def __init__(self,
p=Point3(0, 0, 0),
p1=None,
p2=None,
normal=Vector3(0, 0, 1)):
"""
a plane can be defined either by three non-_collinear points or by a point and a vetor denotin the normal
to the plane. The normal vector has priority over the points. I.e., if both the normal vector and points 1 and 2
are given, p1 and p2 are ignored
:param p: the first point
:param p1: the second point (mutually exclusive with normal)
:param p2: the third point (mutually exclusive with normal)
:param normal: the normal vector (mutually exclusive with point 1 and 2)
"""
self._position = p
if normal and isinstance(normal, Vector3):
self._normal = normal / normal.magnitude(
) # normalize the vector (i.e., magnitude = 1)
elif p1 and isinstance(p1, Point3) and p2 and isinstance(p2, Point3):
if Point3.are_collinear(p, p1, p2):
# if the three points are _collinear raise an error
raise ValueError(
'To define a plane three non-_collinear points must be given'
)
# compute the normal to the triangle p-p1-p2
cross_prod = (p2 - p).cross(p2 - p)
self._normal = cross_prod / cross_prod.magnitude # normalize the vector (i.e., magnitude = 1)
def distance(self, other):
"""
:param other: Point3
:return: the distance between self and other
"""
# NOTE: the distance is the magnitude of the vector from self to other
return Vector3(other - self).magnitude()
def intersection(self, other):
# self= [p,p+r]; other=[q,q+other]
if isinstance(other, Line): # if other is actually a segment
if self.intersects(other): # if the segments intersect
# print("INTERSECT")
p = Vector3(self.vertices[0])
q = Vector3(other.vertices[0])
r = Vector3(self.vertices[1] - self.vertices[0])
s = Vector3(other.vertices[1] - other.vertices[0])
if not self.is_collinear_with(other): # general case
# print("NOT COLLINEAR")
t = (q - p).cross(s)[2] / r.cross(s)[2]
return Point3(p.x() + t * r.x(), p.y() + t * r.y())
else: # special case: collinear segments
# print("COLLINEAR")
# position of other.vertices[0] wrt self.vertices[0],self.vertices[1]
s0_pos = Point3.collinear_position(self.vertices[0],
self.vertices[1],
other.vertices[0])
# position of other.vertices[1] wrt self.vertices[0],self.vertices[1]
s1_pos = Point3.collinear_position(self.vertices[0],
self.vertices[1],
other.vertices[1])
start = None
end = None
# if one vertex of other is before self
if s0_pos == BEFORE or s1_pos == BEFORE:
# print("ONE VERTEX OF S IS BEFORE")
start = self.vertices[
0] # the resulting segment starts at self[0]
if s0_pos == BETWEEN:
# print("S[0] IS BETWEEN")
end = other.vertices[0]
elif s1_pos == BETWEEN:
# print("S[1] IS BETWEEN")
end = other.vertices[1]
elif s0_pos == AFTER or s1_pos == AFTER:
# print("THE OTHER VERTEX IS AFTER")
end = self.vertices[1]
# otherwise, at least one vertex of other must lie between self[0] and self[1]
# (because we know they intersect and are collinear)
else:
# print("ONE VERTEX OF S IS BETWEEN")
# assume other is completely inside self
start = other.vertices[
0] # then the intersection coincides with other
end = other.vertices[1]
if s0_pos == AFTER:
start = self.vertices[
1] # the intersection starts at self[1]
elif s1_pos == AFTER:
end = self.vertices[
1] # the intersection ends at self[1]
# print("start:"+str(start))
# print("end:"+str(end))
if start != end:
return Line(vertices=[Point3(start), Point3(end)])
else:
return Point3(start)
return None
def intersects(self, other):
"""
:param other: Line or Vector
:return: True if the two segments intersect, False otherwise
"""
if isinstance(other, (Vector2, Vector3)):
other = Line([Point3(0, 0, 0), Point3(other)])
if not isinstance(other, Line):
raise TypeError(
"Line.intersects requires a Line or a Vector as parameter")
l1, l2 = self, other # handy aliases for the segments
# in order for two segments to intersect they must lie on the same plane
if not Point3.are_coplanar(l1.start, l1.end, l2.start, l2.end):
print("Segments are not COPLANAR")
return False
# Ok, the segments are COPLANAR!
# now, we have two cases to be treated separately: (1) the segments are COLLINEAR, (2) or not
if l1.is_collinear_with(l2):
# if COLLINEAR we need to check the relative distances between the extremes of the segments
# We have only three possible cases where the segments intersect:
# A) l2 is inside l1 (i.e., both l2 extremes are BETWEEN l1 extremes)
# B) l1 is inside l2
# C) none of the above, but one extreme of one of the two segments is BETWEEN the extremes of the other
# compute the positions of the segments' extremes wrt the other segment
l2_start_position = Point3.collinear_position(
l1.start, l1.end, l2.start)
l2_end_position = Point3.collinear_position(
l1.start, l1.end, l2.end)
l1_start_position = Point3.collinear_position(
l2.start, l2.end, l1.start)
l1_end_position = Point3.collinear_position(
l2.start, l2.end, l1.end)
# case A)
if l2_start_position == BETWEEN and l2_end_position == BETWEEN:
return True
# case B)
elif l1_start_position == BETWEEN and l1_end_position == BETWEEN:
return True
# case C)
elif any(pos == BETWEEN for pos in [
l2_start_position, l2_end_position, l1_start_position,
l1_end_position
]):
return True
else:
return False
else:
# if the two segments are not COLLINEAR, it can still be the case that one of the extremes of
# one segment is COLLINEAR with the other segment. This case must be treated separately
# consider l1 as reference segment and check if the extremes of l2 are COLLINEAR with l1
is_l2_start_collinear = Point3.are_collinear(
l1.start, l1.end, l2.start)
is_l2_end_collinear = Point3.are_collinear(l1.start, l1.end,
l2.end)
# consider l2 as reference segment and check if the extremes of l1 are COLLINEAR with l2
is_l1_start_collinear = Point3.are_collinear(
l2.start, l2.end, l1.start)
is_l1_end_collinear = Point3.are_collinear(l2.start, l2.end,
l1.end)
if is_l2_start_collinear or is_l2_end_collinear or is_l1_start_collinear or is_l1_end_collinear:
# NOTE: at most one of the above variables can be True, otherwise it means that the two
# segments are COLLINEAR and we would have entered the if condition above--l1.is_collinear_with(l2).
# determine which is the reference segment
ref_segment = l1 if is_l2_start_collinear or is_l2_end_collinear else l2
# determine which extreme of the other segment is COLLINEAR with the ref_segment
# assume it is l2.start
coll_extreme = l2.start
if ref_segment is l1:
if is_l2_end_collinear:
coll_extreme = l2.end
else:
if is_l1_start_collinear:
coll_extreme = l1.start
else:
coll_extreme = l2.end
# the only case when the two segments intersect is when the coll_extreme is positioned
# BETWEEN the extremes of the reference segment
return Point3.collinear_position(ref_segment.start,
ref_segment.end,
coll_extreme) == BETWEEN
else:
# the two segments are in general position. So, to determine if they intersect
# we need to check the position of the extremes of each segment wrt the other segment.
# To do this in 3-space we need an observation point that is not on the same plane
# so, compute an observation point that is not on the same plane
# easily, we can take the cross product of the two segments
v1 = Vector3(l1.end - l1.start)
v2 = Vector3(l2.end - l2.start)
v3 = v1.cross(v2)
observation_point = Point3(v3) + l1.start
# in order to intersect the extremes of each segment must lie on opposite sides of the other segment
return \
l1.point_position(l2[0], observation_point) * l1.point_position(l2[1], observation_point) < 0 and \
l2.point_position(l1[0], observation_point) * l2.point_position(l1[1], observation_point) < 0