Visualize the Moeller–Trumbore Algorithm

This example demonstrates the Moeller–Trumbore intersection algorithm using pyvista.

For additional details, please reference the following:

First, define the ray triangle intersection method.

import pyvista as pv
import numpy as np


def ray_triangle_intersection(ray_start, ray_vec, triangle):
    """Moeller–Trumbore intersection algorithm.

    Parameters
    ----------
    ray_start : np.ndarray
        Length three numpy array representing start of point.

    ray_vec : np.ndarray
        Direction of the ray.

    triangle : np.ndarray
        ``3 x 3`` numpy array containing the three vertices of a
        triangle.

    Returns
    -------
    bool
        ``True`` when there is an intersection.

    tuple
        Length three tuple containing the distance ``t``, and the
        intersection in unit triangle ``u``, ``v`` coordinates.  When
        there is no intersection, these values will be:
        ``[np.nan, np.nan, np.nan]``

    """
    # define a null intersection
    null_inter = np.array([np.nan, np.nan, np.nan])

    # break down triangle into the individual points
    v1, v2, v3 = triangle
    eps = 0.000001

    # compute edges
    edge1 = v2 - v1
    edge2 = v3 - v1
    pvec = np.cross(ray_vec, edge2)
    det = edge1.dot(pvec)

    if abs(det) < eps:  # no intersection
        return False, null_inter
    inv_det = 1. / det
    tvec = ray_start - v1
    u = tvec.dot(pvec) * inv_det

    if u < 0. or u > 1.:  # if not intersection
        return False, null_inter

    qvec = np.cross(tvec, edge1)
    v = ray_vec.dot(qvec) * inv_det
    if v < 0. or u + v > 1.:  # if not intersection
        return False, null_inter

    t = edge2.dot(qvec) * inv_det
    if t < eps:
        return False, null_inter

    return True, np.array([t, u, v])
# Create a basic triangle within pyvista
points = np.array([[0, 0, 0],
                   [0, 1, 0],
                   [1, 0, 0]])
faces = np.array([3, 0, 1, 2])
tri = pv.PolyData(points, faces)

# cast a ray above pointed downwards
start = np.array([0.3, 0.25, 1])
direction = np.array([0, 0, -1])

# compute if the intersection exists
inter, tuv = ray_triangle_intersection(start, direction, points)
t, u, v = tuv

print('Intersected', inter)
print('t:', t)
print('u:', u)
print('v:', v)

Out:

Intersected True
t: 1.0
u: 0.25
v: 0.3

Plot the problem setup and the intersection

if inter:

    # reconstruct intersection point in barycentric coordinates.  See
    # https://en.wikipedia.org/wiki/Barycentric_coordinate_system
    a, b, c = (1 - u - v), u, v
    point = tri.points[0]*a + tri.points[1]*b + tri.points[2]*c

    pl = pv.Plotter()
    pl.add_text(f'Intersected at ({point[0]:.3}, {point[0]:.3}, {point[0]:.3})',
                font_size=26)
    pl.add_mesh(tri)
    _ = pl.add_arrows(np.array([start]),
                      np.array([direction]),
                      show_scalar_bar=False,
                      color='r', style='wireframe')
    pl.add_points(np.array([point]), point_size=20, render_points_as_spheres=True,
                  color='b')
    pl.add_point_labels(tri, [f'a = {1 - u - v:.3}', f'b = {u:.3}', f'c = {v:.3}'],
                        font_size=40)
    pl.show_bounds()
    pl.camera_position = 'xy'
    pl.show()

else:  # no intersection
    pl = pv.Plotter()
    pl.add_text('No intersection')
    _ = pl.add_arrows(np.array([start]),
                      np.array([direction]),
                      show_scalar_bar=False,
                      color='r', style='wireframe')
    pl.add_mesh(tri)

    pl.show_bounds()
    pl.camera_position = 'xy'

    pl.show()
ray trace moeller

Total running time of the script: ( 0 minutes 0.120 seconds)

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