def intersection_beam(self, ray: Ray) -> List[Tuple[float, float]]:
"""
Returns all couples :math:`(s, t)` such that there exist
:math:`\\vec{X}` satisfying
.. math::
\\vec{X} = (1-s)^3 \\vec{p_0} + 3 s (1-s)^2 \\vec{p_1}
+ 3 s^2 (1-s) \\vec{p_2} + s^3 \\vec{p_3}
and
.. math::
\\vec{X} = \\vec{o} + t \\vec{d}
with :math:`0 \\lq s \\lq 1` and :math:`t >= 0`
"""
p0, p1, p2, p3 = self._p
a = orthogonal(ray.direction)
a0 = np.dot(a, p0 - ray.origin)
a1 = -3 * np.dot(a, p0 - p1)
a2 = 3 * np.dot(a, p0 - 2 * p1 + p2)
a3 = np.dot(a, -p0 + 3 * p1 - 3 * p2 + p3)
roots = cubic_real_roots(a0, a1, a2, a3)
intersection_points = [(1 - s)**3 * p0 + 3 * s * (1 - s)**2 * p1 +
3 * s**2 * (1 - s) * p2 + s**3 * p3
for s in roots]
travel = [
np.dot(X - ray.origin, ray.direction) for X in intersection_points
]
def valid_domain(s, t):
return 0 <= s <= 1 and t > Ray.min_travel
return [(s, t) for (s, t) in zip(roots, travel) if valid_domain(s, t)]
def normal(self, s: float) -> np.ndarray:
"""Returns a vector normal at the curve at curvilinear coordinate s"""
return orthogonal(self.tangent(s))