def _contains(self, other):
if not is_Tuple(other) or len(other) != 3:
return false
v = Mat(other)
p = v - self.center
if self.closed:
return norm(cross(p, self.direction)) <= self.slope*dot(p, self.direction)
else:
return norm(cross(p, self.direction)) < self.slope*dot(p, self.direction)
def _contains(self, other):
if not is_Tuple(other) or len(other) != 3:
return false
v = Mat(other)
p = v - self.center
if self.closed:
return norm(cross(p, self.direction))**2 <= self.radius**2
else:
return norm(cross(p, self.direction))**2 < self.radius**2
def _contains(self, other):
if not is_Tuple(other) or len(other) != 3:
return false
v = Mat(other)
if self.closed:
return norm(v-self.center)**2 <= self.radius**2
else:
return norm(v-self.center)**2 < self.radius**2
def as_abstract(self):
"""
>>> from sympy import *
>>> from symplus.strplus import init_mprinting
>>> init_mprinting()
>>> Sphere().as_abstract()
{(x, y, z) | x**2 + y**2 + z**2 < 1}
>>> Sphere(3, [1,0,2]).as_abstract()
{(x, y, z) | y**2 + (x - 1)**2 + (z - 2)**2 < 9}
"""
if self.closed:
expr = norm(r-self.center)**2 <= self.radius**2
else:
expr = norm(r-self.center)**2 < self.radius**2
return AbstractSet((x,y,z), expr)
def rquat2rmat(rquat):
"""
>>> from sympy import *
>>> from symplus.strplus import init_mprinting
>>> init_mprinting()
>>> t = Symbol('t', positive=True)
>>> simplify(rquat2rmat(rquat(pi/3, i+j)))
<BLANKLINE>
[ 3/4 1/4 sqrt(6)/4]
[ 1/4 3/4 -sqrt(6)/4]
[-sqrt(6)/4 sqrt(6)/4 1/2]
>>> simplify(rquat2rmat(rquat(t, i)))
<BLANKLINE>
[1 0 0]
[0 cos(t) -sin(t)]
[0 sin(t) cos(t)]
>>> simplify(rquat2rmat(rquat(t, i+k)))
<BLANKLINE>
[ cos(t/2)**2 -sqrt(2)*sin(t)/2 sin(t/2)**2]
[sqrt(2)*sin(t)/2 cos(t) -sqrt(2)*sin(t)/2]
[ sin(t/2)**2 sqrt(2)*sin(t)/2 cos(t/2)**2]
"""
w = rquat[0]
xyz = Mat(rquat[1:])
return (1-2*norm(xyz)**2)*eye3 + 2*xyz*xyz.T + 2*w*cross(xyz)
def zvec2zmat(zvec):
"""
>>> from sympy import *
>>> from symplus.strplus import init_mprinting
>>> init_mprinting()
>>> zvec2zmat(j*3)
<BLANKLINE>
[1 0 0]
[0 3 0]
[0 0 1]
>>> zvec2zmat(i-j)
<BLANKLINE>
[ 1/2 + sqrt(2)/2 -sqrt(2)/2 + 1/2 0]
[-sqrt(2)/2 + 1/2 1/2 + sqrt(2)/2 0]
[ 0 0 1]
>>> t,r,p,z,s = aff2trpzs(augment(m=_))
>>> simplify(r)
[2**(1/4)*3**(3/4)*sqrt(1 + sqrt(2) + sqrt(6))/6 0 0 (-6**(3/4) + 54**(1/4))/(6*sqrt(1 + sqrt(2) + sqrt(6)))]'
>>> simplify(z)
[sqrt(6)/2 2*sqrt(3)/3 1]'
>>> simplify(s)
[-1/3 0 0]'
"""
zvec = Mat(zvec)
nvec = normalize(zvec)
zfac = norm(zvec)
return eye3 + nvec*nvec.T*(zfac-1)
def __new__(cls, func, center=[0,0,0], direction=[0,0,1], **kwargs):
center = Mat(center)
direction = Mat(direction)
if norm(direction) == 0:
raise ValueError
direction = normalize(direction)
return Basic.__new__(cls, func, center, direction)
def as_abstract(self):
"""
>>> from sympy import *
>>> from symplus.strplus import init_mprinting
>>> init_mprinting()
>>> SemiInfiniteCone().as_abstract()
{(x, y, z) | sqrt(x**2 + y**2) < z}
>>> SemiInfiniteCone(5, [0,0,0], [3,4,0]).as_abstract()
{(x, y, z) | sqrt(z**2 + (4*x/5 - 3*y/5)**2) < 3*x + 4*y}
"""
p = r - self.center
if self.closed:
expr = norm(cross(p, self.direction)) <= self.slope*dot(p, self.direction)
else:
expr = norm(cross(p, self.direction)) < self.slope*dot(p, self.direction)
return AbstractSet((x,y,z), expr)
def as_abstract(self):
"""
>>> from sympy import *
>>> from symplus.strplus import init_mprinting
>>> init_mprinting()
>>> InfiniteCylinder().as_abstract()
{(x, y, z) | x**2 + y**2 < 1}
>>> InfiniteCylinder(2, [0,0,0], [0,1,1]).as_abstract()
{(x, y, z) | x**2 + (-sqrt(2)*y/2 + sqrt(2)*z/2)**2 < 4}
"""
p = r - self.center
if self.closed:
expr = norm(cross(p, self.direction))**2 <= self.radius**2
else:
expr = norm(cross(p, self.direction))**2 < self.radius**2
return AbstractSet((x,y,z), expr)
def __new__(cls, radius=1, center=[0,0,0], direction=[0,0,1], closed=False, **kwargs):
"""
>>> from sympy import *
>>> from symplus.strplus import init_mprinting
>>> init_mprinting()
>>> InfiniteCylinder()
InfiniteCylinder(1, [0 0 0]', [0 0 1]', False)
>>> InfiniteCylinder(2, [0,0,0], [0,1,1])
InfiniteCylinder(2, [0 0 0]', [0 -sqrt(2)/2 -sqrt(2)/2]', False)
>>> InfiniteCylinder().contains((1,1,1))
False
>>> InfiniteCylinder(2, [0,0,0], [0,1,1]).contains((1,1,1))
True
"""
normalization = kwargs.pop("normalization", True)
radius = sympify(abs(radius))
direction = Mat(direction)
if normalization:
if norm(direction) == 0:
raise ValueError
direction = simplify(normalize(direction))
direction = max(direction, -direction, key=hash)
center = Mat(center)
if normalization:
center = simplify(center - project(center, direction))
closed = sympify(bool(closed))
return Basic.__new__(cls, radius, center, direction, closed)
def __new__(cls, radius=1, height=2, center=[0,0,0], direction=[0,0,1], closed=False, **kwargs):
normalization = kwargs.pop("normalization", True)
radius = sympify(abs(radius))
height = sympify(abs(height))
center = Mat(center)
direction = Mat(direction)
if normalization:
if norm(direction) == 0:
raise ValueError
direction = simplify(normalize(direction))
direction = max(direction, -direction, key=hash)
closed = sympify(bool(closed))
return Basic.__new__(cls, radius, height, center, direction, closed)
def as_abstract(self):
"""
>>> from sympy import *
>>> from symplus.strplus import init_mprinting
>>> init_mprinting()
>>> Revolution(lambda h, s: h**2<s**2).as_abstract()
{(x, y, z) | z**2 < x**2 + y**2}
>>> Revolution(lambda h, s: h+1<s**2, [0,0,0], [3,4,0]).as_abstract()
{(x, y, z) | 3*x/5 + 4*y/5 + 1 < z**2 + (4*x/5 - 3*y/5)**2}
"""
p = r - self.center
expr = self.func(dot(p, self.direction), norm(cross(p, self.direction)))
return AbstractSet((x,y,z), expr)
def __new__(cls, offset=0, direction=[0,0,1], closed=False, **kwargs):
"""
>>> from sympy import *
>>> from symplus.strplus import init_mprinting
>>> init_mprinting()
>>> Halfspace()
Halfspace(0, [0 0 1]', False)
>>> Halfspace(3, [1,2,0])
Halfspace(3, [sqrt(5)/5 2*sqrt(5)/5 0]', False)
>>> Halfspace().contains((1,2,3))
True
>>> Halfspace(3, [1,2,0]).contains((1,2,3))
False
"""
normalization = kwargs.pop("normalization", True)
offset = sympify(offset)
direction = Mat(direction)
if normalization:
if norm(direction) == 0:
raise ValueError
direction = simplify(normalize(direction))
closed = sympify(bool(closed))
return Basic.__new__(cls, offset, direction, closed)
def __new__(cls, slope=1, center=[0,0,0], direction=[0,0,1], closed=False, **kwargs):
"""
>>> from sympy import *
>>> from symplus.strplus import init_mprinting
>>> init_mprinting()
>>> SemiInfiniteCone()
SemiInfiniteCone(1, [0 0 0]', [0 0 1]', False)
>>> SemiInfiniteCone(5, [0,0,0], [3,4,0])
SemiInfiniteCone(5, [0 0 0]', [3/5 4/5 0]', False)
>>> SemiInfiniteCone().contains((-1,0,2))
True
>>> SemiInfiniteCone(5, [0,0,0], [3,4,0]).contains((-1,0,1))
False
"""
normalization = kwargs.pop("normalization", True)
slope = sympify(abs(slope))
center = Mat(center)
direction = Mat(direction)
if normalization:
if norm(direction) == 0:
raise ValueError
direction = simplify(normalize(direction))
closed = sympify(bool(closed))
return Basic.__new__(cls, slope, center, direction, closed)
def _contains(self, other):
if not is_Tuple(other) or len(other) != 3:
return false
v = Mat(other)
p = v - self.center
return self.func(dot(p, self.direction), norm(cross(p, self.direction)))