def are_coplanar(*points):
"""
This function tests whether passed points are coplanar or not.
It uses the fact that the triple scalar product of three vectors
vanishes if the vectors are coplanar. Which means that the volume
of the solid described by them will have to be zero for coplanarity.
Parameters
==========
A set of points 3D points
Returns
=======
boolean
Examples
========
>>> from sympy import Point3D
>>> p1 = Point3D(1, 2, 2)
>>> p2 = Point3D(2, 7, 2)
>>> p3 = Point3D(0, 0, 2)
>>> p4 = Point3D(1, 1, 2)
>>> Point3D.are_coplanar(p1, p2, p3, p4)
True
>>> p5 = Point3D(0, 1, 3)
>>> Point3D.are_coplanar(p1, p2, p3, p5)
False
"""
from sympy.geometry.plane import Plane
points = list(set(points))
if len(points) < 3:
raise ValueError('At least 3 points are needed to define a plane.')
a, b = points[:2]
for i, c in enumerate(points[2:]):
try:
p = Plane(a, b, c)
for j in (0, 1, i):
points.pop(j)
return all(p.is_coplanar(i) for i in points)
except ValueError:
pass
raise ValueError('At least 3 non-collinear points needed to '
'define plane.')
def are_coplanar(*points):
"""
This function tests whether passed points are coplanar or not.
It uses the fact that the triple scalar product of three vectors
vanishes if the vectors are coplanar. Which means that the volume
of the solid described by them will have to be zero for coplanarity.
Parameters
==========
A set of points 3D points
Returns
=======
boolean
Examples
========
>>> from sympy import Point3D
>>> p1 = Point3D(1, 2, 2)
>>> p2 = Point3D(2, 7, 2)
>>> p3 = Point3D(0, 0, 2)
>>> p4 = Point3D(1, 1, 2)
>>> Point3D.are_coplanar(p1, p2, p3, p4)
True
>>> p5 = Point3D(0, 1, 3)
>>> Point3D.are_coplanar(p1, p2, p3, p5)
False
"""
from sympy.geometry.plane import Plane
points = list(set(points))
if len(points) < 3:
raise ValueError('At least 3 points are needed to define a plane.')
a, b = points[:2]
for i, c in enumerate(points[2:]):
try:
p = Plane(a, b, c)
for j in (0, 1, i):
points.pop(j)
return all(p.is_coplanar(i) for i in points)
except ValueError:
pass
raise ValueError('At least 3 non-collinear points needed to '
'define plane.')
def test_refraction_angle():
n1, n2 = symbols('n1, n2')
m1 = Medium('m1')
m2 = Medium('m2')
r1 = Ray3D(Point3D(-1, -1, 1), Point3D(0, 0, 0))
i = Matrix([1, 1, 1])
n = Matrix([0, 0, 1])
normal_ray = Ray3D(Point3D(0, 0, 0), Point3D(0, 0, 1))
P = Plane(Point3D(0, 0, 0), normal_vector=[0, 0, 1])
assert refraction_angle(r1, 1, 1, n) == Matrix([[1], [1], [-1]])
assert refraction_angle([1, 1, 1], 1, 1, n) == Matrix([[1], [1], [-1]])
assert refraction_angle((1, 1, 1), 1, 1, n) == Matrix([[1], [1], [-1]])
assert refraction_angle(i, 1, 1, [0, 0, 1]) == Matrix([[1], [1], [-1]])
assert refraction_angle(i, 1, 1, (0, 0, 1)) == Matrix([[1], [1], [-1]])
assert refraction_angle(i, 1, 1, normal_ray) == Matrix([[1], [1], [-1]])
assert refraction_angle(i, 1, 1, plane=P) == Matrix([[1], [1], [-1]])
assert refraction_angle(r1, 1, 1, plane=P) == \
Ray3D(Point3D(0, 0, 0), Point3D(1, 1, -1))
assert refraction_angle(r1, m1, 1.33, plane=P) == \
Ray3D(Point3D(0, 0, 0), Point3D(100/133, 100/133, -789378201649271*sqrt(3)/1000000000000000))
assert refraction_angle(r1, 1, m2, plane=P) == \
Ray3D(Point3D(0, 0, 0), Point3D(1, 1, -1))
assert refraction_angle(r1, n1, n2, plane=P) == \
Ray3D(Point3D(0, 0, 0), Point3D(n1/n2, n1/n2, -sqrt(3)*sqrt(-2*n1**2/(3*n2**2) + 1)))
assert refraction_angle(r1, 1.33, 1, plane=P) == 0 # TIR
assert refraction_angle(r1, 1, 1, normal_ray) == \
Ray3D(Point3D(0, 0, 0), direction_ratio=[1, 1, -1])
def test_refraction_angle():
n1, n2 = symbols('n1, n2')
m1 = Medium('m1')
m2 = Medium('m2')
r1 = Ray3D(Point3D(-1, -1, 1), Point3D(0, 0, 0))
i = Matrix([1, 1, 1])
n = Matrix([0, 0, 1])
normal_ray = Ray3D(Point3D(0, 0, 0), Point3D(0, 0, 1))
P = Plane(Point3D(0, 0, 0), normal_vector=[0, 0, 1])
assert refraction_angle(r1, 1, 1, n) == Matrix([[1], [1], [-1]])
assert refraction_angle([1, 1, 1], 1, 1, n) == Matrix([[1], [1], [-1]])
assert refraction_angle((1, 1, 1), 1, 1, n) == Matrix([[1], [1], [-1]])
assert refraction_angle(i, 1, 1, [0, 0, 1]) == Matrix([[1], [1], [-1]])
assert refraction_angle(i, 1, 1, (0, 0, 1)) == Matrix([[1], [1], [-1]])
assert refraction_angle(i, 1, 1, normal_ray) == Matrix([[1], [1], [-1]])
assert refraction_angle(i, 1, 1, plane=P) == Matrix([[1], [1], [-1]])
assert refraction_angle(r1, 1, 1, plane=P) == \
Ray3D(Point3D(0, 0, 0), Point3D(1, 1, -1))
assert refraction_angle(r1, m1, 1.33, plane=P) == \
Ray3D(Point3D(0, 0, 0), Point3D(Rational(100, 133), Rational(100, 133), -789378201649271*sqrt(3)/1000000000000000))
assert refraction_angle(r1, 1, m2, plane=P) == \
Ray3D(Point3D(0, 0, 0), Point3D(1, 1, -1))
assert refraction_angle(r1, n1, n2, plane=P) == \
Ray3D(Point3D(0, 0, 0), Point3D(n1/n2, n1/n2, -sqrt(3)*sqrt(-2*n1**2/(3*n2**2) + 1)))
assert refraction_angle(r1, 1.33, 1, plane=P) == 0 # TIR
assert refraction_angle(r1, 1, 1, normal_ray) == \
Ray3D(Point3D(0, 0, 0), direction_ratio=[1, 1, -1])
assert ae(refraction_angle(0.5, 1, 2), 0.24207, 5)
assert ae(refraction_angle(0.5, 2, 1), 1.28293, 5)
raises(ValueError, lambda: refraction_angle(r1, m1, m2, normal_ray, P))
raises(TypeError, lambda: refraction_angle(m1, m1, m2)
) # can add other values for arg[0]
raises(TypeError, lambda: refraction_angle(r1, m1, m2, None, i))
raises(TypeError, lambda: refraction_angle(r1, m1, m2, m2))
def test_deviation():
n1, n2 = symbols('n1, n2')
r1 = Ray3D(Point3D(-1, -1, 1), Point3D(0, 0, 0))
n = Matrix([0, 0, 1])
i = Matrix([-1, -1, -1])
normal_ray = Ray3D(Point3D(0, 0, 0), Point3D(0, 0, 1))
P = Plane(Point3D(0, 0, 0), normal_vector=[0, 0, 1])
assert deviation(r1, 1, 1, normal=n) == 0
assert deviation(r1, 1, 1, plane=P) == 0
assert deviation(r1, 1, 1.1, plane=P).evalf(3) + 0.119 < 1e-3
assert deviation(i, 1, 1.1, normal=normal_ray).evalf(3) + 0.119 < 1e-3
assert deviation(r1, 1.33, 1, plane=P) is None # TIR
assert deviation(r1, 1, 1, normal=[0, 0, 1]) == 0
assert deviation([-1, -1, -1], 1, 1, normal=[0, 0, 1]) == 0
def are_coplanar(*e):
""" Returns True if the given entities are coplanar otherwise False
Parameters
==========
e: entities to be checked for being coplanar
Returns
=======
Boolean
Examples
========
>>> from sympy import Point3D, Line3D
>>> from sympy.geometry.util import are_coplanar
>>> a = Line3D(Point3D(5, 0, 0), Point3D(1, -1, 1))
>>> b = Line3D(Point3D(0, -2, 0), Point3D(3, 1, 1))
>>> c = Line3D(Point3D(0, -1, 0), Point3D(5, -1, 9))
>>> are_coplanar(a, b, c)
False
"""
from sympy.geometry.line import LinearEntity3D
from sympy.geometry.entity import GeometryEntity
from sympy.geometry.point import Point3D
from sympy.geometry.plane import Plane
# XXX update tests for coverage
e = set(e)
# first work with a Plane if present
for i in list(e):
if isinstance(i, Plane):
e.remove(i)
return all(p.is_coplanar(i) for p in e)
if all(isinstance(i, Point3D) for i in e):
if len(e) < 3:
return False
# remove pts that are collinear with 2 pts
a, b = e.pop(), e.pop()
for i in list(e):
if Point3D.are_collinear(a, b, i):
e.remove(i)
if not e:
return False
else:
# define a plane
p = Plane(a, b, e.pop())
for i in e:
if i not in p:
return False
return True
else:
pt3d = []
for i in e:
if isinstance(i, Point3D):
pt3d.append(i)
elif isinstance(i, LinearEntity3D):
pt3d.extend(i.args)
elif isinstance(
i, GeometryEntity
): # XXX we should have a GeometryEntity3D class so we can tell the difference between 2D and 3D -- here we just want to deal with 2D objects; if new 3D objects are encountered that we didn't handle above, an error should be raised
# all 2D objects have some Point that defines them; so convert those points to 3D pts by making z=0
for p in i.args:
if isinstance(p, Point):
pt3d.append(Point3D(*(p.args + (0, ))))
return are_coplanar(*pt3d)
anterior_point, ["Ant Point"], point_color='red', point_size=20)
p.add_legend()
p.add_axes()
p.show()
raise
# Check if point is lateral or medial
# posterior_point_distance =
posterior_point_distance = sagittal_plane.distance(
Point3D(posterior_point, evaluate=False))
if (posterior_point_distance < 0):
# Find sagittale plane
initial_sagitall_axis = np.array([1, 0, 0])
initial_sagitall_plane = Plane(
femur_cm, normal_vector=initial_sagitall_axis)
max_distance = -100000
for idx, point in enumerate(femur_3D_points):
d = initial_sagitall_plane.distance(point)
if (d > max_distance):
max_distance = d
index_point_max_sagitall_distance = idx
else:
initial_sagitall_axis = np.array([1, 0, 0])
initial_sagitall_plane = Plane(
femur_cm, normal_vector=initial_sagitall_axis)
max_distance = 100000
for idx, point in enumerate(femur_3D_points):
d = initial_sagitall_plane.distance(point)
if (d < max_distance):
max_distance = d