Frank Dellaert
commit
9e4b4b3f72
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@ -1,5 +1,5 @@
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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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#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\lyxformat 474
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\begin_document
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\begin_header
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\textclass article
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@ -15,13 +15,13 @@ theorems-std
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\font_roman times
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\font_sans default
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\output_sync 0
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@ -32,15 +32,24 @@ theorems-std
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\use_hyperref false
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\papersize default
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\use_geometry true
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\use_amsmath 1
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\use_esint 0
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\use_mhchem 1
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\use_mathdots 1
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\use_package amsmath 1
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\use_package amssymb 1
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\use_package cancel 1
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\use_package esint 0
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\use_package mathdots 1
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\use_package mathtools 1
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\use_package mhchem 1
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\use_package stackrel 1
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\use_package stmaryrd 1
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\use_package undertilde 1
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\suppress_date false
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\justification true
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\use_refstyle 0
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\index Index
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\shortcut idx
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@ -96,7 +105,7 @@ We will start with a small example of a robot moving in a plane, parameterized
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2D pose
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\emph default
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\begin_inset Formula $(x,\, y,\,\theta)$
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\begin_inset Formula $(x,\,y,\,\theta)$
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\end_inset
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.
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@ -135,7 +144,7 @@ A similar story holds for translation in the
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direction, and in fact for translations in general:
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\begin_inset Formula
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\[
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(x_{t},\, y_{t},\,\theta_{t})=(x_{0}+v_{x}t,\, y_{0}+v_{y}t,\,\theta_{0})
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(x_{t},\,y_{t},\,\theta_{t})=(x_{0}+v_{x}t,\,y_{0}+v_{y}t,\,\theta_{0})
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\]
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\end_inset
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@ -143,7 +152,7 @@ A similar story holds for translation in the
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Similarly for rotation we have
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\begin_inset Formula
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\[
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(x_{t},\, y_{t},\,\theta_{t})=(x_{0},\, y_{0},\,\theta_{0}+\omega t)
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(x_{t},\,y_{t},\,\theta_{t})=(x_{0},\,y_{0},\,\theta_{0}+\omega t)
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\]
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\end_inset
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@ -175,7 +184,7 @@ status collapsed
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\end_inset
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\begin_inset Caption
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\begin_inset Caption Standard
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\begin_layout Plain Layout
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Robot moving along a circular trajectory.
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@ -196,20 +205,20 @@ However, if we combine translation and rotation, the story breaks down!
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We cannot write
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\begin_inset Formula
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\[
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(x_{t},\, y_{t},\,\theta_{t})=(x_{0}+v_{x}t,\, y_{0}+v_{y}t,\,\theta_{0}+\omega t)
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(x_{t},\,y_{t},\,\theta_{t})=(x_{0}+v_{x}t,\,y_{0}+v_{y}t,\,\theta_{0}+\omega t)
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\]
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\end_inset
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The reason is that, if we move the robot a tiny bit according to the velocity
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vector
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\begin_inset Formula $(v_{x},\, v_{y},\,\omega)$
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\begin_inset Formula $(v_{x},\,v_{y},\,\omega)$
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\end_inset
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, we have (to first order)
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\begin_inset Formula
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\[
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(x_{\delta},\, y_{\delta},\,\theta_{\delta})=(x_{0}+v_{x}\delta,\, y_{0}+v_{y}\delta,\,\theta_{0}+\omega\delta)
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(x_{\delta},\,y_{\delta},\,\theta_{\delta})=(x_{0}+v_{x}\delta,\,y_{0}+v_{y}\delta,\,\theta_{0}+\omega\delta)
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\]
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\end_inset
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@ -255,7 +264,7 @@ status open
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\end_inset
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\begin_inset Caption
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\begin_inset Caption Standard
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\begin_layout Plain Layout
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\begin_inset CommandInset label
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@ -290,7 +299,7 @@ To make progress, we have to be more precise about how the robot behaves.
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as
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\begin_inset Formula
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\[
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T_{1}T_{2}=(x_{1},\, y_{1},\,\theta_{1})(x_{2},\, y_{2},\,\theta_{2})=(x_{1}+\cos\theta_{1}x_{2}-\sin\theta y_{2},\, y_{1}+\sin\theta_{1}x_{2}+\cos\theta_{1}y_{2},\,\theta_{1}+\theta_{2})
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T_{1}T_{2}=(x_{1},\,y_{1},\,\theta_{1})(x_{2},\,y_{2},\,\theta_{2})=(x_{1}+\cos\theta_{1}x_{2}-\sin\theta y_{2},\,y_{1}+\sin\theta_{1}x_{2}+\cos\theta_{1}y_{2},\,\theta_{1}+\theta_{2})
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\]
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\end_inset
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@ -1600,13 +1609,13 @@ Hence, an alternative way of writing down elements of
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\end_inset
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is as the ordered pair
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\begin_inset Formula $(R,\, t)$
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\begin_inset Formula $(R,\,t)$
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\end_inset
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, with composition defined a
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\begin_inset Formula
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\[
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(R_{1},\, t_{1})(R_{2},\, t_{2})=(R_{1}R_{2},\, R{}_{1}t_{2}+t_{1})
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(R_{1},\,t_{1})(R_{2},\,t_{2})=(R_{1}R_{2},\,R{}_{1}t_{2}+t_{1})
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\]
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\end_inset
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@ -2569,13 +2578,13 @@ where
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\end_inset
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is as the ordered pair
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\begin_inset Formula $(R,\, t)$
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\begin_inset Formula $(R,\,t)$
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\end_inset
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, with composition defined as
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\begin_inset Formula
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\[
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(R_{1},\, t_{1})(R_{2},\, t_{2})=(R_{1}R_{2},\, R{}_{1}t_{2}+t_{1})
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(R_{1},\,t_{1})(R_{2},\,t_{2})=(R_{1}R_{2},\,R{}_{1}t_{2}+t_{1})
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\]
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\end_inset
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@ -2944,6 +2953,218 @@ p\\
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\end_inset
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\end_layout
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\begin_layout Section
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3D Similarity Transformations
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\end_layout
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\begin_layout Standard
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The group of 3D similarity transformations
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\begin_inset Formula $Sim(3)$
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\end_inset
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is the set of
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\begin_inset Formula $4\times4$
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\end_inset
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invertible matrices of the form
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\begin_inset Formula
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\[
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T\define\left[\begin{array}{cc}
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R & t\\
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0 & s^{-1}
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\end{array}\right]
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\]
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\end_inset
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where
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\begin_inset Formula $s$
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\end_inset
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is a scalar.
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There are several different conventions in use for the Lie algebra generators,
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but we use
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\begin_inset Formula
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\[
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G^{1}=\left(\begin{array}{cccc}
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0 & 0 & 0 & 0\\
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0 & 0 & -1 & 0\\
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0 & 1 & 0 & 0\\
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0 & 0 & 0 & 0
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\end{array}\right)\mbox{}G^{2}=\left(\begin{array}{cccc}
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0 & 0 & 1 & 0\\
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0 & 0 & 0 & 0\\
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-1 & 0 & 0 & 0\\
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0 & 0 & 0 & 0
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\end{array}\right)\mbox{ }G^{3}=\left(\begin{array}{cccc}
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0 & -1 & 0 & 0\\
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1 & 0 & 0 & 0\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0
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\end{array}\right)
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\]
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\end_inset
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\begin_inset Formula
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\[
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G^{4}=\left(\begin{array}{cccc}
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0 & 0 & 0 & 1\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0
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\end{array}\right)\mbox{}G^{5}=\left(\begin{array}{cccc}
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 1\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0
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\end{array}\right)\mbox{ }G^{6}=\left(\begin{array}{cccc}
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 1\\
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0 & 0 & 0 & 0
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\end{array}\right)\mbox{ }G^{7}=\left(\begin{array}{cccc}
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & -1
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\end{array}\right)
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\]
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\end_inset
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\end_layout
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\begin_layout Subsection
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Actions
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\end_layout
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\begin_layout Standard
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The action of
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\begin_inset Formula $\SEthree$
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\end_inset
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on 3D points is done by embedding the points in
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\begin_inset Formula $\mathbb{R}^{4}$
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\end_inset
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by using homogeneous coordinates
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\begin_inset Formula
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\[
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\hat{q}=\left[\begin{array}{c}
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q\\
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s^{-1}
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\end{array}\right]=\left[\begin{array}{c}
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Rp+t\\
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s^{-1}
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\end{array}\right]=\left[\begin{array}{cc}
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R & t\\
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0 & s^{-1}
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\end{array}\right]\left[\begin{array}{c}
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p\\
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1
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\end{array}\right]=T\hat{p}
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\]
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\end_inset
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The derivative
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\begin_inset Formula $D_{1}f(\xi)$
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\end_inset
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in an incremental change
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\begin_inset Formula $\xi$
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\end_inset
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to
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||||
\begin_inset Formula $T$
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\end_inset
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is given by
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||||
\begin_inset Formula $TH(p)$
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\end_inset
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where
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\begin_inset Formula
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\[
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H(p)=G_{jk}^{i}p^{j}=\left(\begin{array}{ccccccc}
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0 & z & -y & 1 & 0 & 0 & 0\\
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-z & 0 & x & 0 & 1 & 0 & 0\\
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y & -x & 0 & 0 & 0 & 1 & 0\\
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0 & 0 & 0 & 0 & 0 & 0 & -1
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\end{array}\right)
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\]
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\end_inset
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In other words
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\begin_inset Formula
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\[
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D_{1}f(\xi)=\left[\begin{array}{cc}
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R & t\\
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0 & s^{-1}
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\end{array}\right]\left[\begin{array}{ccc}
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-\left[p\right]_{x} & I_{3} & 0\\
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0 & 0 & -1
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\end{array}\right]=\left[\begin{array}{ccc}
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-R\left[p\right]_{x} & R & -t\\
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0 & 0 & -s^{-1}
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\end{array}\right]
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\]
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\end_inset
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This is the derivative for the action on homogeneous coordinates.
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Switching back to non-homogeneous coordinates is done by
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\begin_inset Formula
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\[
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\left[\begin{array}{c}
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q\\
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a
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\end{array}\right]\rightarrow q/a
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\]
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\end_inset
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with derivative
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||||
\begin_inset Formula
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\[
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||||
\left[\begin{array}{cc}
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a^{-1}I_{3} & -qa^{-2}\end{array}\right]
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\]
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\end_inset
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For
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\begin_inset Formula $a=s^{-1}$
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\end_inset
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we obtain
|
||||
\begin_inset Formula
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\[
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D_{1}f(\xi)=\left[\begin{array}{cc}
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sI_{3} & -qs^{2}\end{array}\right]\left[\begin{array}{ccc}
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-R\left[p\right]_{x} & R & -t\\
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0 & 0 & -s^{-1}
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\end{array}\right]=\left[\begin{array}{ccc}
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-sR\left[p\right]_{x} & sR & -st+qs\end{array}\right]=\left[\begin{array}{ccc}
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-sR\left[p\right]_{x} & sR & sRp\end{array}\right]
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\]
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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 Newpage pagebreak
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\end_inset
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\end_layout
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\begin_layout Section
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doc/math.lyx
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@ -1,5 +1,5 @@
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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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||||
#LyX 2.1 created this file. For more info see http://www.lyx.org/
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||||
\lyxformat 474
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||||
\begin_document
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\begin_header
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\textclass article
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@ -17,13 +17,13 @@ theorems-ams-bytype
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\font_roman times
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\font_sans default
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\font_typewriter default
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\font_math auto
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\font_default_family rmdefault
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\graphics default
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\default_output_format default
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\output_sync 0
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@ -34,15 +34,24 @@ theorems-ams-bytype
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\use_hyperref false
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\use_geometry true
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\cite_engine_type default
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\justification true
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\use_refstyle 0
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\index Index
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\shortcut idx
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@ -1603,11 +1612,7 @@ name "sec:Derivatives-of-Actions"
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\end_layout
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\begin_layout Standard
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The (usual) action of an
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\begin_inset Formula $n$
|
||||
\end_inset
|
||||
|
||||
-dimensional matrix group
|
||||
The (usual) action of a matrix group
|
||||
\begin_inset Formula $G$
|
||||
\end_inset
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||||
|
||||
|
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@ -1632,7 +1637,7 @@ Since this is a function defined on the product
|
|||
\end_inset
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||||
|
||||
the derivative is a linear transformation
|
||||
\begin_inset Formula $Df:\Multi{2n}n$
|
||||
\begin_inset Formula $Df:\Multi{m+n}n$
|
||||
\end_inset
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||||
|
||||
with
|
||||
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@ -1643,7 +1648,15 @@ Df_{(T,p)}\left(\xi,\delta p\right)=D_{1}f_{(T,p)}\left(\xi\right)+D_{2}f_{(T,p)
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|
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\end_inset
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||||
|
||||
where
|
||||
\begin_inset Formula $m$
|
||||
\end_inset
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||||
|
||||
is the dimensionality of the manifold
|
||||
\begin_inset Formula $G$
|
||||
\end_inset
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||||
|
||||
.
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||||
\end_layout
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||||
\begin_layout Theorem
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||||
|
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@ -1672,7 +1685,7 @@ H(p) & I_{n}\end{array}\right]
|
|||
\end_inset
|
||||
|
||||
with
|
||||
\begin_inset Formula $H:\Reals n\rightarrow\Reals{n\times n}$
|
||||
\begin_inset Formula $H:\Reals m\rightarrow\Reals{n\times m}$
|
||||
\end_inset
|
||||
|
||||
a linear mapping that depends on
|
||||
|
|
@ -1799,7 +1812,7 @@ with
|
|||
\end_inset
|
||||
|
||||
an
|
||||
\begin_inset Formula $n\times n$
|
||||
\begin_inset Formula $n\times m$
|
||||
\end_inset
|
||||
|
||||
matrix that depends on
|
||||
|
|
@ -3377,7 +3390,7 @@ Retractions
|
|||
|
||||
\begin_layout Standard
|
||||
\begin_inset FormulaMacro
|
||||
\renewcommand{\retract}{\mathcal{R}}
|
||||
\newcommand{\retract}{\mathcal{R}}
|
||||
{\mathcal{R}}
|
||||
\end_inset
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||||
|
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|
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BIN
doc/math.pdf
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doc/math.pdf
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@ -67,19 +67,19 @@ Similarity3 Similarity3::operator*(const Similarity3& T) const {
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}
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Similarity3 Similarity3::inverse() const {
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Rot3 Rt = R_.inverse();
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Point3 sRt = R_.inverse() * (-s_ * t_);
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const Rot3 Rt = R_.inverse();
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const Point3 sRt = Rt * (-s_ * t_);
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return Similarity3(Rt, sRt, 1.0 / s_);
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}
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Point3 Similarity3::transform_from(const Point3& p, //
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OptionalJacobian<3, 7> H1, OptionalJacobian<3, 3> H2) const {
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Point3 q = R_ * p + t_;
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const Point3 q = R_ * p + t_;
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if (H1) {
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const Matrix3 R = R_.matrix();
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Matrix3 DR = s_ * R * skewSymmetric(-p.x(), -p.y(), -p.z());
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// TODO(frank): explain the derivative in lambda
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*H1 << DR, s_ * R, s_ * p.vector();
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// For this derivative, see LieGroups.pdf
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const Matrix3 sR = s_ * R_.matrix();
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const Matrix3 DR = sR * skewSymmetric(-p.x(), -p.y(), -p.z());
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*H1 << DR, sR, sR * p.vector();
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}
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if (H2)
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*H2 = s_ * R_.matrix(); // just 3*3 sub-block of matrix()
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@ -94,7 +94,7 @@ Matrix4 Similarity3::wedge(const Vector7& xi) {
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// http://www.ethaneade.org/latex2html/lie/node29.html
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const auto w = xi.head<3>();
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const auto u = xi.segment<3>(3);
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double lambda = xi[6];
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const double lambda = xi[6];
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Matrix4 W;
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W << skewSymmetric(w), u, 0, 0, 0, -lambda;
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return W;
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@ -106,8 +106,7 @@ Matrix7 Similarity3::AdjointMap() const {
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const Vector3 t = t_.vector();
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const Matrix3 A = s_ * skewSymmetric(t) * R;
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Matrix7 adj;
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adj <<
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R, Z_3x3, Matrix31::Zero(), // 3*7
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adj << R, Z_3x3, Matrix31::Zero(), // 3*7
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A, s_ * R, -s_ * t, // 3*7
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Matrix16::Zero(), 1; // 1*7
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return adj;
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@ -115,72 +114,35 @@ Matrix7 Similarity3::AdjointMap() const {
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Matrix3 Similarity3::GetV(Vector3 w, double lambda) {
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// http://www.ethaneade.org/latex2html/lie/node29.html
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double lambda2 = lambda * lambda;
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double theta2 = w.transpose() * w;
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double theta = sqrt(theta2);
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double A, B, C;
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// TODO(frank): eliminate copy/paste
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if (theta2 > 1e-9 && lambda2 > 1e-9) {
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const double theta2 = w.transpose() * w;
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double Y, Z, W;
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if (theta2 > 1e-9) {
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const double theta = sqrt(theta2);
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const double X = sin(theta) / theta;
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const double Y = (1 - cos(theta)) / theta2;
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const double Z = (1 - X) / theta2;
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const double W = (0.5 - Y) / theta2;
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const double alpha = lambda2 / (lambda2 + theta2);
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const double beta = (exp(-lambda) - 1 + lambda) / lambda2;
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const double gamma = Y - (lambda * Z);
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const double mu = (1 - lambda + (0.5 * lambda2) - exp(-lambda))
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/ (lambda2 * lambda);
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const double upsilon = Z - (lambda * W);
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A = (1 - exp(-lambda)) / lambda;
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B = alpha * (beta - gamma) + gamma;
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C = alpha * (mu - upsilon) + upsilon;
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} else if (theta2 <= 1e-9 && lambda2 > 1e-9) {
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//Taylor series expansions
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const double Y = 0.5 - theta2 / 24.0;
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const double Z = 1.0 / 6.0 - theta2 / 120.0;
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const double W = 1.0 / 24.0 - theta2 / 720.0;
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const double alpha = lambda2 / (lambda2 + theta2);
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const double beta = (exp(-lambda) - 1 + lambda) / lambda2;
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const double gamma = Y - (lambda * Z);
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const double mu = (1 - lambda + (0.5 * lambda2) - exp(-lambda))
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/ (lambda2 * lambda);
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const double upsilon = Z - (lambda * W);
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A = (1 - exp(-lambda)) / lambda;
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B = alpha * (beta - gamma) + gamma;
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C = alpha * (mu - upsilon) + upsilon;
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} else if (theta2 > 1e-9 && lambda2 <= 1e-9) {
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const double X = sin(theta) / theta;
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const double Y = (1 - cos(theta)) / theta2;
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const double Z = (1 - X) / theta2;
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const double W = (0.5 - Y) / theta2;
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const double alpha = lambda2 / (lambda2 + theta2);
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const double beta = 0.5 - lambda / 6.0 + lambda2 / 24.0
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- (lambda * lambda2) / 120;
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const double gamma = Y - (lambda * Z);
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const double mu = 1.0 / 6.0 - lambda / 24 + lambda2 / 120
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- (lambda * lambda2) / 720;
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const double upsilon = Z - (lambda * W);
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if (lambda < 1e-9) {
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A = 1 - lambda / 2.0 + lambda2 / 6.0;
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Y = (1 - cos(theta)) / theta2;
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Z = (1 - X) / theta2;
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W = (0.5 - Y) / theta2;
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} else {
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A = (1 - exp(-lambda)) / lambda;
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// Taylor series expansion for theta=0, X not needed (as is 1)
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Y = 0.5 - theta2 / 24.0;
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Z = 1.0 / 6.0 - theta2 / 120.0;
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W = 1.0 / 24.0 - theta2 / 720.0;
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}
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B = alpha * (beta - gamma) + gamma;
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C = alpha * (mu - upsilon) + upsilon;
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const double lambda2 = lambda * lambda, lambda3 = lambda2 * lambda;
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double A, alpha = 0.0, beta, mu;
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if (lambda2 > 1e-9) {
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A = (1.0 - exp(-lambda)) / lambda;
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alpha = 1.0 / (1.0 + theta2 / lambda2);
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beta = (exp(-lambda) - 1 + lambda) / lambda2;
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mu = (1 - lambda + (0.5 * lambda2) - exp(-lambda)) / lambda3;
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} else {
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const double Y = 0.5 - theta2 / 24.0;
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const double Z = 1.0 / 6.0 - theta2 / 120.0;
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const double W = 1.0 / 24.0 - theta2 / 720.0;
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const double gamma = Y - (lambda * Z);
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const double upsilon = Z - (lambda * W);
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if (lambda < 1e-9) {
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A = 1 - lambda / 2.0 + lambda2 / 6.0;
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} else {
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A = (1 - exp(-lambda)) / lambda;
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}
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B = gamma;
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C = upsilon;
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A = 1.0 - lambda / 2.0 + lambda2 / 6.0;
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beta = 0.5 - lambda / 6.0 + lambda2 / 24.0 - lambda3 / 120.0;
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mu = 1.0 / 6.0 - lambda / 24.0 + lambda2 / 120.0 - lambda3 / 720.0;
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}
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const double gamma = Y - (lambda * Z), upsilon = Z - (lambda * W);
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const double B = alpha * (beta - gamma) + gamma;
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const double C = alpha * (mu - upsilon) + upsilon;
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const Matrix3 Wx = skewSymmetric(w[0], w[1], w[2]);
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return A * I_3x3 + B * Wx + C * Wx * Wx;
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}
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@ -218,7 +180,7 @@ std::ostream &operator<<(std::ostream &os, const Similarity3& p) {
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const Matrix4 Similarity3::matrix() const {
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Matrix4 T;
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T.topRows<3>() << R_.matrix(), t_.vector();
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T.bottomRows<1>() << 0, 0, 0, 1.0/s_;
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T.bottomRows<1>() << 0, 0, 0, 1.0 / s_;
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return T;
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}
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@ -226,4 +188,4 @@ Similarity3::operator Pose3() const {
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return Pose3(R_, s_ * t_);
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}
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}
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} // namespace gtsam
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@ -45,7 +45,8 @@ static const Rot3 R = Rot3::Rodrigues(0.3, 0, 0);
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static const double s = 4;
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static const Similarity3 id;
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static const Similarity3 T1(R, Point3(3.5, -8.2, 4.2), 1);
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static const Similarity3 T2(Rot3::Rodrigues(0.3, 0.2, 0.1), Point3(3.5, -8.2, 4.2), 1);
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static const Similarity3 T2(Rot3::Rodrigues(0.3, 0.2, 0.1),
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Point3(3.5, -8.2, 4.2), 1);
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static const Similarity3 T3(Rot3::Rodrigues(-90, 0, 0), Point3(1, 2, 3), 1);
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static const Similarity3 T4(R, P, s);
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static const Similarity3 T5(R, P, 10);
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@ -187,11 +188,7 @@ TEST(Similarity3, manifold_first_order) {
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// Return as a 4*4 Matrix
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TEST(Similarity3, Matrix) {
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Matrix4 expected;
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expected <<
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1, 0, 0, 1,
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0, 1, 0, 1,
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0, 0, 1, 0,
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0, 0, 0, 0.5;
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expected << 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0.5;
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Matrix4 actual = T6.matrix();
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EXPECT(assert_equal(expected, actual));
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}
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@ -200,12 +197,12 @@ TEST(Similarity3, Matrix) {
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// Exponential and log maps
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TEST(Similarity3, ExpLogMap) {
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Vector7 delta;
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delta << 0.1,0.2,0.3,0.4,0.5,0.6,0.7;
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delta << 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7;
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Vector7 actual = Similarity3::Logmap(Similarity3::Expmap(delta));
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EXPECT(assert_equal(delta, actual));
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Vector7 zeros;
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zeros << 0,0,0,0,0,0,0;
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zeros << 0, 0, 0, 0, 0, 0, 0;
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Vector7 logIdentity = Similarity3::Logmap(Similarity3::identity());
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EXPECT(assert_equal(zeros, logIdentity));
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@ -214,7 +211,8 @@ TEST(Similarity3, ExpLogMap) {
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EXPECT(assert_equal(expZero, ident));
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// Compare to matrix exponential, using expm in Lie.h
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EXPECT(assert_equal(expm<Similarity3>(delta), Similarity3::Expmap(delta), 1e-3));
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EXPECT(
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assert_equal(expm<Similarity3>(delta), Similarity3::Expmap(delta), 1e-3));
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}
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//******************************************************************************
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@ -237,8 +235,8 @@ TEST(Similarity3, GroupAction) {
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EXPECT(assert_equal(Point3(2, 2, 3), Ta.transform_from(pa)));
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EXPECT(assert_equal(Point3(4, 4, 6), Tb.transform_from(pa)));
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Similarity3 Tc(Rot3::Rz(M_PI/2.0), Point3(1, 2, 3), 1.0);
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Similarity3 Td(Rot3::Rz(M_PI/2.0), Point3(1, 2, 3), 2.0);
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Similarity3 Tc(Rot3::Rz(M_PI / 2.0), Point3(1, 2, 3), 1.0);
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Similarity3 Td(Rot3::Rz(M_PI / 2.0), Point3(1, 2, 3), 2.0);
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EXPECT(assert_equal(Point3(1, 3, 3), Tc.transform_from(pa)));
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EXPECT(assert_equal(Point3(2, 6, 6), Td.transform_from(pa)));
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@ -249,10 +247,10 @@ TEST(Similarity3, GroupAction) {
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Point3 q(1, 2, 3);
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for (const auto T : { T1, T2, T3, T4, T5, T6 }) {
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Point3 q(1, 0, 0);
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Matrix H1 = numericalDerivative21<Point3, Similarity3, Point3>(f, T1, q);
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Matrix H2 = numericalDerivative22<Point3, Similarity3, Point3>(f, T1, q);
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Matrix H1 = numericalDerivative21<Point3, Similarity3, Point3>(f, T, q);
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Matrix H2 = numericalDerivative22<Point3, Similarity3, Point3>(f, T, q);
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Matrix actualH1, actualH2;
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T1.transform_from(q, actualH1, actualH2);
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T.transform_from(q, actualH1, actualH2);
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EXPECT(assert_equal(H1, actualH1));
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EXPECT(assert_equal(H2, actualH2));
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}
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@ -387,25 +385,25 @@ TEST(Similarity3, AlignScaledPointClouds) {
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TEST(Similarity3 , Invariants) {
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Similarity3 id;
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EXPECT(check_group_invariants(id,id));
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EXPECT(check_group_invariants(id,T3));
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EXPECT(check_group_invariants(T2,id));
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EXPECT(check_group_invariants(T2,T3));
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EXPECT(check_group_invariants(id, id));
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EXPECT(check_group_invariants(id, T3));
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EXPECT(check_group_invariants(T2, id));
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EXPECT(check_group_invariants(T2, T3));
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EXPECT(check_manifold_invariants(id,id));
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EXPECT(check_manifold_invariants(id,T3));
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EXPECT(check_manifold_invariants(T2,id));
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EXPECT(check_manifold_invariants(T2,T3));
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EXPECT(check_manifold_invariants(id, id));
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EXPECT(check_manifold_invariants(id, T3));
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EXPECT(check_manifold_invariants(T2, id));
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EXPECT(check_manifold_invariants(T2, T3));
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}
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//******************************************************************************
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TEST(Similarity3 , LieGroupDerivatives) {
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Similarity3 id;
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CHECK_LIE_GROUP_DERIVATIVES(id,id);
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CHECK_LIE_GROUP_DERIVATIVES(id,T2);
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CHECK_LIE_GROUP_DERIVATIVES(T2,id);
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CHECK_LIE_GROUP_DERIVATIVES(T2,T3);
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CHECK_LIE_GROUP_DERIVATIVES(id, id);
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CHECK_LIE_GROUP_DERIVATIVES(id, T2);
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CHECK_LIE_GROUP_DERIVATIVES(T2, id);
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CHECK_LIE_GROUP_DERIVATIVES(T2, T3);
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}
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//******************************************************************************
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