SPECIAL_ORTHO_GROUP

Overview

The SPECIAL_ORTHO_GROUP function generates random rotation matrices from the special orthogonal group SO(N), which is the group of all orthogonal N \times N matrices with determinant +1. These matrices represent proper rotations in N-dimensional Euclidean space, preserving both distances and orientation.

This implementation uses the scipy.stats.special_ortho_group distribution from SciPy. The function draws random samples from the Haar distribution, which is the unique uniform distribution on SO(N). The Haar measure, named after mathematician Alfréd Haar, provides a mathematically rigorous way to sample “uniformly at random” from continuous groups like SO(N). For more mathematical background, see the Wikipedia article on orthogonal groups.

An orthogonal matrix Q satisfies the property:

Q Q^T = Q^T Q = I

where I is the identity matrix and Q^T is the transpose of Q. For SO(N) specifically, the additional constraint \det(Q) = +1 ensures the transformation is a pure rotation without reflection.

The algorithm generates a Haar-distributed orthogonal matrix using scipy.stats.ortho_group, then adjusts the result to guarantee the determinant is +1. For 3D rotations specifically, SciPy also provides scipy.spatial.transform.Rotation.random as an alternative.

Random rotation matrices from SO(N) have applications in:

  • 3D graphics and robotics: Generating random orientations for objects or sensors
  • Monte Carlo simulations: Sampling over rotation spaces
  • Statistical physics: Modeling spin systems and lattice gauge theories
  • Machine learning: Random projections and data augmentation with rotational transformations

This example function is provided as-is without any representation of accuracy.

Excel Usage

=SPECIAL_ORTHO_GROUP(dim, size, seed)
  • dim (int, required): Dimension of the square matrix (must be >= 2).
  • size (int, optional, default: null): Number of matrices to generate (must be >= 1). If omitted, returns a single matrix.
  • seed (int, optional, default: null): Random seed for reproducibility.

Returns (list[list]): 2D orthogonal matrix, or error message string.

Example 1: Single 3x3 orthogonal matrix

Inputs:

dim seed
3 42

Excel formula:

=SPECIAL_ORTHO_GROUP(3, 42)

Expected output:

Result
-0.600002 -0.283962 -0.747906
0.167015 -0.958746 0.230026
-0.78237 0.0131045 0.622676
Example 2: Two 3x3 orthogonal matrices stacked

Inputs:

dim size seed
3 2 42

Excel formula:

=SPECIAL_ORTHO_GROUP(3, 2, 42)

Expected output:

Result
-0.600002 0.191744 -0.77668
0.167015 -0.919436 -0.35601
-0.78237 -0.343324 0.519639
-0.977171 0.0152953 0.211902
-0.150232 0.6555 -0.740101
-0.150222 -0.75504 -0.638238
Example 3: Single 4x4 orthogonal matrix

Inputs:

dim seed
4 42

Excel formula:

=SPECIAL_ORTHO_GROUP(4, 42)

Expected output:

Result
-0.286541 -0.817253 0.357279 -0.349775
-0.079761 0.0944496 -0.612368 -0.780847
-0.373634 -0.382451 -0.668876 0.516462
-0.878595 0.420603 0.223519 -0.034671
Example 4: Three 4x4 orthogonal matrices stacked

Inputs:

dim size seed
4 3 42

Excel formula:

=SPECIAL_ORTHO_GROUP(4, 3, 42)

Expected output:

Result
-0.286541 -0.869479 -0.342952 -0.21044
0.079761 -0.0396118 -0.496162 0.863651
0.373634 -0.486955 0.707768 0.349767
-0.878595 0.072888 0.367794 0.295779
0.131049 0.571957 0.770312 -0.24962
0.13104 0.534717 -0.605539 -0.574656
-0.883841 0.39686 -0.0670547 0.238393
0.429512 0.479004 -0.188271 0.742044
0.482576 0.240182 0.0252718 0.841899
-0.557701 0.317068 -0.724877 0.250978
-0.47635 -0.710851 0.218045 0.469294
0.478727 -0.580061 -0.652971 -0.0893223

Python Code

Show Code 
import numpy as np
from scipy.stats import special_ortho_group as scipy_special_ortho_group

def special_ortho_group(dim, size=None, seed=None):
    """
    Draws random samples from the special orthogonal group SO(N), returning orthogonal matrices with determinant +1.

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.special_ortho_group.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        dim (int): Dimension of the square matrix (must be >= 2).
        size (int, optional): Number of matrices to generate (must be >= 1). If omitted, returns a single matrix. Default is None.
        seed (int, optional): Random seed for reproducibility. Default is None.

    Returns:
        list[list]: 2D orthogonal matrix, or error message string.
    """
    def to_2d_list(arr):
        arr = np.asarray(arr)
        if arr.ndim == 2:
            return arr.tolist()
        elif arr.ndim == 3:
            # Multiple matrices: flatten into a single 2D list
            return [row for mat in arr for row in mat.tolist()]
        else:
          return "Error: Invalid output shape from scipy.stats.special_ortho_group."

    def valid_matrix(matrix):
        for row in matrix:
            for val in row:
                if not isinstance(val, (int, float)):
                    return False
                if val != val or val in (float('inf'), float('-inf')):
                    return False
        return True
    try:
        # Validate dim - accept float if it represents an integer
        if isinstance(dim, float) and dim.is_integer():
            dim = int(dim)
        if not isinstance(dim, int) or dim < 2:
            return "Error: Invalid input: dim must be an integer >= 2."

        # Validate size - accept float if it represents an integer
        if size is not None:
            if isinstance(size, float) and size.is_integer():
                size = int(size)
            if not isinstance(size, int) or size < 1:
                return "Error: Invalid input: size must be an integer >= 1 or None."

        try:
            # Draw samples, only pass size if not None
            rng_seed = int(seed) if seed is not None else None
            if size is None:
                result = scipy_special_ortho_group.rvs(dim, random_state=rng_seed)
            else:
                result = scipy_special_ortho_group.rvs(dim, size=size, random_state=rng_seed)
        except Exception as e:
            return f"Error: scipy.stats.special_ortho_group error: {e}"

        out = to_2d_list(result)

        if isinstance(out, list) and all(isinstance(row, list) for row in out):
            if not valid_matrix(out):
                return "Error: Invalid output: matrix contains non-numeric or special values."
            return out

        return out
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Dimension of the square matrix (must be >= 2).
Number of matrices to generate (must be >= 1). If omitted, returns a single matrix.
Random seed for reproducibility.