*{}="1"; \end_layout \begin_layout Plain Layout (9,-16)*{}="2"; \end_layout \begin_layout Plain Layout "1";"2" ** \backslash crv{(3,-10) & (9,-10)}; \end_layout \begin_layout Plain Layout (-3,-16)*{}="1"; \end_layout \begin_layout Plain Layout (-9,-16)*{}="2"; \end_layout \begin_layout Plain Layout "1";"2" ** \backslash crv{(-3,-10) & (-9,-10)}; \end_layout \begin_layout Plain Layout (-15,-16)*{}="A2"; \end_layout \begin_layout Plain Layout (15,-16)*{}="B2"; \end_layout \begin_layout Plain Layout (-3,0)*{}="A"; \end_layout \begin_layout Plain Layout (3,0)*{}="B"; \end_layout \begin_layout Plain Layout (-3,-1)*{}="A1"; \end_layout \begin_layout Plain Layout (3,-1)*{}="B1"; \end_layout \begin_layout Plain Layout "A";"A1" ** \backslash dir{-}; \end_layout \begin_layout Plain Layout "B";"B1" ** \backslash dir{-}; 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\end_layout \begin_layout Plain Layout ?(.08)* \backslash dir{<}; ?(.85)* \backslash dir{<}; \end_layout \begin_layout Plain Layout (7.5,0)*{}="x1"; (-7.5,0)*{}="x2"; \end_layout \begin_layout Plain Layout "x1"; "x2" ** \backslash crv{(7,-10) & (-7,-10)}; \end_layout \begin_layout Plain Layout (16,-20)*{ \backslash bullet}="RDD"+(2.5,-1)*{x^{ \backslash ast}}; \end_layout \begin_layout Plain Layout (-20,-20)*{ \backslash bullet}="LDD"+(-1,-3)*{x}; \end_layout \begin_layout Plain Layout (20,-12.5)*{ \backslash bullet}="RUD"+(3.5,1)*{x^{ \backslash ast}}; \end_layout \begin_layout Plain Layout (-16,-15)*{ \backslash bullet}="LUD"; \end_layout \begin_layout Plain Layout (-16,-2.5)*{}="A"; (16.1,-14.9)*{}="B"; \end_layout \begin_layout Plain Layout "RD"; "RDD" ** \backslash dir{-}; \end_layout \begin_layout Plain Layout "LD"; "LDD" ** \backslash dir{-}; \end_layout \begin_layout Plain Layout "A"; "LUD" ** \backslash dir{.}; \end_layout \begin_layout Plain Layout "RDD"; "LDD" ** \backslash crv{(0,-17)}; ?(.57)* \backslash dir{>}; \end_layout \begin_layout Plain Layout "RU"; "RUD" ** \backslash dir{-}; \end_layout \begin_layout Plain Layout "LU"; "A" ** \backslash dir{-}; \end_layout \begin_layout Plain Layout "B"; "RUD" ** \backslash crv{(18,-14.15)}; \end_layout \begin_layout Plain Layout "B"; "LUD" ** \backslash crv{~*=<4pt>{.}(0,-18)}; ?(.4)* \backslash dir{<}; \end_layout \begin_layout Plain Layout \backslash endxy \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Description Strings \end_layout \begin_layout Standard \begin_inset Preview \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash xy (-15,0)*{};(15,0)*{}; \end_layout \begin_layout Plain Layout \backslash endxy \end_layout \begin_layout Plain Layout \backslash qquad = \backslash qquad \backslash xy \end_layout \begin_layout Plain Layout (-10,12)*{}; (-10,-12)*{}; ** \backslash dir{-}?(.5)* \backslash dir{<}+(-4,0)*{F(x)}; \end_layout \begin_layout Plain Layout (-2,12)*{}="1"; \end_layout \begin_layout Plain Layout (6,-7) *{}="2"; \end_layout \begin_layout Plain Layout (14,-7) *{}="3"; \end_layout \begin_layout Plain Layout (14,-13)*{}="3'"; \end_layout \begin_layout Plain Layout (18,0) *={}=""; \end_layout \begin_layout Plain Layout "1";"2" ** \backslash crv{(-4,-21)&(5,-15)}; \end_layout \begin_layout Plain Layout ?(.15)* \backslash dir{<}+(5,10)*{ \backslash overline{F(x)}}; \end_layout \begin_layout Plain Layout "2";"3" ** \backslash crv{(6,1)&(14,1)}; \end_layout \begin_layout Plain Layout ?(0)* \backslash dir{<}+(4,8)*{ \backslash widetilde{F(i_x)}} ; \end_layout \begin_layout Plain Layout ?(.95)* \backslash dir{<}+(5,-2)*{F( \backslash overline x)}; \end_layout \begin_layout Plain Layout "3";"3'" ** \backslash dir{-}; \end_layout \begin_layout Plain Layout (-2,6)*{}="X1"; \end_layout \begin_layout Plain Layout (-10,6)*{}="X2"; \end_layout \begin_layout Plain Layout "X1";"X2" ** \backslash crv{~*=<2pt>{.}(-2,-1)&(-10,-1)}; \end_layout \begin_layout Plain Layout (-2,-6)*{}="XA"; \end_layout \begin_layout Plain Layout (-10,-6)*{}="XB"; \end_layout \begin_layout Plain Layout "XA";"XB" ** \backslash crv{~*=<2pt>{.}(-2,1)&(-10,1)}; \end_layout \begin_layout Plain Layout \backslash endxy \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Description Categories \end_layout \begin_layout Standard \begin_inset Preview \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Plain Layout \backslash xymatrix{ \end_layout \begin_layout Plain Layout (x \backslash otimes 1) \backslash otimes y \end_layout \begin_layout Plain Layout \backslash ar[rr]^{a_{x,1,y}} \end_layout \begin_layout Plain Layout \backslash ar[dr]_{r_x \backslash otimes 1_y} \end_layout \begin_layout Plain Layout && x \backslash otimes(1 \backslash otimes y) \end_layout \begin_layout Plain Layout \backslash ar[dl]^{1_x \backslash otimes \backslash ell_y } \backslash \backslash \end_layout \begin_layout Plain Layout & x \backslash otimes y } \end_layout \end_inset \end_layout \end_inset \begin_inset Preview \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Plain Layout \backslash xy \end_layout \begin_layout Plain Layout (-9,0)*+{ \backslash bullet}="1"; \end_layout \begin_layout Plain Layout (0,0)*+{ \backslash bullet}="2"; \end_layout \begin_layout Plain Layout { \backslash ar@/^.9pc/ "1";"2"}; \end_layout \begin_layout Plain Layout { \backslash ar@/_.9pc/ "1";"2"}; \end_layout \begin_layout Plain Layout { \backslash ar@/^2pc/ "1";"2"}; \end_layout \begin_layout Plain Layout { \backslash ar@/_2pc/ "1";"2"}; \end_layout \begin_layout Plain Layout { \backslash ar@{=>} (-4.5,1.5)*{};(-4.5,-1.5)*{}} ; \end_layout \begin_layout Plain Layout { \backslash ar@{=>} (-4.5,7.25)*{};(-4.5,4.75)*{}} ; \end_layout \begin_layout Plain Layout { \backslash ar@{=>} (-4.5,-4.75)*{};(-4.5,-7.25)*{}} ; \end_layout \begin_layout Plain Layout (0,0)*+{ \backslash bullet}="1"; \end_layout \begin_layout Plain Layout (9,0)*+{ \backslash bullet}="2"; \end_layout \begin_layout Plain Layout (18,0)*+{ \backslash bullet}="3"; \end_layout \begin_layout Plain Layout { \backslash ar "1";"2"}; \end_layout \begin_layout Plain Layout { \backslash ar "2";"3"}; \end_layout \begin_layout Plain Layout { \backslash ar@/^1pc/ "1";"2"}; \end_layout \begin_layout Plain Layout { \backslash ar@/_1pc/ "1";"2"}; \end_layout \begin_layout Plain Layout { \backslash ar@{=>} (4.5,3)*{};(4.5,.75)*{}} ; \end_layout \begin_layout Plain Layout { \backslash ar@{=>} (4.5,-.75)*{};(4.5,-3)*{}} ; \end_layout \begin_layout Plain Layout \backslash endxy \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Description Transformations \end_layout \begin_layout Standard \begin_inset Preview \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Plain Layout \backslash xy %FIG.19. Natural transformation between functors. \end_layout \begin_layout Plain Layout (-26,-2)*+{ \backslash bullet}="a"; \end_layout \begin_layout Plain Layout (-12,3)*+{ \backslash bullet}="b"; \end_layout \begin_layout Plain Layout (-16,-5)*+{ \backslash bullet}="c"; \end_layout \begin_layout Plain Layout { \backslash ar@/^.25pc/ "a";"b"}; \end_layout \begin_layout Plain Layout { \backslash ar@/_.25pc/"a";"c"}; \end_layout \begin_layout Plain Layout { \backslash ar@/_.15pc/ "b";"c"}; \end_layout \begin_layout Plain Layout %START TOP \end_layout \begin_layout Plain Layout (12,10)*+{ \backslash bullet}="ta"; \end_layout \begin_layout Plain Layout (26,15)*+{ \backslash bullet}="tb"; \end_layout \begin_layout Plain Layout (22,8)*+{ \backslash bullet}="tc"; \end_layout \begin_layout Plain Layout { \backslash ar@/^.25pc/ "ta";"tb"}; \end_layout \begin_layout Plain Layout { \backslash ar@/_.25pc/ "ta";"tc"}; \end_layout \begin_layout Plain Layout { \backslash ar@/_.15pc/ "tb";"tc"}; \end_layout \begin_layout Plain Layout %START bottom \end_layout \begin_layout Plain Layout (12,-10)*+{ \backslash bullet}="ba"; \end_layout \begin_layout Plain Layout (26,-5)*+{ \backslash bullet}="bb"; \end_layout \begin_layout Plain Layout (22,-12)*+{ \backslash bullet}="bc"; \end_layout \begin_layout Plain Layout { \backslash ar@/^.25pc/@{.>} "ba";"bb"}; \end_layout \begin_layout Plain Layout { \backslash ar@/_.25pc/ "ba";"bc"}; \end_layout \begin_layout Plain Layout { \backslash ar@/_.15pc/ "bb";"bc"}; \end_layout \begin_layout Plain Layout %START CONNECT \end_layout \begin_layout Plain Layout { \backslash ar@/_.15pc/ "ta";"ba"}; \end_layout \begin_layout Plain Layout { \backslash ar@/_.15pc/ "tb";"bb"}; \end_layout \begin_layout Plain Layout { \backslash ar@/_.15pc/ "tc";"bc"}; \end_layout \begin_layout Plain Layout %START ARROWS AND LABELS \end_layout \begin_layout Plain Layout (-12,6)*{}="DF"; \end_layout \begin_layout Plain Layout (6,10)*{}="CF"; \end_layout \begin_layout Plain Layout { \backslash ar@/^.35pc/^F "DF";"CF"}; \end_layout \begin_layout Plain Layout (-12,-6)*{}="DF"; \end_layout \begin_layout Plain Layout (6,-10)*{}="CF"; \end_layout \begin_layout Plain Layout { \backslash ar@/_.35pc/_G "DF";"CF"}; \end_layout \begin_layout Plain Layout (-2,5)*{}="DF"; \end_layout \begin_layout Plain Layout (-2,-5)*{}="CF"; \end_layout \begin_layout Plain Layout { \backslash ar@{=>}^{ \backslash alpha} "DF";"CF"}; \end_layout \begin_layout Plain Layout (29,3)*{D}; \end_layout \begin_layout Plain Layout (-23,3)*{C}; \end_layout \begin_layout Plain Layout \backslash endxy \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Section IPA \end_layout \begin_layout Standard \begin_inset Preview \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Plain Layout \backslash textipa{Ekspl@"neIS@n} \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Section QTree \end_layout \begin_layout Standard \begin_inset Preview \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Plain Layout \backslash Tree[.S [.N This ] [.V is ] [.NP [.Det a ] [.N test ] ] ] \end_layout \end_inset \end_layout \end_inset \begin_inset Preview \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Plain Layout \backslash Tree [.$ \backslash sigma$ $ \backslash mu$ $ \backslash mu$ ] \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Section PSTricks \begin_inset CommandInset label LatexCommand label name "sec:PSTricks" \end_inset \end_layout \begin_layout Standard Some simple graphing examples. \end_layout \begin_layout Subsection Function plotting \end_layout \begin_layout Standard \begin_inset Preview \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash begin{pspicture*}(-1,-3.5)(10.5,3.5) \end_layout \begin_layout Plain Layout \backslash psaxes{->}(0,0)(0,-3)(10,3) \end_layout \begin_layout Plain Layout \backslash uput[-90](10,0){$x$} \backslash uput[0](0,3){$f(x)$} \end_layout \begin_layout Plain Layout \backslash psplot[linecolor=red,linewidth=1pt]{0.1}{10}{ x ln } \end_layout \begin_layout Plain Layout \backslash end{pspicture*} \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection Inline data plotting \end_layout \begin_layout Standard \align center \begin_inset Preview \begin_layout Standard \align center \begin_inset ERT status open \begin_layout Plain Layout \backslash begin{pspicture*}(-1.5,-1.5)(11,5) \end_layout \begin_layout Plain Layout \backslash psaxes{->}(0,0)(0,0)(10,5) \end_layout \begin_layout Plain Layout \backslash psset{linewidth=2.5pt} \end_layout \begin_layout Plain Layout \backslash rput(5.0,-1.0){Speed [MPH]} \end_layout \begin_layout Plain Layout \backslash rput(-1.0,1.0){ \backslash begin{rotate}{90}Cost [Dollars] \backslash end{rotate}} \end_layout \begin_layout Plain Layout \backslash savedata{ \backslash mydata}[{{0,0},{1.3,0.32},{5.5,1.1},{6.5,1.15},{8.1,1.28},{10,4}}] \end_layout \begin_layout Plain Layout \backslash dataplot[plotstyle=curve,showpoints=true,dotstyle=triangle,linecolor=blue]{ \backslash mydata} \end_layout \begin_layout Plain Layout \backslash end{pspicture*} \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection Lissajous curves \end_layout \begin_layout Standard \begin_inset Preview \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash begin{pspicture*}(-1.1,-1.1)(1.1,1.1) \end_layout \begin_layout Plain Layout \backslash parametricplot[plotstyle=line,plotpoints=50,linecolor=red]{0}{8}{50 t mul cos 100 t mul sin} \end_layout \begin_layout Plain Layout \backslash end{pspicture*} \end_layout \end_inset \end_layout \end_inset \begin_inset space \hfill{} \end_inset \begin_inset Preview \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash begin{pspicture*}(-1.1,-1.1)(1.1,1.1) \end_layout \begin_layout Plain Layout \backslash parametricplot[plotstyle=line,plotpoints=100,linecolor=red]{0}{8}{150 t mul cos 100 t mul sin} \end_layout \begin_layout Plain Layout \backslash end{pspicture*} \end_layout \end_inset \end_layout \end_inset \begin_inset space \hfill{} \end_inset \begin_inset Preview \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash begin{pspicture*}(-1.1,-1.1)(1.1,1.1) \end_layout \begin_layout Plain Layout \backslash parametricplot[plotstyle=line,plotpoints=150,linecolor=red]{0}{8}{150 t mul cos 200 t mul sin} \end_layout \begin_layout Plain Layout \backslash end{pspicture*} \end_layout \end_inset \end_layout \end_inset \begin_inset space \hfill{} \end_inset \begin_inset Preview \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash begin{pspicture*}(-1.1,-1.1)(1.1,1.1) \end_layout \begin_layout Plain Layout \backslash parametricplot[plotstyle=line,plotpoints=200,linecolor=red]{0}{8}{250 t mul cos 200 t mul sin} \end_layout \begin_layout Plain Layout \backslash end{pspicture*} \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Section TikZ \end_layout \begin_layout Standard \begin_inset Flex URL status collapsed \begin_layout Plain Layout http://www.texample.net/media/tikz/examples/TEX/free-body-diagrams.tex \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Preview \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Plain Layout \backslash def \backslash iangle{35} % Angle of the inclined plane \end_layout \begin_layout Plain Layout \backslash def \backslash down{-90} \end_layout \begin_layout Plain Layout \backslash def \backslash arcr{0.5cm} % Radius of the arc used to indicate angles \end_layout \begin_layout Plain Layout \end_layout \begin_layout Plain Layout \backslash begin{tikzpicture}[ \end_layout \begin_layout Plain Layout force/.style={>=latex,draw=blue,fill=blue}, \end_layout \begin_layout Plain Layout axis/.style={densely dashed,gray,font= \backslash small}, \end_layout \begin_layout Plain Layout M/.style={rectangle,draw,fill=lightgray,minimum size=0.5cm,thin}, \end_layout \begin_layout Plain Layout m/.style={rectangle,draw=black,fill=lightgray,minimum size=0.3cm,thin}, \end_layout \begin_layout Plain Layout plane/.style={draw=black,fill=blue!10}, \end_layout \begin_layout Plain Layout string/.style={draw=red, thick}, \end_layout \begin_layout Plain Layout pulley/.style={thick}, \end_layout \begin_layout Plain Layout ] \end_layout \begin_layout Plain Layout \end_layout \begin_layout Plain Layout \backslash matrix[column sep=1cm] { \end_layout \begin_layout Plain Layout %% Sketch \end_layout \begin_layout Plain Layout \backslash draw[plane] (0,-1) coordinate (base) \end_layout \begin_layout Plain Layout -- coordinate[pos=0.5] (mid) ++( \backslash iangle:3) coordinate (top) \end_layout \begin_layout Plain Layout |- (base) -- cycle; \end_layout \begin_layout Plain Layout \backslash path (mid) node[M,rotate= \backslash iangle,yshift=0.25cm] (M) {}; \end_layout \begin_layout Plain Layout \backslash draw[pulley] (top) -- ++( \backslash iangle:0.25) circle (0.25cm) \end_layout \begin_layout Plain Layout ++ (90- \backslash iangle:0.5) coordinate (pulley); \end_layout \begin_layout Plain Layout \backslash draw[string] (M.east) -- ++( \backslash iangle:1.5cm) arc (90+ \backslash iangle:0:0.25) \end_layout \begin_layout Plain Layout -- ++(0,-1) node[m] {}; \end_layout \begin_layout Plain Layout \end_layout \begin_layout Plain Layout \backslash draw[->] (base)++( \backslash arcr,0) arc (0: \backslash iangle: \backslash arcr); \end_layout \begin_layout Plain Layout \backslash path (base)++( \backslash iangle*0.5: \backslash arcr+5pt) node {$ \backslash alpha$}; \end_layout \begin_layout Plain Layout %% \end_layout \begin_layout Plain Layout \end_layout \begin_layout Plain Layout & \end_layout \begin_layout Plain Layout %% Free body diagram of M \end_layout \begin_layout Plain Layout \backslash begin{scope}[rotate= \backslash iangle] \end_layout \begin_layout Plain Layout \backslash node[M,transform shape] (M) {}; \end_layout \begin_layout Plain Layout % Draw axes and help lines \end_layout \begin_layout Plain Layout \end_layout \begin_layout Plain Layout {[axis,->] \end_layout \begin_layout Plain Layout \backslash draw (0,-1) -- (0,2) node[right] {$+y$}; \end_layout \begin_layout Plain Layout \backslash draw (M) -- ++(2,0) node[right] {$+x$}; \end_layout \begin_layout Plain Layout % Indicate angle. The code is a bit awkward. \end_layout \begin_layout Plain Layout \end_layout \begin_layout Plain Layout \backslash draw[solid,shorten >=0.5pt] ( \backslash down- \backslash iangle: \backslash arcr) \end_layout \begin_layout Plain Layout arc( \backslash down- \backslash iangle: \backslash down: \backslash arcr); \end_layout \begin_layout Plain Layout \backslash node at ( \backslash down-0.5* \backslash iangle:1.3* \backslash arcr) {$ \backslash alpha$}; \end_layout \begin_layout Plain Layout } \end_layout \begin_layout Plain Layout \end_layout \begin_layout Plain Layout % Forces \end_layout \begin_layout Plain Layout {[force,->] \end_layout \begin_layout Plain Layout % Assuming that Mg = 1. The normal force will therefore be cos(alpha) \end_layout \begin_layout Plain Layout \backslash draw (M.center) -- ++(0,{cos( \backslash iangle)}) node[above right] {$N$}; \end_layout \begin_layout Plain Layout \backslash draw (M.west) -- ++(-1,0) node[left] {$f_R$}; \end_layout \begin_layout Plain Layout \backslash draw (M.east) -- ++(1,0) node[above] {$T$}; \end_layout \begin_layout Plain Layout } \end_layout \begin_layout Plain Layout \end_layout \begin_layout Plain Layout \backslash end{scope} \end_layout \begin_layout Plain Layout % Draw gravity force. The code is put outside the rotated \end_layout \begin_layout Plain Layout % scope for simplicity. No need to do any angle calculations. \end_layout \begin_layout Plain Layout \backslash draw[force,->] (M.center) -- ++(0,-1) node[below] {$Mg$}; \end_layout \begin_layout Plain Layout %% \end_layout \begin_layout Plain Layout \end_layout \begin_layout Plain Layout & \end_layout \begin_layout Plain Layout %%% \end_layout \begin_layout Plain Layout % Free body diagram of m \end_layout \begin_layout Plain Layout \backslash node[m] (m) {}; \end_layout \begin_layout Plain Layout \backslash draw[axis,->] (m) -- ++(0,-2) node[left] {$+$}; \end_layout \begin_layout Plain Layout {[force,->] \end_layout \begin_layout Plain Layout \backslash draw (m.north) -- ++(0,1) node[above] {$T'$}; \end_layout \begin_layout Plain Layout \backslash draw (m.south) -- ++(0,-1) node[right] {$mg$}; \end_layout \begin_layout Plain Layout } \end_layout \begin_layout Plain Layout \end_layout \begin_layout Plain Layout \backslash \backslash \end_layout \begin_layout Plain Layout }; \end_layout \begin_layout Plain Layout \backslash end{tikzpicture} \end_layout \begin_layout Plain Layout \end_layout \end_inset \end_layout \end_inset \end_layout \end_body \end_document