From bad1c38fe68ea427f421d4eafad6d18de540af52 Mon Sep 17 00:00:00 2001
From: Frank Dellaert <dellaert@gmail.com>
Date: Sun, 15 Dec 2013 21:29:20 +0000
Subject: [PATCH] Revived document about manifold geometry of the sphere,
 deleted by Can a while ago :-(

---
 doc/sphere.lyx | 422 +++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 422 insertions(+)
 create mode 100644 doc/sphere.lyx

diff --git a/doc/sphere.lyx b/doc/sphere.lyx
new file mode 100644
index 000000000..35d6a47fb
--- /dev/null
+++ b/doc/sphere.lyx
@@ -0,0 +1,422 @@
+#LyX 2.0 created this file. For more info see http://www.lyx.org/
+\lyxformat 413
+\begin_document
+\begin_header
+\textclass article
+\use_default_options true
+\maintain_unincluded_children false
+\language english
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman default
+\font_sans default
+\font_typewriter default
+\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 default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize 11
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_amsmath 1
+\use_esint 1
+\use_mhchem 1
+\use_mathdots 1
+\cite_engine basic
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\use_refstyle 1
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 3cm
+\topmargin 3cm
+\rightmargin 3cm
+\bottommargin 3cm
+\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 false
+\end_header
+
+\begin_body
+
+\begin_layout Title
+Retraction on a Sphere
+\end_layout
+
+\begin_layout Author
+Frank, Can, and Manohar
+\end_layout
+
+\begin_layout Standard
+\begin_inset FormulaMacro
+\newcommand{\xihat}{\hat{\xi}}
+{\hat{\xi}}
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection*
+Retraction
+\end_layout
+
+\begin_layout Standard
+Suppose we have a point 
+\begin_inset Formula $p\in S^{2}$
+\end_inset
+
+ and a 3-vector 
+\begin_inset Formula $\xihat$
+\end_inset
+
+, Absil 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Absil07book"
+
+\end_inset
+
+ tells us we can simply add 
+\begin_inset Formula $\xihat$
+\end_inset
+
+ to 
+\begin_inset Formula $p$
+\end_inset
+
+ and renormalize to get a new point 
+\begin_inset Formula $q$
+\end_inset
+
+ on the sphere.
+ This is what he calls a 
+\series bold
+retraction 
+\family roman
+\series medium
+\shape up
+\size normal
+\emph off
+\bar no
+\strikeout off
+\uuline off
+\uwave off
+\noun off
+\color none
+
+\begin_inset Formula $R_{p}(\xihat)$
+\end_inset
+
+,
+\family default
+\series default
+\shape default
+\size default
+\emph default
+\bar default
+\strikeout default
+\uuline default
+\uwave default
+\noun default
+\color inherit
+ 
+\begin_inset Formula 
+\[
+q=R_{p}(\xihat)=\frac{p+\xihat}{\left|p+\xihat\right|}=\frac{p+\xihat}{\alpha}
+\]
+
+\end_inset
+
+with 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ the norm of 
+\begin_inset Formula $p+\xihat$
+\end_inset
+
+.
+ The only restriction on 
+\begin_inset Formula $\xihat$
+\end_inset
+
+ is that it is in the tangent space 
+\begin_inset Formula $T_{p}S^{2}$
+\end_inset
+
+ at 
+\begin_inset Formula $p$
+\end_inset
+
+, i.e., 
+\begin_inset Formula $p^{T}\xihat=0$
+\end_inset
+
+.
+ Multiplying with 
+\begin_inset Formula $p^{T}$
+\end_inset
+
+ on both sides we have
+\begin_inset Formula 
+\[
+\alpha p^{T}q=p^{T}p+p^{T}\xihat
+\]
+
+\end_inset
+
+and (since 
+\begin_inset Formula $p^{T}p=1$
+\end_inset
+
+ and 
+\begin_inset Formula $p^{T}\xihat=0$
+\end_inset
+
+) we have 
+\begin_inset Formula $\alpha=1/(p^{T}q)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsubsection*
+Inverse
+\end_layout
+
+\begin_layout Standard
+Suppose we are given points 
+\begin_inset Formula $p$
+\end_inset
+
+ and 
+\begin_inset Formula $q$
+\end_inset
+
+ on the sphere, what is the tangent vector 
+\begin_inset Formula $\xihat$
+\end_inset
+
+ that takes 
+\begin_inset Formula $p$
+\end_inset
+
+ to 
+\begin_inset Formula $q$
+\end_inset
+
+? We can find a basis 
+\begin_inset Formula $B$
+\end_inset
+
+ for the tangent space, with 
+\begin_inset Formula $B=\left[b_{1}|b_{2}\right]$
+\end_inset
+
+ a 
+\begin_inset Formula $3\times2$
+\end_inset
+
+ matrix, by either
+\end_layout
+
+\begin_layout Enumerate
+Decompose 
+\begin_inset Formula $p=QR$
+\end_inset
+
+, with 
+\begin_inset Formula $Q$
+\end_inset
+
+ orthonormal and 
+\begin_inset Formula $R$
+\end_inset
+
+ of the form 
+\begin_inset Formula $[1\,0\,0]^{T}$
+\end_inset
+
+, and hence 
+\begin_inset Formula $p=Q_{1}$
+\end_inset
+
+.
+ The basis 
+\begin_inset Formula $B=\left[Q_{2}|Q_{3}\right]$
+\end_inset
+
+, i.e., the last two columns of 
+\begin_inset Formula $Q$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Form 
+\begin_inset Formula $b_{1}=p\times a$
+\end_inset
+
+, with 
+\begin_inset Formula $a$
+\end_inset
+
+ (consistently) chosen to be non-parallel to 
+\begin_inset Formula $p$
+\end_inset
+
+, and 
+\begin_inset Formula $b_{2}=p\times b_{1}$
+\end_inset
+
+.
+ 
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+To choose 
+\begin_inset Formula $a$
+\end_inset
+
+, one way is to divide the sphere into regions, e.g., pick the axis 
+\begin_inset Formula $e_{i}$
+\end_inset
+
+ such that 
+\begin_inset Formula $e_{i}^{T}p$
+\end_inset
+
+ is smallest.
+ However, that leads to discontinuous boundaries.
+ Since 
+\begin_inset Formula $0\leq\left|e_{i}^{T}p\right|\leq1$
+\end_inset
+
+ for all 
+\begin_inset Formula $p\in S^{2}$
+\end_inset
+
+, a better idea might be to use a mixture, e.g.,
+\begin_inset Formula 
+\[
+a=\frac{1}{2(x^{2}+y^{2}+z^{2})}\left[\begin{array}{c}
+y^{2}+z^{2}\\
+x^{2}+z^{2}\\
+x^{2}+y^{2}
+\end{array}\right]
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Now, if 
+\begin_inset Formula $\xihat=B\xi$
+\end_inset
+
+ with 
+\begin_inset Formula $\xi\in R^{2}$
+\end_inset
+
+ the 2D coordinate in the tangent plane basis 
+\begin_inset Formula $B$
+\end_inset
+
+, we have 
+\begin_inset Formula 
+\[
+\alpha q=p+\xihat=p+B\xi
+\]
+
+\end_inset
+
+If we multiply both sides with 
+\begin_inset Formula $B^{T}$
+\end_inset
+
+ (project on the basis 
+\begin_inset Formula $B$
+\end_inset
+
+) we obtain
+\begin_inset Formula 
+\[
+\alpha B^{T}q=B^{T}p+B^{T}B\xi
+\]
+
+\end_inset
+
+and because 
+\begin_inset Formula $B^{T}p=0$
+\end_inset
+
+ and 
+\begin_inset Formula $B^{T}B=I$
+\end_inset
+
+ we trivially obtain 
+\begin_inset Formula $\xi$
+\end_inset
+
+ as the scaled projection 
+\begin_inset Formula $B^{T}q$
+\end_inset
+
+:
+\begin_inset Formula 
+\[
+\xi=\alpha B^{T}q=\frac{B^{T}q}{p^{T}q}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset bibtex
+LatexCommand bibtex
+bibfiles "../../../papers/refs"
+options "plain"
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document