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 transforms
T_combined = SE3.multiply(T1, T2)
T_inverse = SE3.inverse(T_combined)

# 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)
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)

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)
log(se3)	(...,7) -> (...,6)
exp(tangent)	(...,6) -> (...,7)