diff options
| -rw-r--r-- | 2-point.tex | 115 |
1 files changed, 67 insertions, 48 deletions
diff --git a/2-point.tex b/2-point.tex index 5c40278..e20792e 100644 --- a/2-point.tex +++ b/2-point.tex @@ -21,7 +21,7 @@ \usepackage{anyfontsize,authblk} \usepackage{tikz} -\usetikzlibrary{calc,fadings,decorations.pathreplacing,perspective,3d} +\usetikzlibrary{calc,fadings,decorations.pathreplacing,calligraphy} \addbibresource{2-point.bib} @@ -62,11 +62,12 @@ \tikzset{#1/.style={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} % } \newcommand\TangentPlane[5][current plane]{% - \pgfmathsinandcos\sinEl\cosEl{#3} % elevation - \pgfmathsinandcos\sint\cost{#4} % latitude - \pgfmathsinandcos\sinu\cosu{#5} % azimuth - \pgfmathsetmacro\yshift{#2*\cosEl*\sint} - \tikzset{#1/.style={cm={\cost*\cosu,\cost*\sinu*\sinEl,0,\cost*\sinEl*\cosEl,(0,0)}}} % + \pgfmathsinandcos\sint\cost{#3} % elevation + \pgfmathsinandcos\sinb\cosb{-#4} % latitude + \pgfmathsinandcos\sina\cosa{#5+90} % azimuth + \pgfmathsetmacro\xshift{\cosb*\sina} + \pgfmathsetmacro\yshift{-\cost*\sinb-\cosa*\cosb*\sint} + \tikzset{#1/.style={cm={-\sina*\sinb,\cosa*\sinb*\sint-\cost*\cosb,\cosa,\sina*\sint,(#2*\xshift,#2*\yshift)}}} % } \newcommand\DrawLongitudeCircle[2][1]{ \LongitudePlane{\angEl}{#2} @@ -91,50 +92,9 @@ \tikzset{% >=latex, % option for nice arrows inner sep=0pt,% - outer sep=2pt,% - mark coordinate/.style={inner sep=0pt,outer sep=0pt,minimum size=3pt, - fill=black,circle}% + outer sep=2pt% } -\begin{tikzpicture} % "THE GLOBE" showcase - \def\R{4 } % sphere radius - \def\angEl{20} % elevation angle - \def\angAz{-20} % azimuth angle - \filldraw[ball color=white] (0,0) circle (\R); - \filldraw[fill=white] (0,0) circle (\R); - - \foreach \t in {0,45} { \DrawLatitudeCircle[\R]{\t} } - \foreach \t in {-120} { \DrawLongitudeCircle[\R]{\t} } - - \pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole - \coordinate (O) at (0,0); - \node[circle,draw,black,scale=0.3] at (0,0) {}; - \coordinate (N) at (0,\H); - \draw[left] node at (0,\H){$\pmb\sigma_1$}; - \draw[thick, ->](O)--(N); - - \NewLatitudePlane[planeP]{\R}{\angEl}{45}; - \path[planeP] (-120:\R) coordinate (P); - \draw[left] node at (P){$\mathbf s_1$}; - - \NewLatitudePlane[equator]{\R}{\angEl}{00}; - \path[equator] (-30:\R) coordinate (Pprime); - \draw[right] node at (Pprime){$\pmb\sigma_b$}; - - \NewLatitudePlane[sbplane]{\R}{\angEl}{45}; - \path[sbplane] (20:\R) coordinate (sb); - \draw[right] node at (sb){$\mathbf s_b$}; - - \TangentPlane[tplane]{\R}{\angEl}{45}{-120}; - \draw[tplane,fill=gray,fill opacity=0.3] circle (1); - \draw[shift={(P)},rotate around y=45,canvas is xy plane at z=0,->,thick] (0,0) -> ({sin(10)},{cos(10)}); - - \draw[thick, ->] (O)->(P); - \draw[thick, ->] (O)->(Pprime); - \draw[thick, ->] (O)->(sb); - - -\end{tikzpicture} \cite{Ros_2020_Distribution, Ros_2019_Complex, Ros_2019_Complexity} @@ -613,6 +573,65 @@ $\sigma$ replicas constrained to lie at fixed overlap with \emph{all} the $\mathbf s$ replicas, and the second is the only of the $\mathbf s$ replicas at which the Hessian is evaluated. +\begin{figure} + \centering + \begin{tikzpicture} + \def\R{4 } % sphere radius + \def\Rt{2 } % tangent plane radius + \def\angEl{15} % elevation angle + \def\angsa{-160} % azimuth of s_1 + \def\angq{40} % elevation of constraint circle + \filldraw[ball color=white] (0,0) circle (\R); + % \filldraw[fill=white] (0,0) circle (\R); + + \foreach \t in {0,\angq} { \DrawLatitudeCircle[\R]{\t} } + %\foreach \t in {\angsa} { \DrawLongitudeCircle[\R]{\t} } + + \pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole + \coordinate (O) at (0,0); + \node[circle,draw,black,scale=0.3] at (0,0) {}; + \coordinate (N) at (0,\H); + \draw node[right=10,below] at (0,\H){$\pmb\sigma_1$}; + \draw[thick, ->](O)--(N); + + \NewLatitudePlane[planeP]{\R}{\angEl}{\angq}; + \path[planeP] (\angsa:\R) coordinate (P); + \path[planeP] (0:1.5*\R) coordinate (Q); + \path[planeP] (0:\R) coordinate (Q2); + \draw[left] node at (P){$\mathbf s_1$}; + + \NewLatitudePlane[equator]{\R}{\angEl}{00}; + \path[equator] (-30:\R) coordinate (Pprime); + \path[equator] (0:{1.5*cos(\angq)*\R}) coordinate (Qe); + \path[equator] (0:\R) coordinate (Qe2); + \draw node[right=5,below] at (Pprime){$\pmb\sigma_c$}; + + \NewLatitudePlane[sbplane]{\R}{\angEl}{\angq}; + \path[sbplane] (20:\R) coordinate (sb); + \draw node[right=3,above=1] at (sb){$\mathbf s_b$}; + + \TangentPlane[tplane]{\R}{\angEl}{\angq}{\angsa}; + \draw[tplane,fill=gray,fill opacity=0.3] circle (\Rt); + \draw[tplane,->,thick] (0,0) -> ({\Rt*cos(160)},{\Rt*sin(160)}) node[above=1.5,right] {$\mathbf x_a$}; + + \draw[thick, ->] (O)->(P); + \draw[thick, ->] (O)->(Pprime); + \draw[thick, ->] (O)->(sb); + + \draw[dotted] (Qe) -- (Qe2); + \draw[dotted] (Q2) -- (Q); + \draw[decorate, decoration = {brace,raise=3}] (Q) -- (Qe) node[pos=0.5,right=7]{$q$}; + \end{tikzpicture} + \caption{ + A sketch of the vectors involved in the calculation of the isolated + eigenvalue. All replicas $\mathbf x$ sit in an $N-2$ sphere corresponding + with the tangent plane of the first $\mathbf s$ replica (not to scale). All of the + $\mathbf s$ replicas lie on the sphere, constrained to be at fixed overlap + $q$ with the first of the $\pmb\sigma$ replicas, the reference + configuration. All of the $\pmb\sigma$ replicas lie on the sphere. + } +\end{figure} + Using the same methodology as above, the disorder-dependant terms are captured in the linear operator \begin{equation} \mathcal O(\mathbf t)= |