Revived document about manifold geometry of the sphere, deleted by Can a while ago :-(
Frank Dellaert
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#LyX 2.0 created this file. For more info see http://www.lyx.org/
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\lyxformat 413
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\begin_document
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\begin_header
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\index Index
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\shortcut idx
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\color #008000
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\end_index
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\leftmargin 3cm
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\topmargin 3cm
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\rightmargin 3cm
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\end_header
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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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\end_layout
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\begin_layout Author
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Frank, Can, and Manohar
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\end_layout
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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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\end_inset
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\end_layout
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\begin_layout Subsubsection*
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Retraction
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\end_layout
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\begin_layout Standard
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Suppose we have a point
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\begin_inset Formula $p\in S^{2}$
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\end_inset
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and a 3-vector
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\begin_inset Formula $\xihat$
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\end_inset
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, Absil
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\begin_inset CommandInset citation
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LatexCommand cite
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key "Absil07book"
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\end_inset
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tells us we can simply add
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\begin_inset Formula $\xihat$
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\end_inset
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to
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\begin_inset Formula $p$
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\end_inset
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and renormalize to get a new point
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\begin_inset Formula $q$
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\end_inset
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on the sphere.
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This is what he calls a
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\series bold
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retraction
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\family roman
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\series medium
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\shape up
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\size normal
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\emph off
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\bar no
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\uuline off
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\uwave off
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\noun off
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\color none
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\begin_inset Formula $R_{p}(\xihat)$
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\end_inset
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,
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\family default
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\series default
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\size default
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\emph default
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\uuline default
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\uwave default
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\noun default
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\color inherit
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\begin_inset Formula
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\[
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q=R_{p}(\xihat)=\frac{p+\xihat}{\left|p+\xihat\right|}=\frac{p+\xihat}{\alpha}
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\]
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\end_inset
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with
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\begin_inset Formula $\alpha$
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\end_inset
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the norm of
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\begin_inset Formula $p+\xihat$
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\end_inset
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.
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The only restriction on
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\begin_inset Formula $\xihat$
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\end_inset
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is that it is in the tangent space
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\begin_inset Formula $T_{p}S^{2}$
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\end_inset
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at
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\begin_inset Formula $p$
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\end_inset
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, i.e.,
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\begin_inset Formula $p^{T}\xihat=0$
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\end_inset
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.
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Multiplying with
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\begin_inset Formula $p^{T}$
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\end_inset
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on both sides we have
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\begin_inset Formula
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\[
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\alpha p^{T}q=p^{T}p+p^{T}\xihat
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\]
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\end_inset
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and (since
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\begin_inset Formula $p^{T}p=1$
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\end_inset
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and
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\begin_inset Formula $p^{T}\xihat=0$
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\end_inset
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) we have
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\begin_inset Formula $\alpha=1/(p^{T}q)$
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\end_inset
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.
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\end_layout
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\begin_layout Subsubsection*
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Inverse
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\end_layout
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\begin_layout Standard
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Suppose we are given points
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\begin_inset Formula $p$
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\end_inset
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and
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\begin_inset Formula $q$
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\end_inset
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on the sphere, what is the tangent vector
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\begin_inset Formula $\xihat$
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\end_inset
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that takes
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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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? We can find a basis
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\begin_inset Formula $B$
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\end_inset
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for the tangent space, with
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\begin_inset Formula $B=\left[b_{1}|b_{2}\right]$
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\end_inset
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a
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\begin_inset Formula $3\times2$
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\end_inset
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matrix, by either
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\end_layout
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\begin_layout Enumerate
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Decompose
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\begin_inset Formula $p=QR$
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\end_inset
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, with
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\begin_inset Formula $Q$
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\end_inset
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orthonormal and
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\begin_inset Formula $R$
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\end_inset
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of the form
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\begin_inset Formula $[1\,0\,0]^{T}$
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\end_inset
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, and hence
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\begin_inset Formula $p=Q_{1}$
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\end_inset
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.
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The basis
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\begin_inset Formula $B=\left[Q_{2}|Q_{3}\right]$
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\end_inset
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, i.e., the last two columns of
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\begin_inset Formula $Q$
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\end_inset
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.
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\end_layout
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\begin_layout Enumerate
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Form
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\begin_inset Formula $b_{1}=p\times a$
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\end_inset
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, with
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\begin_inset Formula $a$
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\end_inset
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(consistently) chosen to be non-parallel to
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\begin_inset Formula $p$
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\end_inset
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, and
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\begin_inset Formula $b_{2}=p\times b_{1}$
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\end_inset
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.
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\begin_inset Note Note
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status collapsed
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\begin_layout Plain Layout
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To choose
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\begin_inset Formula $a$
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\end_inset
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, one way is to divide the sphere into regions, e.g., pick the axis
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\begin_inset Formula $e_{i}$
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\end_inset
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such that
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\begin_inset Formula $e_{i}^{T}p$
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\end_inset
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is smallest.
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However, that leads to discontinuous boundaries.
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Since
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\begin_inset Formula $0\leq\left|e_{i}^{T}p\right|\leq1$
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\end_inset
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for all
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\begin_inset Formula $p\in S^{2}$
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\end_inset
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, a better idea might be to use a mixture, e.g.,
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\begin_inset Formula
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\[
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a=\frac{1}{2(x^{2}+y^{2}+z^{2})}\left[\begin{array}{c}
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y^{2}+z^{2}\\
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x^{2}+z^{2}\\
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x^{2}+y^{2}
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\end{array}\right]
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\]
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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Now, if
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\begin_inset Formula $\xihat=B\xi$
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\end_inset
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with
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\begin_inset Formula $\xi\in R^{2}$
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\end_inset
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the 2D coordinate in the tangent plane basis
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\begin_inset Formula $B$
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\end_inset
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, we have
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\begin_inset Formula
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\[
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\alpha q=p+\xihat=p+B\xi
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\]
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\end_inset
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If we multiply both sides with
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\begin_inset Formula $B^{T}$
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\end_inset
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(project on the basis
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\begin_inset Formula $B$
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\end_inset
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) we obtain
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\begin_inset Formula
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\[
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\alpha B^{T}q=B^{T}p+B^{T}B\xi
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\]
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\end_inset
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and because
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\begin_inset Formula $B^{T}p=0$
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\end_inset
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and
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\begin_inset Formula $B^{T}B=I$
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\end_inset
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we trivially obtain
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\begin_inset Formula $\xi$
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\end_inset
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as the scaled projection
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\begin_inset Formula $B^{T}q$
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\end_inset
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:
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\begin_inset Formula
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\[
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\xi=\alpha B^{T}q=\frac{B^{T}q}{p^{T}q}
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\]
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\end_inset
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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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bibfiles "../../../papers/refs"
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options "plain"
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\end_inset
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\end_layout
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\end_body
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\end_document
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