Exponential map
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				|  | @ -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 | ||||
|  |  | |||
										
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