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# Copyright (C) 2004, Thomas Hamelryck (thamelry@binf.ku.dk)
# This code is part of the Biopython distribution and governed by its
# license. Please see the LICENSE file that should have been included
# as part of this package.
"""Vector class, including rotation-related functions."""
from __future__ import print_function
import numpy
def m2rotaxis(m):
"""
Return angles, axis pair that corresponds to rotation matrix m.
"""
# Angle always between 0 and pi
# Sense of rotation is defined by axis orientation
t = 0.5 * (numpy.trace(m) - 1)
t = max(-1, t)
t = min(1, t)
angle = numpy.arccos(t)
if angle < 1e-15:
# Angle is 0
return 0.0, Vector(1, 0, 0)
elif angle < numpy.pi:
# Angle is smaller than pi
x = m[2, 1] - m[1, 2]
y = m[0, 2] - m[2, 0]
z = m[1, 0] - m[0, 1]
axis = Vector(x, y, z)
axis.normalize()
return angle, axis
else:
# Angle is pi - special case!
m00 = m[0, 0]
m11 = m[1, 1]
m22 = m[2, 2]
if m00 > m11 and m00 > m22:
x = numpy.sqrt(m00 - m11 - m22 + 0.5)
y = m[0, 1] / (2 * x)
z = m[0, 2] / (2 * x)
elif m11 > m00 and m11 > m22:
y = numpy.sqrt(m11 - m00 - m22 + 0.5)
x = m[0, 1] / (2 * y)
z = m[1, 2] / (2 * y)
else:
z = numpy.sqrt(m22 - m00 - m11 + 0.5)
x = m[0, 2] / (2 * z)
y = m[1, 2] / (2 * z)
axis = Vector(x, y, z)
axis.normalize()
return numpy.pi, axis
def vector_to_axis(line, point):
"""
Returns the vector between a point and
the closest point on a line (ie. the perpendicular
projection of the point on the line).
@type line: L{Vector}
@param line: vector defining a line
@type point: L{Vector}
@param point: vector defining the point
"""
line = line.normalized()
np = point.norm()
angle = line.angle(point)
return point - line ** (np * numpy.cos(angle))
def rotaxis2m(theta, vector):
"""
Calculate a left multiplying rotation matrix that rotates
theta rad around vector.
Example:
>>> m=rotaxis(pi, Vector(1, 0, 0))
>>> rotated_vector=any_vector.left_multiply(m)
@type theta: float
@param theta: the rotation angle
@type vector: L{Vector}
@param vector: the rotation axis
@return: The rotation matrix, a 3x3 Numeric array.
"""
vector = vector.copy()
vector.normalize()
c = numpy.cos(theta)
s = numpy.sin(theta)
t = 1 - c
x, y, z = vector.get_array()
rot = numpy.zeros((3, 3))
# 1st row
rot[0, 0] = t * x * x + c
rot[0, 1] = t * x * y - s * z
rot[0, 2] = t * x * z + s * y
# 2nd row
rot[1, 0] = t * x * y + s * z
rot[1, 1] = t * y * y + c
rot[1, 2] = t * y * z - s * x
# 3rd row
rot[2, 0] = t * x * z - s * y
rot[2, 1] = t * y * z + s * x
rot[2, 2] = t * z * z + c
return rot
rotaxis = rotaxis2m
def refmat(p, q):
"""
Return a (left multiplying) matrix that mirrors p onto q.
Example:
>>> mirror=refmat(p, q)
>>> qq=p.left_multiply(mirror)
>>> print(q)
>>> print(qq) # q and qq should be the same
@type p,q: L{Vector}
@return: The mirror operation, a 3x3 Numeric array.
"""
p.normalize()
q.normalize()
if (p - q).norm() < 1e-5:
return numpy.identity(3)
pq = p - q
pq.normalize()
b = pq.get_array()
b.shape = (3, 1)
i = numpy.identity(3)
ref = i - 2 * numpy.dot(b, numpy.transpose(b))
return ref
def rotmat(p, q):
"""
Return a (left multiplying) matrix that rotates p onto q.
Example:
>>> r=rotmat(p, q)
>>> print(q)
>>> print(p.left_multiply(r))
@param p: moving vector
@type p: L{Vector}
@param q: fixed vector
@type q: L{Vector}
@return: rotation matrix that rotates p onto q
@rtype: 3x3 Numeric array
"""
rot = numpy.dot(refmat(q, -p), refmat(p, -p))
return rot
def calc_angle(v1, v2, v3):
"""
Calculate the angle between 3 vectors
representing 3 connected points.
@param v1, v2, v3: the tree points that define the angle
@type v1, v2, v3: L{Vector}
@return: angle
@rtype: float
"""
v1 = v1 - v2
v3 = v3 - v2
return v1.angle(v3)
def calc_dihedral(v1, v2, v3, v4):
"""
Calculate the dihedral angle between 4 vectors
representing 4 connected points. The angle is in
]-pi, pi].
@param v1, v2, v3, v4: the four points that define the dihedral angle
@type v1, v2, v3, v4: L{Vector}
"""
ab = v1 - v2
cb = v3 - v2
db = v4 - v3
u = ab ** cb
v = db ** cb
w = u ** v
angle = u.angle(v)
# Determine sign of angle
try:
if cb.angle(w) > 0.001:
angle = -angle
except ZeroDivisionError:
# dihedral=pi
pass
return angle
class Vector(object):
"3D vector"
def __init__(self, x, y=None, z=None):
if y is None and z is None:
# Array, list, tuple...
if len(x) != 3:
raise ValueError("Vector: x is not a "
"list/tuple/array of 3 numbers")
self._ar = numpy.array(x, 'd')
else:
# Three numbers
self._ar = numpy.array((x, y, z), 'd')
def __repr__(self):
x, y, z = self._ar
return "<Vector %.2f, %.2f, %.2f>" % (x, y, z)
def __neg__(self):
"Return Vector(-x, -y, -z)"
a = -self._ar
return Vector(a)
def __add__(self, other):
"Return Vector+other Vector or scalar"
if isinstance(other, Vector):
a = self._ar + other._ar
else:
a = self._ar + numpy.array(other)
return Vector(a)
def __sub__(self, other):
"Return Vector-other Vector or scalar"
if isinstance(other, Vector):
a = self._ar - other._ar
else:
a = self._ar - numpy.array(other)
return Vector(a)
def __mul__(self, other):
"Return Vector.Vector (dot product)"
return sum(self._ar * other._ar)
def __div__(self, x):
"Return Vector(coords/a)"
a = self._ar / numpy.array(x)
return Vector(a)
def __pow__(self, other):
"Return VectorxVector (cross product) or Vectorxscalar"
if isinstance(other, Vector):
a, b, c = self._ar
d, e, f = other._ar
c1 = numpy.linalg.det(numpy.array(((b, c), (e, f))))
c2 = -numpy.linalg.det(numpy.array(((a, c), (d, f))))
c3 = numpy.linalg.det(numpy.array(((a, b), (d, e))))
return Vector(c1, c2, c3)
else:
a = self._ar * numpy.array(other)
return Vector(a)
def __getitem__(self, i):
return self._ar[i]
def __setitem__(self, i, value):
self._ar[i] = value
def __contains__(self, i):
return (i in self._ar)
def norm(self):
"Return vector norm"
return numpy.sqrt(sum(self._ar * self._ar))
def normsq(self):
"Return square of vector norm"
return abs(sum(self._ar * self._ar))
def normalize(self):
"Normalize the Vector"
self._ar = self._ar / self.norm()
def normalized(self):
"Return a normalized copy of the Vector"
v = self.copy()
v.normalize()
return v
def angle(self, other):
"Return angle between two vectors"
n1 = self.norm()
n2 = other.norm()
c = (self * other) / (n1 * n2)
# Take care of roundoff errors
c = min(c, 1)
c = max(-1, c)
return numpy.arccos(c)
def get_array(self):
"Return (a copy of) the array of coordinates"
return numpy.array(self._ar)
def left_multiply(self, matrix):
"Return Vector=Matrix x Vector"
a = numpy.dot(matrix, self._ar)
return Vector(a)
def right_multiply(self, matrix):
"Return Vector=Vector x Matrix"
a = numpy.dot(self._ar, matrix)
return Vector(a)
def copy(self):
"Return a deep copy of the Vector"
return Vector(self._ar)
if __name__ == "__main__":
from numpy.random import random
v1 = Vector(0, 0, 1)
v2 = Vector(0, 0, 0)
v3 = Vector(0, 1, 0)
v4 = Vector(1, 1, 0)
v4.normalize()
print(v4)
print(calc_angle(v1, v2, v3))
dih = calc_dihedral(v1, v2, v3, v4)
# Test dihedral sign
assert(dih > 0)
print("DIHEDRAL %f" % dih)
ref = refmat(v1, v3)
rot = rotmat(v1, v3)
print(v3)
print(v1.left_multiply(ref))
print(v1.left_multiply(rot))
print(v1.right_multiply(numpy.transpose(rot)))
# -
print(v1 - v2)
print(v1 - 1)
print(v1 + (1, 2, 3))
# +
print(v1 + v2)
print(v1 + 3)
print(v1 - (1, 2, 3))
# *
print(v1 * v2)
# /
print(v1 / 2)
print(v1 / (1, 2, 3))
# **
print(v1 ** v2)
print(v1 ** 2)
print(v1 ** (1, 2, 3))
# norm
print(v1.norm())
# norm squared
print(v1.normsq())
# setitem
v1[2] = 10
print(v1)
# getitem
print(v1[2])
print(numpy.array(v1))
print("ROT")
angle = random() * numpy.pi
axis = Vector(random(3) - random(3))
axis.normalize()
m = rotaxis(angle, axis)
cangle, caxis = m2rotaxis(m)
print(angle - cangle)
print(axis - caxis)
print("")
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