# Visualize the Moeller–Trumbore Algorithm¶

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

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()
font_size=26)
np.array([direction]),
show_scalar_bar=False,
color='r', style='wireframe')
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()
np.array([direction]),
show_scalar_bar=False,
color='r', style='wireframe')