On the Continuity of Rotation Representations in Neural Networks

Yi ZhouConnelly BarnesJingwan LuJimei YangHao Li

article2018CVPR1,933 citations

Proves that widely used 3D rotation representations like quaternions and Euler angles are topologically discontinuous and introduces continuous 5D and 6D alternatives that substantially improve neural network learning across vision and graphics tasks.

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Deep learning models are increasingly deployed to predict rotations in computer vision, robotics, and 3D animation, covering tasks such as object pose estimation, camera localization, and human motion synthesis. A persistent challenge in these applications is the occurrence of severe, unpredictable orientation errors—often reaching up to 180 degrees—even after extensive model training. The article set out to identify the mathematical source of these failures and demonstrate new rotation formats that prevent these large learning errors.

To address this issue, the article evaluated rotation formats through topological analysis and empirical experiments. The authors developed a formal definition of continuity tailored to neural networks and proved that representing the complete three-dimensional rotation space continuously in four or fewer dimensions is mathematically impossible. They constructed continuous five-dimensional and six-dimensional representations using Gram-Schmidt orthogonalization and stereographic projection, generalizing the concept to higher-dimensional rotations. To validate their approach, the authors benchmarked these continuous representations against standard methods (quaternions, Euler angles, axis-angle formats, and direct matrix prediction) across three tasks: an autoencoder sanity test, 3D point cloud rotation estimation, and human pose inverse kinematics using large motion-capture datasets.

Across all benchmarks, the findings show that standard three-dimensional and four-dimensional representations—including Euler angles, axis-angles, and quaternions—suffer from inherent mathematical discontinuities that cause models to converge slowly and generate massive errors near critical rotation boundaries. In the autoencoder sanity test, models using discontinuous representations exhibited mean errors between 6 and 14 times higher than continuous ones, frequently producing errors up to 180 degrees, whereas the proposed continuous representations capped maximum errors under 2 degrees. In the 3D point cloud experiment, the 6D representation produced a mean error of 2.85 degrees, outperforming quaternions (9.03 degrees) and Euler angles (14.13 degrees). Similarly, in human pose inverse kinematics, the 6D format yielded the lowest mean error (1.9 centimeters) and completely avoided the severe anatomical distortions caused by quaternions and direct matrix regression.

These findings indicate that large orientation failures in production models are frequently caused by mathematical mismatches in representation choice rather than inadequate network architectures or insufficient data. Adopting continuous formats provides substantial practical benefits, including faster training convergence, lower compute overhead, and the elimination of catastrophic flip errors in safety-critical robotics and automated vision systems. Furthermore, using a 6D representation directly inside the network resolves the orthogonality drift inherent in unconstrained 3x3 matrix prediction without requiring disruptive post-processing steps.

For technical leaders and engineering teams deploying neural networks for 3D rotation tasks, the article strongly recommends adopting the 6D continuous representation as the default training target. If downstream systems strictly require quaternions or Euler angles for storage or interpolation, pipelines should output the 6D format during network learning and perform downstream conversions afterward. While the 5D representation is mathematically continuous, it proved harder for networks to optimize due to projection-induced gradient distortion, making the 6D format the superior choice in practice. Given that these continuous mappings assume full, unconstrained rotation spaces, specialized applications operating within strictly limited angular ranges may still function with traditional formats, but standardizing on 6D removes edge-case failure risks with minimal computational overhead.

Adobe Pinscreen University of Southern California autoencoders human pose inverse kinematics rotation representations continuous rotation homeomorphic variational auto-encoding

arXiv: 1812.07035

Paper: Animating rotation with quaternion curves, Ken Shoemake (1985). This foundational paper establishes how 3D rotations can be parameterized and smoothly interpolated using unit quaternions on spherical manifolds, defining the classical continuous and discontinuous representations that the source paper analyzes.
Paper: Least-Squares Estimation of Transformation Parameters Between Two Point Patterns, S. Umeyama (1991). It provides the fundamental closed-form SVD-based mathematical formulation for estimating proper 3D orthogonal rotation matrices, which informs the orthonormalization and continuous mapping techniques discussed in the source.
Paper: PoseCNN: A Convolutional Neural Network for 6D Object Pose Estimation in Cluttered Scenes, Yu Xiang et al. (2017). It details standard deep learning practices for 3D rotation and 6D pose estimation that rely on discontinuous rotation representations such as quaternions, serving as a primary baseline and motivation for the source.

Paper: Efficient Geometry-aware 3D Generative Adversarial Networks, Eric R. Chan et al. (2022). This paper advances 3D generative modeling and camera pose conditioning in neural networks, directly benefiting from continuous 3D rotation parameterizations during training.
Paper: GIRAFFE: Representing Scenes as Compositional Generative Neural Feature Fields, Michael Niemeyer et al. (2021). It builds upon continuous 3D coordinate and rotation parameterizations to compose and control disentangled 3D object poses and feature fields in generative networks.

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Abstract

In neural networks, it is often desirable to work with various representations of the same space. For example, 3D rotations can be represented with quaternions or Euler angles. In this paper, we advance a definition of a continuous representation, which can be helpful for training deep neural networks. We relate this to topological concepts such as homeomorphism and embedding. We then investigate what are continuous and discontinuous representations for 2D, 3D, and n-dimensional rotations. We demonstrate that for 3D rotations, all representations are discontinuous in the real Euclidean spaces of four or fewer dimensions. Thus, widely used representations such as quaternions and Euler angles are discontinuous and difficult for neural networks to learn. We show that the 3D rotations have continuous representations in 5D and 6D, which are more suitable for learning. We also present continuous representations for the general case of the n-dimensional rotation group SO(n). While our main focus is on rotations, we also show that our constructions apply to other groups such as the orthogonal group and similarity transforms. We finally present empirical results, which show that our continuous rotation representations outperform discontinuous ones for several practical problems in graphics and vision, including a simple autoencoder sanity test, a rotation estimator for 3D point clouds, and an inverse kinematics solver for 3D human poses.

Topics

autoencoders human pose inverse kinematics point cloud network architecture recurrent neural networks visual representations

Keywords

rotation representations continuous rotation homeomorphic variational auto-encoding unit quaternions rotation matrices rotation autoencoder network approximation theory forward kinematics

1 Introduction
2 Related Work
3 Definition of Continuous Representation
4 Rotation Representation Analysis
4.1 Discontinuous Representations
4.2 Continuous Representations
5 Empirical Results
5.1 Sanity Test
5.2 Pose Estimation for 3D Point Clouds
5.3 Inverse Kinematics for Human Poses
6 Conclusion
7 Acknowledgements
References
A Overview of the Supplemental Document
B 6D Representation for the 3D Rotations
C Proof that Case 4 gives a Continuous Representation
D The Unit Quaternions are a Discontinuous Representation for the 3D Rotations
E Interaction Between 5D and 6D Continuous Representations and Discontinuous Ones
F Visualizing Discontinuities in 3D Rotations
G Additional Empirical Results
G.1 Visualization of Inverse Kinematics Test Result
G.2 Additional Sanity test

Knowls

Knowl 1 — Continuous Representation for Neural Networks
definition
Let $X$ be a compact topological space, representing the original target domain (such as a manifold of rotations or poses). Let $R \subseteq \mathbb{R}^d$ be a subset of a real Euclidean vector space equipped with the standard Euclidean topology, representing the network representation space.

A pair of functions $(f, g)$ is defined as a representation of $X$ in $R$ if $g: X \to R$ maps from the original space to the representation space, and $f: R \to X$ is a left inverse of $g$ , satisfying: $f(g(x)) = x \quad \forall x \in X$

The representation $(f, g)$ is defined to be continuous if the mapping $g: X \to R$ is continuous with respect to the topology of $X$ and the Euclidean topology of $R$ .

Because $X$ is compact and $R$ is Hausdorff under the Euclidean topology, any continuous injective map $g: X \to R$ is a topological embedding, meaning $X$ is homeomorphic to its image $g(X) \subseteq R$ . Consequently, a continuous representation $(f, g)$ exists if and only if $X$ topologically embeds into the Euclidean space $\mathbb{R}^d$ .
Knowl 2 — Non-Existence of Continuous 3D Rotation Representations in Four or Fewer Dimensions
theoretical result
The 3D special orthogonal group $SO(3)$ is homeomorphic to the real projective space $\mathbb{R}P^3$ . By topological embedding theorems, $\mathbb{R}P^3$ embeds into Euclidean space $\mathbb{R}^5$ equipped with the Euclidean topology, but cannot embed into $\mathbb{R}^d$ for any $d \le 4$ .

Because the existence of a continuous representation $(f, g)$ of $SO(3)$ in $\mathbb{R}^d$ requires $g$ to be a topological embedding into $\mathbb{R}^d$ , there is no continuous representation of $SO(3)$ in four or fewer dimensions ( $d \le 4$ ). Consequently, all 3D representations (Euler angles, axis-angle, Rodrigues vectors) and 4D representations (unit quaternions and unnormalized quaternions) are fundamentally discontinuous mappings from $SO(3)$ under the Euclidean topology.
Knowl 3 — Continuous 6D Representation of 3D Rotations via Gram-Schmidt Orthogonalization
model/method
A continuous 6D representation $(f_{GS}, g_{GS})$ for the 3D rotation group $SO(3)$ is constructed by dropping the last column of a $3 \times 3$ rotation matrix and recovering the full orthogonal matrix via Gram-Schmidt orthogonalization and the vector cross product.

For any rotation matrix $M = [a_1 \mid a_2 \mid a_3] \in SO(3)$ with column vectors $a_1, a_2, a_3 \in \mathbb{R}^3$ , the mapping to the 6D representation space $g_{GS}: SO(3) \to \mathbb{R}^{3 \times 2} \cong \mathbb{R}^6$ is: $g_{GS}(M) = [a_1 \mid a_2]$

Given a 6D representation matrix $[a_1 \mid a_2] \in \mathbb{R}^{3 \times 2}$ where $a_1$ and $a_2$ are linearly independent, the reconstruction mapping $f_{GS}: \mathbb{R}^{3 \times 2} \to SO(3)$ recovers the orthonormal matrix $[b_1 \mid b_2 \mid b_3] \in SO(3)$ via: $b_1 = N(a_1) = \frac{a_1}{\|a_1\|}$ $b_2 = N(a_2 - (b_1 \cdot a_2) b_1) = \frac{a_2 - (b_1 \cdot a_2) b_1}{\|a_2 - (b_1 \cdot a_2) b_1\|}$ $b_3 = b_1 \times b_2$ where $N(v) = v / \|v\|$ denotes vector normalization.

This mapping satisfies $f_{GS}(g_{GS}(M)) = M$ for all $M \in SO(3)$ and is continuous on $\mathbb{R}^{3 \times 2} \setminus D$ , where $D$ is the measure-zero set of linearly dependent pairs $(a_1, a_2)$ .
Knowl 4 — Continuous 5D Representation of 3D Rotations via Stereographic Projection
model/method
The 6D Gram-Schmidt representation of $SO(3)$ can be reduced to a continuous 5D representation without introducing topological discontinuities by composing it with a normalized stereographic projection.

Let the normalized stereographic projection $P: \mathbb{R}^m \to \mathbb{R}^{m-1}$ from the projection point $(1, 0, \dots, 0)$ be defined for $u \in \mathbb{R}^m$ by: $P(u) = \left[ \frac{v_2}{1 - v_1}, \frac{v_3}{1 - v_1}, \dots, \frac{v_m}{1 - v_1} \right]^T, \quad \text{where } v = \frac{u}{\|u\|}$

Let the stereographic un-projection $Q: \mathbb{R}^{m-1} \to \mathbb{R}^m$ be defined for $w \in \mathbb{R}^{m-1}$ by: $Q(w) = \frac{1}{\|w\|} \left[ \frac{1}{2}(\|w\|^2 - 1), w_1, w_2, \dots, w_{m-1} \right]^T$

For $M \in SO(3)$ , let $\gamma(M) = [a_1^T, a_2^T]^T \in \mathbb{R}^6$ denote the flattened first two column vectors of $M$ . The forward mapping $g_P: SO(3) \to \mathbb{R}^5$ projects the last 4 components of $\gamma(M)$ : $g_P(M) = [\gamma_{1:2}(M), P(\gamma_{3:6}(M))]$ where $\gamma_{1:2} = [\gamma_1, \gamma_2]$ and $\gamma_{3:6} = [\gamma_3, \gamma_4, \gamma_5, \gamma_6]$ .

The inverse mapping $f_P: \mathbb{R}^5 \to SO(3)$ un-projects the last 3 components and evaluates the 6D Gram-Schmidt map $f_{GS}$ : $f_P(u) = f_{GS}\left( [u_{1:2}, Q(u_{3:5})] \text{ reshaped to } 3 \times 2 \right)$

Because $\gamma_{3:6}$ is a column of an orthonormal matrix, its normalized first element $v_1$ never equals $1$ , guaranteeing that $g_P$ is globally continuous on $SO(3)$ and $f_P(g_P(M)) = M$ for all $M \in SO(3)$ .
Knowl 5 — Continuous (n^2 - n)-Dimensional Representation for SO(n)
model/method
For the $n$ -dimensional special orthogonal group $SO(n)$ , a continuous representation $(f_{GS}, g_{GS})$ in $\mathbb{R}^{n \times (n-1)} = \mathbb{R}^{n^2 - n}$ is constructed by dropping the final column of the matrix and applying an $n$ -dimensional Gram-Schmidt reconstruction.

The mapping $g_{GS}: SO(n) \to \mathbb{R}^{n \times (n-1)}$ drops the $n$ -th column vector of $M = [a_1 \mid a_2 \mid \dots \mid a_n] \in SO(n)$ : $g_{GS}([a_1 \mid a_2 \mid \dots \mid a_n]) = [a_1 \mid a_2 \mid \dots \mid a_{n-1}]$

The inverse mapping $f_{GS}: \mathbb{R}^{n \times (n-1)} \to SO(n)$ computes an orthonormal basis $[b_1 \mid b_2 \mid \dots \mid b_n] \in SO(n)$ by: $b_1 = N(a_1)$ $b_i = N\left(a_i - \sum_{j=1}^{i-1} (b_j \cdot a_i) b_j\right) \quad \text{for } 2 \le i < n$ $b_n = \det\begin{bmatrix} e_1 & b_1 & \dots & b_{n-1} \\ e_2 & b_1 & \dots & b_{n-1} \\ \vdots & \vdots & \ddots & \vdots \\ e_n & b_1 & \dots & b_{n-1} \end{bmatrix}^T$ where $N(v) = v / \|v\|$ and $e_1, \dots, e_n$ are the canonical basis vectors in $\mathbb{R}^n$ . The determinant computes the generalized cross product in $\mathbb{R}^n$ . This representation is continuous and satisfies $f_{GS}(g_{GS}(M)) = M$ for all $M \in SO(n)$ on the domain where $a_1, \dots, a_{n-1}$ are linearly independent.
Knowl 6 — Continuous (n^2 - 2n + 2)-Dimensional Representation for SO(n) via Multiple Projections
model/method
The $(n^2 - n)$ -dimensional Gram-Schmidt representation of $SO(n)$ can be reduced by up to $n-2$ dimensions to yield a continuous representation in $\mathbb{R}^{n^2 - 2n + 2}$ .

Let $\gamma(M) \in \mathbb{R}^{n(n-1)}$ be the vectorized representation formed by concatenating the first $n-1$ columns $a_1, \dots, a_{n-1} \in \mathbb{R}^n$ of $M \in SO(n)$ . For each projection index $i \in \{1, \dots, n-2\}$ , an $(n+1)$ -dimensional vector is formed by prepending one component of the first column ( $\gamma_{n+1-i}$ ) to the $(i+1)$ -th column ( $a_{i+1}$ ).

Each of these $n-2$ vectors is mapped through the normalized stereographic projection $P: \mathbb{R}^{n+1} \to \mathbb{R}^n$ : $P(u) = \left[ \frac{v_2}{1 - v_1}, \dots, \frac{v_{n+1}}{1 - v_1} \right]^T, \quad v = \frac{u}{\|u\|}$

The $n-2$ projected $n$ -dimensional vectors are concatenated with the remaining 2 unprojected components of $a_1$ , yielding an output vector in $\mathbb{R}^{n^2 - 2n + 2}$ . The inverse mapping applies the stereographic un-projection $Q: \mathbb{R}^n \to \mathbb{R}^{n+1}$ to each projected segment and then evaluates $f_{GS}$ on the reshaped $n \times (n-1)$ matrix.
Knowl 7 — Continuous Representations for Orthogonal and Similarity Transformation Groups
model/method
The continuous representations of $SO(n)$ generalize to the full orthogonal group $O(n)$ and the similarity transformation group $\text{Sim}(n) = \{ \rho(x) = \alpha R x + u \mid \alpha > 0, R \in O(n), u \in \mathbb{R}^n \}$ :
1. Orthogonal Group $O(n)$ : The matrix is parameterized using the continuous $(n^2 - n)$ - or $(n^2 - 2n + 2)$ -dimensional representation together with a sign scalar $s = \det(M) \in \{-1, +1\}$ . In the decoding map $f_{GS}$ , the computed final basis vector $b_n$ is multiplied by $s$ , yielding $s \cdot b_n$ .
2. Similarity Transforms $\text{Sim}(n)$ : The translation vector $u \in \mathbb{R}^n$ is output directly. The scaled orthogonal matrix $\alpha R$ is parameterized using the continuous representation without prior normalization, where the positive scale factor is recovered as $\alpha = \|a_1\|$ , and the matrix reconstructed by $f_{GS}$ is multiplied by $\alpha$ .
Knowl 8 — Discontinuity of Quaternion Parameterizations of SO(3)
theoretical result
Standard mappings from $SO(3)$ to unit quaternions in Euclidean space $\mathbb{R}^4$ possess unavoidable jump discontinuities at $180^\circ$ rotations (matrices $M \in SO(3)$ with trace $\text{Tr}(M) = -1$ ).

For the unit quaternion mapping $g_u: SO(3) \to \mathbb{R}^4$ defined by: $g_u(M) = \begin{bmatrix} \text{copysign}\left(\frac{1}{2}\sqrt{1 + M_{11} - M_{22} - M_{33}}, M_{32} - M_{23}\right) \\ \text{copysign}\left(\frac{1}{2}\sqrt{1 - M_{11} + M_{22} - M_{33}}, M_{13} - M_{31}\right) \\ \text{copysign}\left(\frac{1}{2}\sqrt{1 - M_{11} - M_{22} + M_{33}}, M_{21} - M_{12}\right) \\ \frac{1}{2}\sqrt{1 + M_{11} + M_{22} + M_{33}} \end{bmatrix}$ where $\text{copysign}(a, b) = \text{sgn}(b)|a|$ , consider the planar rotation matrix parameterized by angle $\theta$ : $B(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$

As $\theta \to \pi^-$ , the third component of $g_u(B(\theta))$ approaches $+\sqrt{(1 - \cos\theta)/2} = +1$ . As $\theta \to \pi^+$ , the third component approaches $-\sqrt{(1 - \cos\theta)/2} = -1$ . Because the two directional limits differ, $g_u$ is discontinuous at $\theta = \pi$ .

Knowl 9 — Autoencoder Sanity Test Benchmark Across Rotation Representations

data/table

In an autoencoder sanity test, a 4-layer MLP encoder (128 hidden units per layer, Leaky ReLU activations) maps $SO(3)$ rotation matrices to candidate representations, and a fixed representation-specific decoder recovers the $SO(3)$ matrix. The network is trained with Adam using an $L_2$ matrix loss and evaluated on $10^5$ rotation matrices using the geodesic angular error $\mathcal{L}_{\text{angle}} = \arccos((\text{Tr}(M M'^{-1}) - 1) / 2)$ .

The table reports the mean, maximum, and standard deviation of angular errors (in degrees) after 500k iterations:

Representation	Mean ( $^\circ$ )	Max ( $^\circ$ )	Std ( $^\circ$ )
6D (Gram-Schmidt)	0.49	1.98	0.27
5D (Stereographic)	0.49	1.99	0.27
Quaternion	3.32	179.93	5.97
Axis-Angle	3.69	179.22	5.99
Euler Angles	6.98	179.95	17.31

The continuous 6D and 5D representations produce maximum errors below $2^\circ$ , whereas all discontinuous representations produce severe maximum errors near $180^\circ$ due to discontinuities at boundary rotations.

Knowl 10 — 3D Point Cloud Rotation Estimation Benchmark

data/table

In a 3D point cloud pose estimation experiment on ShapeNet airplane models (2,290 training and 400 test point clouds), a Siamese PointNet architecture estimates the 3D rotation between reference and target point clouds. Models are trained for $2.6 \times 10^6$ iterations minimizing an $L_2$ rotation matrix loss.

The table reports geodesic rotation errors (in degrees) on held-out test models augmented with 100 random rotations:

Representation	Mean ( $^\circ$ )	Max ( $^\circ$ )	Std ( $^\circ$ )
6D (Gram-Schmidt)	2.85	179.83	9.16
5D (Stereographic)	4.78	179.87	12.25
$3 \times 3$ Matrix (post-hoc GS)	4.21	180.00	9.44
Quaternion	9.03	179.66	16.33
Axis-Angle	11.93	179.70	21.35
Euler Angles	14.13	179.67	23.80

The 6D continuous representation attains the lowest mean error ( $2.85^\circ$ ) and smallest standard deviation, with ~95% of test samples having errors under $5^\circ$ . Directly regressing full $3 \times 3$ matrices with post-hoc Gram-Schmidt orthogonalization performs worse ( $4.21^\circ$ ) than integrating Gram-Schmidt within the network.

Knowl 11 — Human Pose Inverse Kinematics Benchmark

data/table

In a 3D human pose inverse kinematics (IK) task using 865 motion clips from the CMU Motion Capture Database (1.14M training frames, 107k test frames across 37 categories), a 4-layer MLP with 1,024 hidden neurons predicts joint rotations from 3D joint positions. A differentiable forward kinematics function $\Pi(T, R)$ reconstructs 3D joint coordinates, penalized by an $L_2$ position loss weighted 10x higher on hip-adjacent joints.

The table reports joint position reconstruction errors (in cm) after 1,960k training iterations:

Representation	Mean (cm)	Max (cm)	Std (cm)
6D (Gram-Schmidt)	1.9	28.7	1.2
5D (Stereographic)	2.0	33.3	1.4
Euler Angles	2.7	48.7	2.1
Axis-Angle	3.0	120.0	2.3
Quaternion	3.3	87.1	3.1
$3 \times 3$ Matrix (unconstrained)	22.9	53.6	4.0

The 6D and 5D continuous representations achieve the lowest mean errors ( $1.9\text{ cm}$ and $2.0\text{ cm}$ ) and substantially lower maximum errors than quaternions ( $87.1\text{ cm}$ ) and axis-angle ( $120.0\text{ cm}$ ). Regressing unconstrained $3 \times 3$ matrices causes bone scaling during forward kinematics, leading to large positional error ( $22.9\text{ cm}$ ).

Coverage note — Visual PCA curves of rotation representations and qualitative human skeleton renderings were omitted as they illustrate the discontinuity theorems and quantitative benchmark tables.

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Citation

MLA

Zhou, Y., et al. “On the Continuity of Rotation Representations in Neural Networks”. arXiv, 2018, http://arxiv.org/abs/1812.07035v4.

APA

Zhou, Y., Barnes, C., Lu, J., Yang, J., & Li, H. (2018). On the Continuity of Rotation Representations in Neural Networks. arXiv. http://arxiv.org/abs/1812.07035v4

Chicago

Zhou, Y., C. Barnes, J. Lu, J. Yang, and H. Li. 2018. “On the Continuity of Rotation Representations in Neural Networks”. arXiv. http://arxiv.org/abs/1812.07035v4.

Harvard

Zhou, Y. et al. (2018) “On the Continuity of Rotation Representations in Neural Networks”, arXiv [Preprint]. Available at: http://arxiv.org/abs/1812.07035v4.

Vancouver

1. Zhou Y, Barnes C, Lu J, Yang J, Li H (2018) On the Continuity of Rotation Representations in Neural Networks. arXiv

BibTeX

@article{zhou2018the,
  title = {On the Continuity of Rotation Representations in Neural Networks},
  author = {Zhou, Yi and Barnes, Connelly and Lu, Jingwan and Yang, Jimei and Li, Hao},
  year = {2018},
  journal = {arXiv},
  url = {http://arxiv.org/abs/1812.07035v4},
  eprint = {1812.07035}
}

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On the Continuity of Rotation Representations in Neural Networks

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Knowls

Knowl 1 — Continuous Representation for Neural Networks

Knowl 2 — Non-Existence of Continuous 3D Rotation Representations in Four or Fewer Dimensions

Knowl 3 — Continuous 6D Representation of 3D Rotations via Gram-Schmidt Orthogonalization

Knowl 4 — Continuous 5D Representation of 3D Rotations via Stereographic Projection

Knowl 5 — Continuous (n^2 - n)-Dimensional Representation for SO(n)

Knowl 6 — Continuous (n^2 - 2n + 2)-Dimensional Representation for SO(n) via Multiple Projections

Knowl 7 — Continuous Representations for Orthogonal and Similarity Transformation Groups

Knowl 8 — Discontinuity of Quaternion Parameterizations of SO(3)

Knowl 9 — Autoencoder Sanity Test Benchmark Across Rotation Representations

Knowl 10 — 3D Point Cloud Rotation Estimation Benchmark

Knowl 11 — Human Pose Inverse Kinematics Benchmark

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