Exponential map
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@ -65,7 +65,11 @@
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\begin_body
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\begin_layout Title
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Retraction on a Sphere
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Manifold Geometry of the Sphere
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\begin_inset Formula $S^{2}$
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\end_inset
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\end_layout
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\begin_layout Author
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@ -74,8 +78,8 @@ Frank, Can, and Manohar
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\begin_layout Standard
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\begin_inset FormulaMacro
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\newcommand{\xihat}{\hat{\xi}}
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{\hat{\xi}}
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\newcommand{\xihat}{z}
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{z}
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\end_inset
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@ -147,7 +151,7 @@ retraction
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\begin_inset Formula
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\[
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q=R_{p}(\xihat)=\frac{p+\xihat}{\left|p+\xihat\right|}=\frac{p+\xihat}{\alpha}
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q=R_{p}(\xihat)=\frac{p+\xihat}{\left\Vert p+z\right\Vert }=\frac{p+\xihat}{\alpha}
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\]
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\end_inset
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@ -407,6 +411,55 @@ and because
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\end_layout
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\begin_layout Subsubsection*
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Exponential Map
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\end_layout
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\begin_layout Standard
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The exponential map itself is not so difficult, and is given in Ma01ijcv,
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as well as in this CVPR tutorial by Anuj Srivastava:
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\begin_inset CommandInset href
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LatexCommand href
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name "http://stat.fsu.edu/~anuj/CVPR_Tutorial/Part2.pdf"
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\end_inset
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.
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\begin_inset Formula
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\[
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\exp_{p}\xihat=\cos\left(\left\Vert \xihat\right\Vert \right)p+\sin\left(\left\Vert \xihat\right\Vert \right)\frac{\xihat}{\left\Vert \xihat\right\Vert }
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\]
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\end_inset
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The latter also gives the inverse, i.e., get the tangent vector
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\begin_inset Formula $z$
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\end_inset
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to go from
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\begin_inset Formula $p$
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\end_inset
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to
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\begin_inset Formula $q$
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\end_inset
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:
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\begin_inset Formula
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\[
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z=\log_{p}q=\frac{\theta}{\sin\theta}\left(q-p\cos\theta\right)p
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\]
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\end_inset
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with
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\begin_inset Formula $\theta=\cos^{-1}\left(p^{T}q\right)$
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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\begin_inset CommandInset bibtex
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LatexCommand bibtex
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