Geometry Utilities

The geometry module provides essential utilities for handling rotations, coordinate transformations, and angular operations in 2D and 3D space. These functions are fundamental for processing vehicle poses, sensor data, and trajectory representations.

Overview

Autonomous driving systems must handle:

3D rotations: Vehicle orientation in 3D space (SO(3))
2D rotations: Planar motion on roads
Yaw extraction: Converting 3D orientation to heading angle
Coordinate transforms: Moving between reference frames
Angle wrapping: Handling angular discontinuities at ±π

The geometry utilities support both NumPy and PyTorch, enabling seamless integration across data processing and model inference.

Rotation Representations

SO(3) to Yaw Conversion

Extract the yaw (heading) angle from a 3D rotation matrix:

import torch
from alpamayo_r1.geometry.rotation import so3_to_yaw_torch

# Vehicle rotation matrices: (batch, 3, 3)
rot_matrices = torch.tensor([...])  # SO(3) rotation matrices

# Extract yaw angles
yaw = so3_to_yaw_torch(rot_matrices)  # (batch,)

This assumes rotations are described in XYZ Euler order. The implementation from rotation.py:25-38:

def so3_to_yaw_torch(rot_mat: torch.Tensor) -> torch.Tensor:
    """Computes the yaw angle given an SO(3) rotation matrix.
    
    Args:
        rot_mat: [..., 3, 3] rotation matrices
        
    Returns:
        [...] yaw angles in radians
    """
    cos_th_cos_phi = rot_mat[..., 0, 0]
    cos_th_sin_phi = rot_mat[..., 1, 0]
    return torch.atan2(cos_th_sin_phi, cos_th_cos_phi)

A NumPy version is also available as so3_to_yaw_np() at rotation.py:41-53.

2D Rotation Matrices

Create 2D rotation matrices from angles:

from alpamayo_r1.geometry.rotation import rotation_matrix_torch

# Angles in radians
angles = torch.tensor([0.0, np.pi/4, np.pi/2])  # (3,)

# Create rotation matrices
rot_mats = rotation_matrix_torch(angles)  # (3, 2, 2)

# rot_mats[1] ≈ [[0.707, -0.707],
#                 [0.707,  0.707]]  (45° rotation)

Implementation from rotation.py:109-125:

def rotation_matrix_torch(angle: torch.Tensor) -> torch.Tensor:
    """Creates 2D rotation matrices.
    
    Args:
        angle: [...] angles in radians
        
    Returns:
        [..., 2, 2] rotation matrices
    """
    rotmat = torch.stack(
        [
            torch.stack([torch.cos(angle), -torch.sin(angle)], dim=-1),
            torch.stack([torch.sin(angle), torch.cos(angle)], dim=-1),
        ],
        dim=-2,
    )
    return rotmat

NumPy version: rotation_matrix() at rotation.py:85-106.

Converting Between 2D and 3D

3D → 2D Projection

Project a 3D rotation to the XY plane:

from alpamayo_r1.geometry.rotation import rot_3d_to_2d

# 3D rotation matrices: (..., 3, 3)
rot_3d = torch.tensor([...])  

# Project to 2D (XY plane)
rot_2d = rot_3d_to_2d(rot_3d)  # (..., 2, 2)

This extracts the X and Y axes, projects them to the XY plane, and applies Gram-Schmidt orthogonalization for numerical stability. See rotation.py:177-194.

2D → 3D Embedding

Embed a 2D rotation into 3D space (flat ground plane assumption):

from alpamayo_r1.geometry.rotation import rot_2d_to_3d

# 2D rotation matrices: (..., 2, 2)
rot_2d = rotation_matrix_torch(yaw_angles)

# Embed to 3D (Z-axis unchanged)
rot_3d = rot_2d_to_3d(rot_2d)  # (..., 3, 3)

# rot_3d[..., :2, :2] == rot_2d
# rot_3d[..., 2, :] == [0, 0, 1]  (Z-axis)
# rot_3d[..., :, 2] == [0, 0, 1]  (Z-axis)

From rotation.py:197-213.

Coordinate Transformations

2D Coordinate Transform

Apply rotation and translation to 2D points:

from alpamayo_r1.geometry.rotation import transform_coords_2d_np
import numpy as np

# Points in local frame: (N, 2)
local_coords = np.array([[1.0, 0.0], [0.0, 1.0]])

# Transform to global frame
global_coords = transform_coords_2d_np(
    coords=local_coords,
    offset=np.array([10.0, 5.0]),  # Translation
    angle=np.pi / 4,                # Rotation (45°)
)

# Or provide pre-computed rotation matrix
rot_mat = rotation_matrix(np.pi / 4)
global_coords = transform_coords_2d_np(
    coords=local_coords,
    offset=np.array([10.0, 5.0]),
    rot_mat=rot_mat,  # If provided, angle is ignored
)

From rotation.py:128-153. The transformation applies rotation first, then translation:

def transform_coords_2d_np(
    coords: np.ndarray,              # (..., 2)
    offset: Optional[np.ndarray],    # (..., 2)
    angle: Optional[np.ndarray],     # (...)
    rot_mat: Optional[np.ndarray],   # (..., 2, 2)
) -> np.ndarray:
    if rot_mat is None and angle is not None:
        rot_mat = rotation_matrix(angle)
    
    if rot_mat is not None:
        coords = np.einsum("...ij,...j->...i", rot_mat, coords)
    
    if offset is not None:
        coords += offset
    
    return coords

Angular Operations

Angle Wrapping

Normalize angles to the range [-π, π):

from alpamayo_r1.geometry.rotation import angle_wrap, round_2pi_torch
import numpy as np
import torch

# NumPy version
angles_np = np.array([0.0, 3.5, -4.0, 2*np.pi])
wrapped_np = angle_wrap(angles_np)
# Output: [0.0, -2.78..., 2.28..., 0.0]

# PyTorch version (numerically robust)
angles_torch = torch.tensor([0.0, 3.5, -4.0, 2*np.pi])
wrapped_torch = round_2pi_torch(angles_torch)

Two implementations are provided:

Basic wrapping (angle_wrap at rotation.py:71-82):

def angle_wrap(radians: TensorOrNDArray) -> TensorOrNDArray:
    return (radians + np.pi) % (2 * np.pi) - np.pi

Robust wrapping (round_2pi_torch at rotation.py:237-246):

def round_2pi_torch(x: torch.Tensor) -> torch.Tensor:
    return torch.atan2(torch.sin(x), torch.cos(x))

The robust version using atan2(sin, cos) handles edge cases better but is slightly slower.

Robust Arctan2

Stable arctan2 that avoids NaN when both inputs are zero:

from alpamayo_r1.geometry.rotation import ratan2

# Standard torch.arctan2(0, 0) = NaN
# ratan2(0, 0) = 0

angle = ratan2(
    s=torch.tensor([0.0, 1.0]),  # sin values
    c=torch.tensor([0.0, 1.0]),  # cos values
)  # Returns: [0.0, 0.785...]

From rotation.py:216-222, this adds a small epsilon when c is near zero to prevent division issues.

Euler Angles

Convert Euler angles to SO(3) rotation matrices (NumPy only):

from alpamayo_r1.geometry.rotation import euler_2_so3
import numpy as np

# Euler angles: (N, 3) in [roll, pitch, yaw] order
euler_angles = np.array([
    [0.0, 0.0, 0.0],        # No rotation
    [0.0, 0.0, np.pi/2],    # 90° yaw
])

# Convert to rotation matrices
rot_matrices = euler_2_so3(
    euler_angles,
    degrees=False,  # Input in radians
    seq="xyz",      # Rotation order
)  # (2, 3, 3)

This wraps scipy.spatial.transform.Rotation. See rotation.py:56-68.

Advanced: Gram-Schmidt Orthogonalization

Stably orthonormalize two 3D vectors:

from alpamayo_r1.geometry.rotation import stable_gramschmidt

# Two non-orthogonal 3D vectors
vectors = torch.tensor([[
    [1.0, 0.1, 0.0],  # x vector (slightly off)
    [0.1, 1.0, 0.0],  # y vector (slightly off)
]])  # (1, 3, 2)

# Orthonormalize to create full rotation matrix
rot_matrix = stable_gramschmidt(vectors)  # (1, 3, 3)

# rot_matrix contains orthonormal (x, y, z) where z = x × y

From rotation.py:156-174, this:

Normalizes the first vector
Projects the second vector onto the orthogonal complement
Normalizes the second vector
Computes the third vector as the cross product

Useful for reconstructing rotation matrices from noisy or incomplete data.

Usage Examples

Example 1: Transform Trajectory to Vehicle Frame

import torch
from alpamayo_r1.geometry.rotation import (
    so3_to_yaw_torch,
    rotation_matrix_torch,
    rot_2d_to_3d,
)

# Global trajectory: (batch, time, 3)
global_xyz = torch.tensor([...])  # (B, T, 3)
global_rot = torch.tensor([...])  # (B, T, 3, 3)

# Current vehicle pose (last timestep)
vehicle_xyz = global_xyz[:, -1]  # (B, 3)
vehicle_rot = global_rot[:, -1]  # (B, 3, 3)
vehicle_yaw = so3_to_yaw_torch(vehicle_rot)  # (B,)

# Transform trajectory to vehicle frame
relative_xyz = global_xyz - vehicle_xyz.unsqueeze(1)

# Rotate to vehicle frame (inverse rotation)
inv_yaw = -vehicle_yaw
rot_2d = rotation_matrix_torch(inv_yaw)  # (B, 2, 2)
relative_xy = torch.einsum(
    "bij,btj->bti",
    rot_2d,
    relative_xyz[..., :2],
)  # (B, T, 2)

Example 2: Heading Smoothing

import torch
from alpamayo_r1.geometry.rotation import round_2pi_torch

# Raw heading angles with discontinuities
raw_headings = torch.tensor([3.1, 3.14, -3.1, -3.0, -2.9])

# Unwrap by tracking jumps
unwrapped = torch.zeros_like(raw_headings)
unwrapped[0] = raw_headings[0]

for i in range(1, len(raw_headings)):
    diff = raw_headings[i] - raw_headings[i-1]
    diff = round_2pi_torch(diff)  # Wrap difference
    unwrapped[i] = unwrapped[i-1] + diff

# Now unwrapped is continuous: [3.1, 3.14, 3.18, 3.28, 3.38]

Example 3: Batch Coordinate Transforms

import numpy as np
from alpamayo_r1.geometry.rotation import transform_coords_2d_np

# Waypoints in vehicle frame: (batch, waypoints, 2)
local_waypoints = np.random.randn(32, 64, 2)

# Vehicle poses: (batch, 3) - [x, y, yaw]
vehicle_poses = np.random.randn(32, 3)

# Transform all waypoints to global frame
global_waypoints = transform_coords_2d_np(
    coords=local_waypoints,
    offset=vehicle_poses[:, None, :2],  # (32, 1, 2)
    angle=vehicle_poses[:, 2],           # (32,)
)  # (32, 64, 2)

Best Practices

Use robust angle operations: Prefer round_2pi_torch() over manual modulo for numerical stability
Avoid gimbal lock: When possible, work with rotation matrices rather than Euler angles
Unwrap angles: For trajectory smoothing, unwrap angles to avoid ±π discontinuities
Batch operations: Vectorize coordinate transforms to leverage GPU parallelism
Numerical stability: Use stable_gramschmidt() when reconstructing rotation matrices
Type consistency: Match NumPy/PyTorch usage with your data pipeline

Common Pitfalls

Forgetting to wrap angles: Always wrap angles after arithmetic operations

# Wrong
new_angle = old_angle + delta  # Can go outside [-π, π]

# Correct
new_angle = round_2pi_torch(old_angle + delta)

Incorrect transform order: Rotation must happen before translation

# Wrong
coords = coords + offset  # Translate first
coords = rot @ coords     # Then rotate

# Correct
coords = rot @ coords     # Rotate first
coords = coords + offset  # Then translate

Mixing 2D and 3D: Ensure dimensional consistency

# Wrong
rot_2d = rotation_matrix_torch(yaw)  # (2, 2)
xyz = rot_2d @ xyz  # xyz is (3,) - dimension mismatch!

# Correct
rot_2d = rotation_matrix_torch(yaw)
xy = rot_2d @ xyz[:2]  # Only transform x, y
xyz = torch.cat([xy, xyz[2:3]])  # Preserve z

Performance Notes

NumPy functions: Optimized for CPU, good for preprocessing
PyTorch functions: GPU-accelerated, use for training/inference
Batched operations: Always prefer batched operations over loops
- rotation_matrix_torch(angles) where angles.shape = (B,) → (B, 2, 2)
- Much faster than looping over batch dimension

API Reference

Rotation Conversions

Function	Input	Output	Description
`so3_to_yaw_torch`	(…, 3, 3)	(…,)	Extract yaw from 3D rotation
`so3_to_yaw_np`	(…, 3, 3)	(…,)	NumPy version
`euler_2_so3`	(N, 3)	(N, 3, 3)	Euler angles → rotation matrices
`rot_2d_to_3d`	(…, 2, 2)	(…, 3, 3)	Embed 2D rotation in 3D
`rot_3d_to_2d`	(…, 3, 3)	(…, 2, 2)	Project 3D rotation to 2D

Rotation Matrices

Function	Input	Output	Description
`rotation_matrix_torch`	(…,)	(…, 2, 2)	Create 2D rotation matrices
`rotation_matrix`	scalar or (…,)	(2, 2) or (…, 2, 2)	NumPy version
`stable_gramschmidt`	(…, 3, 2)	(…, 3, 3)	Orthonormalize vectors

Coordinate Transforms

Function	Input	Output	Description
`transform_coords_2d_np`	(…, 2) + offset/angle	(…, 2)	Rotate and translate points

Angular Operations

Function	Input	Output	Description
`angle_wrap`	(…,)	(…,)	Wrap to [-π, π)
`round_2pi_torch`	(…,)	(…,)	Robust angle wrapping
`round_2pi`	(…,)	(…,)	NumPy version
`ratan2`	sin, cos	angle	Robust arctan2

For complete implementation details, see geometry/rotation.py.

Get Started

Core Concepts

Guides

Model Components

Overview

Rotation Representations

SO(3) to Yaw Conversion

2D Rotation Matrices

Converting Between 2D and 3D

3D → 2D Projection

2D → 3D Embedding

Coordinate Transformations

2D Coordinate Transform

Angular Operations

Angle Wrapping

Robust Arctan2

Euler Angles

Advanced: Gram-Schmidt Orthogonalization

Usage Examples

Example 1: Transform Trajectory to Vehicle Frame

Example 2: Heading Smoothing

Example 3: Batch Coordinate Transforms

Best Practices

Common Pitfalls

Performance Notes

API Reference

Rotation Conversions

Rotation Matrices

Coordinate Transforms

Angular Operations

Build docs developers (and LLMs) love

Get Started

Core Concepts

Guides

Model Components

Documentation Index

​Overview

​Rotation Representations

​SO(3) to Yaw Conversion

​2D Rotation Matrices

​Converting Between 2D and 3D

​3D → 2D Projection

​2D → 3D Embedding

​Coordinate Transformations

​2D Coordinate Transform

​Angular Operations

​Angle Wrapping

​Robust Arctan2

​Euler Angles

​Advanced: Gram-Schmidt Orthogonalization

​Usage Examples

​Example 1: Transform Trajectory to Vehicle Frame

​Example 2: Heading Smoothing

​Example 3: Batch Coordinate Transforms

​Best Practices

​Common Pitfalls

​Performance Notes

​API Reference

​Rotation Conversions

​Rotation Matrices

​Coordinate Transforms

​Angular Operations

Build docs developers (and LLMs) love

Overview

Rotation Representations

SO(3) to Yaw Conversion

2D Rotation Matrices

Converting Between 2D and 3D

3D → 2D Projection

2D → 3D Embedding

Coordinate Transformations

2D Coordinate Transform

Angular Operations

Angle Wrapping

Robust Arctan2

Euler Angles

Advanced: Gram-Schmidt Orthogonalization

Usage Examples

Example 1: Transform Trajectory to Vehicle Frame

Example 2: Heading Smoothing

Example 3: Batch Coordinate Transforms

Best Practices

Common Pitfalls

Performance Notes

API Reference

Rotation Conversions

Rotation Matrices

Coordinate Transforms

Angular Operations