On the eigenstructure of rotations and poses: commonalities and peculiarities
Bibliographic record
Abstract
Locating vehicles, targets and objects in three-dimensional space is key to many fields of science and engineering such as robotics, aerospace, computer vision and graphics. Rotations and poses (position plus orientation) of bodies can be expressed in a variety of ways. Rotation matrices constitute one of the classic matrix Lie groups, the special orthogonal group— <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext mathvariant="italic">SO</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> . Poses can likewise be represented by matrices. One such representation is embodied in a 4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo> </mml:mo> <mml:mo>×</mml:mo> <mml:mo> </mml:mo> </mml:math> 4 matrix establishing another famous matrix Lie group, the special Euclidean group— <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext mathvariant="italic">SE</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> . An alternative representation of pose uses 6 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo> </mml:mo> <mml:mo>×</mml:mo> <mml:mo> </mml:mo> </mml:math> 6 matrices and is referred to as the group of pose adjoints— <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mtext>Ad</mml:mtext> </mml:mstyle> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mtext mathvariant="italic">SE</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> . The eigenstructures of these representations reveal much about them from Euler’s theorem for rotations to the Mozzi–Chasles theorem for the general displacement of a rigid body. While the eigenstructure of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext mathvariant="italic">SO</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> has been extensively studied, those of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext mathvariant="italic">SE</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mtext>Ad</mml:mtext> </mml:mstyle> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mtext mathvariant="italic">SE</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:math> have hardly received the same scrutiny yet their structure is much richer. Motivated by their importance in kinematics and dynamics, we provide here a complete characterization of rotations and poses in terms of the eigenstructure of their matrix Lie group representations. An eigendecomposition of pose matrices reveals that they can be cast into a form similar to that of rotations although the structure of the former can vary depending on the nature of the pose involved. In particular, the pose matrices of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext mathvariant="italic">SE</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mtext>Ad</mml:mtext> </mml:mstyle> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mtext mathvariant="italic">SE</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:math> cannot generally be diagonalized as can rotation matrices but they of course do yield to a Jordan normal form, from which we can identify a principal-axis pose in much the same manner that we can a principal-axis rotation. We also address the minimal polynomials for poses and derive a novel expression for the Jacobian in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mtext>Ad</mml:mtext> </mml:mstyle> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mtext mathvariant="italic">SE</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:math> . Finally, we argue that the true counterpart to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext mathvariant="italic">SO</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> for poses is not <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext mathvariant="italic">SE</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> but <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mtext>Ad</mml:mtext> </mml:mstyle> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mtext mathvariant="italic">SE</mml:m
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".