Moeller-Trumboreアルゴリズムの可視化

この例では，pyvistaを使ってMoeller-Trumbore intersectionアルゴリズムを実演しています．

詳細については，以下を参照してください:

• Möller–Trumbore intersection algorithm

• Fast Minimum Storage Ray Triangle Intersectio

まず，光線3角交差法を定義します．

import numpy as np

import pyvista as pv


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 = 1.0 / det
    tvec = ray_start - v1
    u = tvec.dot(pvec) * inv_det

    if u < 0.0 or u > 1.0:  # if not intersection
        return False, null_inter

    qvec = np.cross(tvec, edge1)
    v = ray_vec.dot(qvec) * inv_det
    if v < 0.0 or u + v > 1.0:  # 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)

/home/runner/work/pyvista-docs-dev-ja/pyvista-docs-dev-ja/pyvista-doc-translations/pyvista/pyvista/core/utilities/points.py:52: UserWarning: Points is not a float type. This can cause issues when transforming or applying filters. Casting to ``np.float32``. Disable this by passing ``force_float=False``.
  warnings.warn(
Intersected True
t: 1.0
u: 0.25
v: 0.3

問題の設定と交点のプロット

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()