From: Leonardo Florez-Valencia Date: Tue, 8 Mar 2016 23:00:06 +0000 (-0500) Subject: More simple testing data X-Git-Tag: v0.1~231 X-Git-Url: https://git.creatis.insa-lyon.fr/pubgit/?a=commitdiff_plain;h=c3b9ceab97e788a8052273644a5840de65c59a8b;p=cpPlugins.git More simple testing data --- diff --git a/data/circles.png b/data/circles.png new file mode 100644 index 0000000..8788dc6 Binary files /dev/null and b/data/circles.png differ diff --git a/data/inertia.lyx b/data/inertia.lyx new file mode 100644 index 0000000..51740ff --- /dev/null +++ b/data/inertia.lyx @@ -0,0 +1,454 @@ +#LyX 2.1 created this file. For more info see http://www.lyx.org/ +\lyxformat 474 +\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_math auto +\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 default +\use_hyperref false +\papersize default +\use_geometry false +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 1 +\index Index +\shortcut idx +\color #008000 +\end_index +\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 Standard +\begin_inset Formula +\[ +\boldsymbol{\iota}=\left[\begin{array}{ccc} +\iota_{00} & \iota_{01} & \iota_{02}\\ +\iota_{10} & \iota_{11} & \iota_{12}\\ +\iota_{20} & \iota_{21} & \iota_{22} +\end{array}\right] +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\left[\begin{array}{ccc} +{\displaystyle \sum_{i=1}^{n}m_{i}\left(x_{i1}^{2}+x_{i2}^{2}\right)} & -{\displaystyle \sum_{i=1}^{n}m_{i}x_{i0}x_{i1}} & -{\displaystyle \sum_{i=1}^{n}m_{i}x_{i0}x_{i2}}\\ +-{\displaystyle \sum_{i=1}^{n}m_{i}x_{i0}x_{i1}} & {\displaystyle \sum_{i=1}^{n}m_{i}\left(x_{i0}^{2}+x_{i2}^{2}\right)} & -{\displaystyle \sum_{i=1}^{n}m_{i}x_{i1}x_{i2}}\\ +-{\displaystyle \sum_{i=1}^{n}m_{i}x_{i0}x_{i2}} & -{\displaystyle \sum_{i=1}^{n}m_{i}x_{i1}x_{i2}} & {\displaystyle \sum_{i=1}^{n}}m_{i}\left(x_{i0}^{2}+x_{i1}^{2}\right) +\end{array}\right] +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left[\begin{array}{ccc} +\left(x_{i1}^{2}+x_{i2}^{2}\right) & -x_{i0}x_{i1} & -x_{i0}x_{i2}\\ +-x_{i0}x_{i1} & \left(x_{i0}^{2}+x_{i2}^{2}\right) & -x_{i1}x_{i2}\\ +-x_{i0}x_{i2} & -x_{i1}x_{i2} & \left(x_{i0}^{2}+x_{i1}^{2}\right) +\end{array}\right] +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left[\begin{array}{ccc} +\left|\mathbf{p}_{i}\right|^{2}-x_{i0}^{2} & -x_{i0}x_{i1} & -x_{i0}x_{i2}\\ +-x_{i0}x_{i1} & \left|\mathbf{p}_{i}\right|^{2}-x_{i1}^{2} & -x_{i1}x_{i2}\\ +-x_{i0}x_{i2} & -x_{i1}x_{i2} & \left|\mathbf{p}_{i}\right|^{2}-x_{i2}^{2} +\end{array}\right] +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\left[\begin{array}{ccc} +\left|\mathbf{p}_{i}\right|^{2} & 0 & 0\\ +0 & \left|\mathbf{p}_{i}\right|^{2} & 0\\ +0 & 0 & \left|\mathbf{p}_{i}\right|^{2} +\end{array}\right]-\left[\begin{array}{ccc} +x_{i0}^{2} & x_{i0}x_{i1} & x_{i0}x_{i2}\\ +x_{i0}x_{i1} & x_{i1}^{2} & x_{i1}x_{i2}\\ +x_{i0}x_{i2} & x_{i1}x_{i2} & x_{i2}^{2} +\end{array}\right]\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\left|\mathbf{p}_{i}\right|^{2}\mathbf{I}-\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left|\mathbf{p}_{i}\right|^{2}\mathbf{I}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top} +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}\mathbf{I}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top} +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)^{\top}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)\mathbf{I}-\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)^{\top} +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}^{\top}-\boldsymbol{\mu}^{\top}\right)\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)\mathbf{I}-\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)\left(\mathbf{p}_{i}^{\top}-\boldsymbol{\mu}^{\top}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}^{\top}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)-\boldsymbol{\mu}^{\top}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)\right)\mathbf{I}-\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}\left(\mathbf{p}_{i}^{\top}-\boldsymbol{\mu}^{\top}\right)-\boldsymbol{\mu}\left(\mathbf{p}_{i}^{\top}-\boldsymbol{\mu}^{\top}\right)\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\mathbf{p}_{i}^{\top}\boldsymbol{\mu}-\boldsymbol{\mu}^{\top}\mathbf{p}_{i}+\boldsymbol{\mu}^{\top}\boldsymbol{\mu}\right)\mathbf{I}-\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-\mathbf{p}_{i}\boldsymbol{\mu}^{\top}-\boldsymbol{\mu}\mathbf{p}_{i}^{\top}+\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\boldsymbol{\mu}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}^{\top}\mathbf{p}_{i}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}^{\top}\boldsymbol{\mu}\right)\mathbf{I}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\boldsymbol{\mu}^{\top}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\right)\boldsymbol{\mu}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}^{\top}\mathbf{p}_{i}+\left(\sum_{i=1}^{n}m_{i}\right)\boldsymbol{\mu}^{\top}\boldsymbol{\mu}\right)\mathbf{I}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\right)\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\right)\boldsymbol{\mu}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}^{\top}\mathbf{p}_{i}+\left(\sum_{i=1}^{n}m_{i}\right)\boldsymbol{\mu}^{\top}\boldsymbol{\mu}\right)\mathbf{I}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-\left(\sum_{i=1}^{n}m_{i}\right)\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\mu}=\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{{\displaystyle \sum_{k=1}^{n}}m_{k}}=\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\right)\boldsymbol{\mu}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}^{\top}\mathbf{p}_{i}+M\boldsymbol{\mu}^{\top}\boldsymbol{\mu}\right)\mathbf{I}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\right)\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}-\sum_{i=1}^{n}m_{i}\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)^{\top}\mathbf{p}_{i}+M\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)^{\top}\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)\right)\mathbf{I}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{\left({\displaystyle \sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)}{M}-\sum_{i=1}^{n}m_{i}\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)^{\top}\mathbf{p}_{i}+\frac{\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}^{\top}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)}{M}\right)\mathbf{I}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\sum_{i=1}^{n}m_{i}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}^{\top}\right)\mathbf{p}_{i}\right)\mathbf{I}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\mathbf{I}^{\top}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\sum_{i=1}^{n}m_{i}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}^{\top}\right)\mathbf{p}_{i}\right)^{\top}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\mathbf{I}^{\top}\left(\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}\right)^{\top}-\left(\frac{1}{M}\sum_{i=1}^{n}m_{i}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}^{\top}\right)\mathbf{p}_{i}\right)^{\top}\right)+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\mathbf{I}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)+\left(\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\mathbf{I}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)+\left(\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)^{\top}+\sum_{i=1}^{n}m_{i}\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-M\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)^{\top}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\mathbf{I}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)+\left(\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)^{\top}+\sum_{i=1}^{n}m_{i}\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-\frac{1}{M}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)^{\top}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\mathbf{I}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)+\left(\frac{1}{M}\sum_{i=1}^{n}m_{i}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}^{\top}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)\mathbf{I}+\left(\frac{1}{M}\sum_{i=1}^{n}m_{i}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right)^{\top} +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)\mathbf{I}+\left(\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +-------------------------------------------------------------------------------- +---------------------------------------- +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\boldsymbol{\iota}_{n}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M_{n}}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)\mathbf{I}+\left(\frac{1}{M_{n}}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}_{n}=\left(\sum_{i=1}^{n}m_{i}\left|\mathbf{p}_{i}\right|^{2}-\frac{1}{M_{n}}\left|{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right|^{2}\right)\mathbf{I}+\left(\frac{1}{M_{n}}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}_{n}=\left(\alpha_{n}-\frac{1}{M_{n}}\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}\right)\mathbf{I}+\left(\frac{1}{M_{n}}\boldsymbol{\beta}_{n}\boldsymbol{\beta}_{n}^{\top}-\boldsymbol{\gamma}_{n}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}_{n+1}=\left(\alpha_{n+1}-\frac{1}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n+1}^{\top}\boldsymbol{\beta}_{n+1}\right)\mathbf{I}+\left(\frac{1}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n+1}\boldsymbol{\beta}_{n+1}^{\top}-\boldsymbol{\gamma}_{n+1}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}_{n+1}-\boldsymbol{\iota}_{n}=\left(\alpha_{n+1}-\alpha_{n}\right)\mathbf{I}-\frac{1}{M_{n+1}}\left(\boldsymbol{\beta}_{n+1}^{\top}\boldsymbol{\beta}_{n+1}\mathbf{I}-\boldsymbol{\beta}_{n+1}\boldsymbol{\beta}_{n+1}^{\top}\right)+\frac{1}{M_{n}}\left(\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}\mathbf{I}-\boldsymbol{\beta}_{n}\boldsymbol{\beta}_{n}^{\top}\right)-\left(\boldsymbol{\gamma}_{n+1}-\boldsymbol{\gamma}_{n}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}_{n+1}-\boldsymbol{\iota}_{n}=\left(\alpha_{n+1}-\alpha_{n}\right)\mathbf{I}-\frac{M_{n}}{M_{n+1}}\frac{1}{M_{n}}\left(\boldsymbol{\beta}_{n+1}^{\top}\boldsymbol{\beta}_{n+1}\mathbf{I}-\boldsymbol{\beta}_{n+1}\boldsymbol{\beta}_{n+1}^{\top}\right)+\frac{1}{M_{n}}\left(\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}\mathbf{I}-\boldsymbol{\beta}_{n}\boldsymbol{\beta}_{n}^{\top}\right)-\left(\boldsymbol{\gamma}_{n+1}-\boldsymbol{\gamma}_{n}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\boldsymbol{\iota}_{n+1}-\boldsymbol{\iota}_{n}=\left(\alpha_{n+1}-\alpha_{n}\right)\mathbf{I}-\frac{1}{M_{n}}\left[\left(\frac{M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n+1}^{\top}\boldsymbol{\beta}_{n+1}+\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}\right)\mathbf{I}-\left(\frac{M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n+1}\boldsymbol{\beta}_{n+1}^{\top}+\boldsymbol{\beta}_{n}\boldsymbol{\beta}_{n}^{\top}\right)\right]-\left(\boldsymbol{\gamma}_{n+1}-\boldsymbol{\gamma}_{n}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\alpha_{n+1}-\alpha_{n}=\sum_{i=1}^{n+1}m_{i}\left|\mathbf{p}_{i}\right|^{2}-\sum_{i=1}^{n}m_{i}\left|\mathbf{p}_{i}\right|^{2} +\] + +\end_inset + + +\begin_inset Formula +\[ +\alpha_{n+1}-\alpha_{n}=m_{n+1}\left|\mathbf{p}_{n+1}\right|^{2} +\] + +\end_inset + + +\begin_inset Formula +\[ +\frac{M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n+1}^{\top}\boldsymbol{\beta}_{n+1}+\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}=\frac{M_{n}}{M_{n}+m_{n+1}}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}+m_{n+1}\mathbf{p}_{n+1}\right)^{\top}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}+m_{n+1}\mathbf{p}_{n+1}\right)+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +=\frac{M_{n}}{M_{n}+m_{n+1}}\left(\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}+m_{n+1}\mathbf{p}_{n+1}^{\top}\right)\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}+m_{n+1}\mathbf{p}_{n+1}\right)+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +=\frac{M_{n}}{M_{n}+m_{n+1}}\left(\boldsymbol{\beta}_{n}^{\top}+m_{n+1}\mathbf{p}_{n+1}^{\top}\right)\left(\boldsymbol{\beta}_{n}+m_{n+1}\mathbf{p}_{n+1}\right)+\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n} +\] + +\end_inset + + +\begin_inset Formula +\[ +=\frac{M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}+2m_{n+1}\frac{M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n}^{\top}\mathbf{p}_{n+1}+m_{n+1}^{2}\frac{M_{n}}{M_{n}+m_{n+1}}\mathbf{p}_{n+1}^{\top}\mathbf{p}_{n+1}+\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n} +\] + +\end_inset + + +\begin_inset Formula +\[ +=\left(\frac{2M_{n}+m_{n+1}}{M_{n}+m_{n+1}}\right)\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}+\frac{2m_{n+1}M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n}^{\top}\mathbf{p}_{n+1}+\frac{m_{n+1}^{2}M_{n}}{M_{n}+m_{n+1}}\mathbf{p}_{n+1}^{\top}\mathbf{p}_{n+1} +\] + +\end_inset + + +\begin_inset Formula +\[ +=\frac{1}{M_{n}+m_{n+1}}\left[\boldsymbol{\beta}_{n}^{\top}\left(\left(2M_{n}+m_{n+1}\right)\boldsymbol{\beta}_{n}+2m_{n+1}M_{n}\mathbf{p}_{n+1}\right)+m_{n+1}^{2}M_{n}\mathbf{p}_{n+1}^{\top}\mathbf{p}_{n+1}\right] +\] + +\end_inset + + +\end_layout + +\end_body +\end_document diff --git a/data/line_rad_25.png b/data/line_rad_25.png new file mode 100644 index 0000000..8d4ca75 Binary files /dev/null and b/data/line_rad_25.png differ diff --git a/data/test.png b/data/test.png new file mode 100644 index 0000000..e54ce70 Binary files /dev/null and b/data/test.png differ diff --git a/data/test2.png b/data/test2.png new file mode 100644 index 0000000..7453ca4 Binary files /dev/null and b/data/test2.png differ diff --git a/data/test3.png b/data/test3.png new file mode 100644 index 0000000..2d29c73 Binary files /dev/null and b/data/test3.png differ