From 6aae6b67fb01f9dea259342ebf30f9a0833c56b5 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 17 May 2023 17:49:39 +0200
Subject: Good draft of sphere figure.

---
 2-point.tex | 115 +++++++++++++++++++++++++++++++++++-------------------------
 1 file changed, 67 insertions(+), 48 deletions(-)

(limited to '2-point.tex')

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)=
-- 
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