How to check a point is inside an ellipsoid with orientation?

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07-10-2024

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Determining if a Point Lies Within an Ellipsoid with Orientation

The Problem: Imagine you have an ellipsoid – a 3D shape resembling a stretched sphere – that's not perfectly aligned with the coordinate axes. You want to know if a given point lies inside this ellipsoid. This problem arises in various fields like computer graphics, physics simulations, and even in GPS navigation.

The Solution: While it's straightforward to check if a point is inside a standard, axis-aligned ellipsoid, things get more complex when the ellipsoid is rotated. This article will explore how to handle this challenge.

The Scenario: Let's say we have an ellipsoid centered at the origin, with semi-axes lengths a, b, and c along the x, y, and z axes respectively. This ellipsoid is rotated by an angle θ around the z-axis. We want to determine whether a point P(x, y, z) lies inside the ellipsoid.

The Original Code: A straightforward approach is to transform the point P into the ellipsoid's coordinate system, then check if it satisfies the standard ellipsoid equation. This code snippet demonstrates this concept using Python:

import numpy as np

def is_point_inside_ellipsoid(point, center, radii, rotation_angle):
  """
  Checks if a point is inside an ellipsoid.

  Args:
    point: The point to check (x, y, z).
    center: The ellipsoid's center (x, y, z).
    radii: The semi-axes lengths of the ellipsoid (a, b, c).
    rotation_angle: The angle of rotation around the z-axis (in radians).

  Returns:
    True if the point is inside the ellipsoid, False otherwise.
  """

  # Translate the point to the ellipsoid's coordinate system
  point_trans = point - center

  # Create a rotation matrix
  rotation_matrix = np.array([
      [np.cos(rotation_angle), -np.sin(rotation_angle), 0],
      [np.sin(rotation_angle), np.cos(rotation_angle), 0],
      [0, 0, 1]
  ])

  # Rotate the point
  point_rotated = np.dot(rotation_matrix, point_trans)

  # Check if the rotated point satisfies the ellipsoid equation
  x_prime, y_prime, z_prime = point_rotated
  return (x_prime**2 / radii[0]**2 + y_prime**2 / radii[1]**2 + z_prime**2 / radii[2]**2) <= 1

Analysis and Clarification:

1. Transformation: We first translate the point to the ellipsoid's coordinate system by subtracting the center's coordinates.

2. Rotation: We then rotate the point using a rotation matrix around the z-axis. This matrix rotates the point by the specified angle.

3. Ellipsoid Equation: Finally, we check if the rotated point satisfies the standard ellipsoid equation, which is:

   (x²/a² + y²/b² + z²/c²) <= 1

Examples:

• If point = (1, 1, 1), center = (0, 0, 0), radii = (2, 3, 4), and rotation_angle = np.pi/4, the function will return True indicating the point is inside the ellipsoid.

• If point = (5, 5, 5), with the same other parameters, the function will return False because the point is outside the ellipsoid.

Additional Value:

The code snippet provided is a basic implementation and can be further enhanced by:

1. Handling General Rotations: For rotations around arbitrary axes, you need to construct a more general rotation matrix using Euler angles or quaternions.

2. Performance Optimization: For large datasets of points, you can optimize the code by vectorizing the operations.

References:

• Ellipsoid
• Rotation Matrix

Conclusion: By understanding the basic concepts of transformation, rotation, and the ellipsoid equation, you can effectively determine whether a point lies inside an ellipsoid with orientation. The code snippet and analysis provided offer a practical starting point for tackling this problem in your own applications.