From 6f537b2b44d589974511399dfaaf51ce26f690d0 Mon Sep 17 00:00:00 2001 From: Frank Dellaert Date: Sun, 15 Dec 2013 21:52:03 +0000 Subject: [PATCH] Exponential map --- doc/sphere.lyx | 61 +++++++++++++++++++++++++++++++++++++++++++++---- doc/sphere.pdf | Bin 0 -> 44587 bytes 2 files changed, 57 insertions(+), 4 deletions(-) create mode 100644 doc/sphere.pdf diff --git a/doc/sphere.lyx b/doc/sphere.lyx index 35d6a47fb..0ddb9f156 100644 --- a/doc/sphere.lyx +++ b/doc/sphere.lyx @@ -65,7 +65,11 @@ \begin_body \begin_layout Title -Retraction on a Sphere +Manifold Geometry of the Sphere +\begin_inset Formula $S^{2}$ +\end_inset + + \end_layout \begin_layout Author @@ -74,8 +78,8 @@ Frank, Can, and Manohar \begin_layout Standard \begin_inset FormulaMacro -\newcommand{\xihat}{\hat{\xi}} -{\hat{\xi}} +\newcommand{\xihat}{z} +{z} \end_inset @@ -147,7 +151,7 @@ retraction \begin_inset Formula \[ -q=R_{p}(\xihat)=\frac{p+\xihat}{\left|p+\xihat\right|}=\frac{p+\xihat}{\alpha} +q=R_{p}(\xihat)=\frac{p+\xihat}{\left\Vert p+z\right\Vert }=\frac{p+\xihat}{\alpha} \] \end_inset @@ -407,6 +411,55 @@ and because \end_layout +\begin_layout Subsubsection* +Exponential Map +\end_layout + +\begin_layout Standard +The exponential map itself is not so difficult, and is given in Ma01ijcv, + as well as in this CVPR tutorial by Anuj Srivastava: +\begin_inset CommandInset href +LatexCommand href +name "http://stat.fsu.edu/~anuj/CVPR_Tutorial/Part2.pdf" + +\end_inset + +. + +\begin_inset Formula +\[ +\exp_{p}\xihat=\cos\left(\left\Vert \xihat\right\Vert \right)p+\sin\left(\left\Vert \xihat\right\Vert \right)\frac{\xihat}{\left\Vert \xihat\right\Vert } +\] + +\end_inset + +The latter also gives the inverse, i.e., get the tangent vector +\begin_inset Formula $z$ +\end_inset + + to go from +\begin_inset Formula $p$ +\end_inset + + to +\begin_inset Formula $q$ +\end_inset + +: +\begin_inset Formula +\[ +z=\log_{p}q=\frac{\theta}{\sin\theta}\left(q-p\cos\theta\right)p +\] + +\end_inset + +with +\begin_inset Formula $\theta=\cos^{-1}\left(p^{T}q\right)$ +\end_inset + +. +\end_layout + \begin_layout Standard \begin_inset CommandInset bibtex LatexCommand bibtex diff --git a/doc/sphere.pdf b/doc/sphere.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e82ac3327f5e4bbca1a6e0c3dad6517dfa5f5931 GIT binary patch literal 44587 zcmb@t1ymi`+AW%(!67&Sf;&Mr?h@RCySqCixVyVcu;A|QPH=a3*8p#m?(|7_|L@*& z?|Y-ht|hg+R?Rg(*~HQx1*z#iFhCP;o*vyo(=p=F;#umLKyz?F(+C=y+3DM;(g>Pq z+v$JQ*R|BsSBIt%*S9dVGs2^%XP}4X;)1rdv(eW!hjyADR#6RKXh3p1DILoW5iD{- zB1dH*Yv3H_g-!txfxW~}hU_j>Kj?~Us$g0eENbRF43Tw(B^Ge7e z=jTb7HH0e^@7B!Pxz4Rhs~wmdK)gp-C>l^mpLzDOPptfa96~nxsr2^y@(yke zQQ%NZjMbJ%N0tF}rK}HtI737@YZj)Ch>vk%xkP7Q(fI5N&TRDSb9hx*c2yk!6tQ(N zSH2QA6e?b$4Gd{`qtzhUtkp|IQ{lC&n?j{+2?-8wUO;S~@Q>K`k8KKzbXO zSFvA{*=T|d?zv~U4c_dT@v%aIo1wzIB^ET7^5-Wsbe^0?TXfejE*N%7-bYwTvHWCP zKDz1KBEPUf=13|i(p78@@4$B`9uR0^2{?2vgMOFDbcP}l(^B}x5SKI%n}Y^&5XOAc 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