Choosing a Rotation Representation¶

pybvh supports five rotation representations. Each has trade-offs that make it better suited for different tasks. This page is the decision guide — the conversion API, shapes, and SE(3) layer live in Rotation Representations & SE(3).

Decision table¶

Representation	Shape	Continuous?	Best for
Euler angles	`(*, 3)`	No (gimbal lock, discontinuities)	BVH native format, human-readable inspection, storage
Rotation matrices	`(*, 3, 3)`	Yes	Forward kinematics, composing rotations, intermediate conversions
6D (Zhou et al.)	`(*, 6)`	Yes	Neural network output layers (no discontinuity penalty during training)
Quaternions	`(*, 4)`	No (antipodal equivalence)	Smooth interpolation (SLERP), compact storage, physics engines
Axis-angle	`(*, 3)`	No (wraps at 2pi)	Small rotations, angular velocity, rotation visualization

When to use each¶

Euler angles¶

The native BVH format. Use when reading/writing files or when you need human-readable rotation values. Avoid for ML training (discontinuities at gimbal lock cause gradient problems) and for interpolation (linear interpolation in Euler space is not physically meaningful).

6D rotation (Zhou et al.)¶

The recommended representation for neural networks. It is continuous — nearby rotations map to nearby 6D vectors — so gradient-based optimization works smoothly. Use bvh.to_6d() for feature extraction and bvh.from_6d() to convert predictions back.

The continuity difference in one picture — a smooth rotation crossing the ±180° boundary, read out in three representations:

A smooth rotation crossing 180 degrees: the Euler readout jumps, the canonical quaternion flips the sign of z, 6D stays perfectly smooth

Euler jumps, the canonical quaternion flips sign where w crosses zero, 6D stays smooth. (Gallery, section 2.)

Quaternions¶

Best for interpolation via rotations.quat_slerp(). Compact (4 values vs. 9 for matrices). However, quaternions have double cover (q and -q represent the same rotation), which can cause discontinuities during training. pybvh canonicalizes to w >= 0.

Rotation matrices¶

Useful as an intermediate representation for composing rotations or for forward kinematics. Rarely used directly as ML features due to redundancy (9 values with 6 constraints).

Axis-angle¶

Compact (3 values) and intuitive (direction = axis, magnitude = angle). Good for representing small perturbations or angular velocities. Discontinuous at 2*pi wrapping.

Converting between representations¶

Any pair converts directly — see Rotation Representations & SE(3) for the conversion API and shape conventions, and the Gallery for the continuity figure that shows why 6D wins for training.

Typical ML pipeline¶

# Extract 6D features for training
root_pos, rot6d = bvh.to_6d()  # (F, J, 6)

# After model prediction, convert back
bvh_pred = bvh.from_6d(pred_root_pos, pred_rot6d)

# Evaluate with quaternion distance or rotation matrix comparison
_, q_gt = bvh.to_quat()
_, q_pred = bvh_pred.to_quat()