nanomanifold 0.4.1

SO3/SE3 operations on any backend

nanomanifold

Fast, batched and differentiable SO(3)/SE(3) transforms for any backend (NumPy, PyTorch, JAX, ...)

Works directly on arrays, defined as:

• SO(3): unit quaternions [w, x, y, z] for 3D rotations, shape (..., 4)
• SE(3): concatenated [quat, translation], shape (..., 7)

import numpy as np
from nanomanifold import SO3, SE3

# Rotations stored as quaternion arrays [w,x,y,z]
q = SO3.from_axis_angle(np.array([0, 0, 1]), np.pi/4)  # 45° around Z
points = np.array([[1, 0, 0], [0, 1, 0]])
rotated = SO3.rotate_points(q, points)

# Rigid transforms stored as 7D arrays [quat, translation]
T = SE3.from_rt(q, np.array([1, 0, 0]))  # rotation + translation
transformed = SE3.transform_points(T, points)

Installation

pip install nanomanifold

Quick Start

Rotations (SO3)

from nanomanifold import SO3

# Create rotations
q1 = SO3.from_axis_angle([1, 0, 0], np.pi/2)    # 90° around X
q2 = SO3.from_euler([0, 0, np.pi/4])            # 45° around Z
q3 = SO3.from_matrix(rotation_matrix)

# Compose and interpolate
q_combined = SO3.multiply(q1, q2)
q_halfway = SO3.slerp(q1, q2, t=0.5)

# Apply to points
points = np.array([[1, 0, 0], [0, 1, 0]])
rotated = SO3.rotate_points(q_combined, points)

Rigid Transforms (SE3)

from nanomanifold import SE3

# Create transforms
T1 = SE3.from_rt(q1, [1, 2, 3])               # rotation + translation
T2 = SE3.from_matrix(transformation_matrix)

# Compose and interpolate
T_combined = SE3.multiply(T1, T2)
T_inverse = SE3.inverse(T_combined)
T_halfway = SE3.slerp(T1, T2, t=0.5)

# Apply to points
transformed = SE3.transform_points(T_combined, points)

API Reference

All functions are available via nanomanifold.SO3 and nanomanifold.SE3. Shapes follow the Array API convention and accept arbitrarily batched inputs.

SO3 (3D Rotations)

Function	Signature
canonicalize(q)	(...,4) -> (...,4)
to_axis_angle(q)	(...,4) -> (...,3)
from_axis_angle(axis_angle)	(...,3) -> (...,4)
to_euler(q, convention="ZYX")	(...,4) -> (...,3)
from_euler(euler, convention="ZYX")	(...,3) -> (...,4)
convert(x, src=..., dst=...)	dynamic
identity_as(ref, batch_dims=..., rotation_type=...)	dynamic
to_matrix(q)	(...,4) -> (...,3,3)
from_matrix(R)	(...,3,3) -> (...,4)
from_quat_xyzw(quat)	(...,4) -> (...,4)
to_quat_xyzw(quat)	(...,4) -> (...,4)
to_6d(q)	(...,4) -> (...,6)
from_6d(d6)	(...,6) -> (...,4)
multiply(q1, q2)	(...,4), (...,4) -> (...,4)
inverse(q)	(...,4) -> (...,4)
rotate_points(q, points)	(...,4), (...,N,3) -> (...,N,3)
slerp(q1, q2, t)	(...,4), (...,4), (...,N) -> (...,N,4)
distance(q1, q2)	(...,4), (...,4) -> (...)
log(q)	(...,4) -> (...,3)
exp(tangent)	(...,3) -> (...,4)
hat(w)	(...,3) -> (...,3,3)
vee(W)	(...,3,3) -> (...,3)
weighted_mean(quats, weights)	sequence of (...,4), (...,N) -> (...,4)
mean(quats)	sequence of (...,4) -> (...,4)
random(*shape)	(...,4)

SE3 (Rigid Transforms)

Function	Signature
canonicalize(se3)	(...,7) -> (...,7)
from_rt(quat, translation)	(...,4), (...,3) -> (...,7)
to_rt(se3)	(...,7) -> (quat, translation)
from_matrix(T)	(...,4,4) -> (...,7)
to_matrix(se3)	(...,7) -> (...,4,4)
multiply(se3_1, se3_2)	(...,7), (...,7) -> (...,7)
inverse(se3)	(...,7) -> (...,7)
transform_points(se3, points)	(...,7), (...,N,3) -> (...,N,3)
slerp(se3_1, se3_2, t)	(...,7), (...,7), (...,N) -> (...,N,7)
log(se3)	(...,7) -> (...,6)
exp(tangent)	(...,6) -> (...,7)
hat(v)	(...,6) -> (...,4,4)
vee(M)	(...,4,4) -> (...,6)
weighted_mean(transforms, weights)	sequence of (...,7), (...,N) -> (...,7)
mean(transforms)	sequence of (...,7) -> (...,7)
random(*shape)	(...,7)

Pairwise Conversions (SO3.conversions)

Convert directly between any two rotation representations without going through quaternions manually. All 30 pairwise functions follow the naming pattern from_{source}_to_{target}.

Representations: axis_angle, euler, matrix, quat_wxyz, quat_xyzw, sixd.

Function	Signature
SO3.conversions.from_axis_angle_to_matrix(aa)	(...,3) -> (...,3,3)
SO3.conversions.from_axis_angle_to_euler(aa, convention)	(...,3) -> (...,3)
SO3.conversions.from_axis_angle_to_quat_wxyz(aa)	(...,3) -> (...,4)
SO3.conversions.from_axis_angle_to_quat_xyzw(aa)	(...,3) -> (...,4)
SO3.conversions.from_axis_angle_to_sixd(aa)	(...,3) -> (...,6)
SO3.conversions.from_euler_to_axis_angle(e, convention)	(...,3) -> (...,3)
SO3.conversions.from_euler_to_matrix(e, convention)	(...,3) -> (...,3,3)
SO3.conversions.from_euler_to_quat_wxyz(e, convention)	(...,3) -> (...,4)
SO3.conversions.from_euler_to_quat_xyzw(e, convention)	(...,3) -> (...,4)
SO3.conversions.from_euler_to_sixd(e, convention)	(...,3) -> (...,6)
SO3.conversions.from_matrix_to_axis_angle(R)	(...,3,3) -> (...,3)
SO3.conversions.from_matrix_to_euler(R, convention)	(...,3,3) -> (...,3)
SO3.conversions.from_matrix_to_quat_wxyz(R)	(...,3,3) -> (...,4)
SO3.conversions.from_matrix_to_quat_xyzw(R)	(...,3,3) -> (...,4)
SO3.conversions.from_matrix_to_sixd(R)	(...,3,3) -> (...,6)
SO3.conversions.from_quat_wxyz_to_axis_angle(q)	(...,4) -> (...,3)
SO3.conversions.from_quat_wxyz_to_euler(q, convention)	(...,4) -> (...,3)
SO3.conversions.from_quat_wxyz_to_matrix(q)	(...,4) -> (...,3,3)
SO3.conversions.from_quat_wxyz_to_quat_xyzw(q)	(...,4) -> (...,4)
SO3.conversions.from_quat_wxyz_to_sixd(q)	(...,4) -> (...,6)
SO3.conversions.from_quat_xyzw_to_axis_angle(q)	(...,4) -> (...,3)
SO3.conversions.from_quat_xyzw_to_euler(q, convention)	(...,4) -> (...,3)
SO3.conversions.from_quat_xyzw_to_matrix(q)	(...,4) -> (...,3,3)
SO3.conversions.from_quat_xyzw_to_quat_wxyz(q)	(...,4) -> (...,4)
SO3.conversions.from_quat_xyzw_to_sixd(q)	(...,4) -> (...,6)
SO3.conversions.from_sixd_to_axis_angle(d6)	(...,6) -> (...,3)
SO3.conversions.from_sixd_to_euler(d6, convention)	(...,6) -> (...,3)
SO3.conversions.from_sixd_to_matrix(d6)	(...,6) -> (...,3,3)
SO3.conversions.from_sixd_to_quat_wxyz(d6)	(...,6) -> (...,4)
SO3.conversions.from_sixd_to_quat_xyzw(d6)	(...,6) -> (...,4)

For runtime-selected conversions, use SO3.convert. Euler uses the usual axis-order convention strings, while quaternion order is controlled via "wxyz" or "xyzw" and defaults to the repo convention "wxyz":

matrix = SO3.convert(axis_angle, src="axis_angle", dst="matrix")
quat_xyzw = SO3.convert(euler, src="euler", dst="quat", src_convention="XYZ", dst_convention="xyzw")
quat_wxyz = SO3.convert(quat_xyzw, src="quat", dst="quat", src_convention="xyzw")
euler = SO3.convert(matrix, src="matrix", dst="euler", dst_convention="ZYX")

Backend-Explicit Mode

By default, nanomanifold auto-detects the array backend via array_api_compat. Every function also accepts an optional xp keyword argument to specify the backend explicitly. This is required for torch.compile(fullgraph=True), since Dynamo cannot trace the dynamic dispatch:

import torch
from nanomanifold import SO3, SE3

@torch.compile(fullgraph=True)
def forward(q1, q2, T1, T2):
    q_mid = SO3.slerp(q1, q2, torch.tensor([0.5]), xp=torch)
    T_mid = SE3.slerp(T1, T2, torch.tensor([0.5]), xp=torch)
    return q_mid, T_mid