def equation(self, x='x', y='y'):
"""The equation of the line: ax + by + c.
Parameters
----------
x : str, optional
The name to use for the x-axis, default value is 'x'.
y : str, optional
The name to use for the y-axis, default value is 'y'.
Returns
-------
equation : sympy expression
Examples
--------
>>> from sympy import Point, Line
>>> p1, p2 = Point(1, 0), Point(5, 3)
>>> l1 = Line(p1, p2)
>>> l1.equation()
-3*x + 4*y + 3
"""
x, y = _symbol(x), _symbol(y)
p1, p2 = self.points
if p1[0] == p2[0]:
return x - p1[0]
elif p1[1] == p2[1]:
return y - p1[1]
a, b, c = self.coefficients
return simplify(a*x + b*y + c)
def equation(self, x='x', y='y'):
"""The equation of the circle.
Parameters
==========
x : str or Symbol, optional
Default value is 'x'.
y : str or Symbol, optional
Default value is 'y'.
Returns
=======
equation : SymPy expression
Examples
========
>>> from sympy import Point, Circle
>>> c1 = Circle(Point(0, 0), 5)
>>> c1.equation()
x**2 + y**2 - 25
"""
x = _symbol(x)
y = _symbol(y)
t1 = (x - self.center.x)**2
t2 = (y - self.center.y)**2
return t1 + t2 - self.major**2
def equation(self, x='x', y='y'):
"""The equation of the ellipse.
Parameters
==========
x : str, optional
Label for the x-axis. Default value is 'x'.
y : str, optional
Label for the y-axis. Default value is 'y'.
Returns
=======
equation : sympy expression
See Also
========
arbitrary_point : Returns parameterized point on ellipse
Examples
========
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(1, 0), 3, 2)
>>> e1.equation()
y**2/4 + (x/3 - 1/3)**2 - 1
"""
x = _symbol(x)
y = _symbol(y)
t1 = ((x - self.center.x) / self.hradius)**2
t2 = ((y - self.center.y) / self.vradius)**2
return t1 + t2 - 1
def equation(self, x="x", y="y"):
"""The equation of the ellipse.
Parameters
----------
x : str, optional
Label for the x-axis. Default value is 'x'.
y : str, optional
Label for the y-axis. Default value is 'y'.
Returns
-------
equation : sympy expression
Examples
--------
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(1, 0), 3, 2)
>>> e1.equation()
y**2/4 + (x/3 - 1/3)**2 - 1
"""
x = _symbol(x)
y = _symbol(y)
t1 = ((x - self.center[0]) / self.hradius) ** 2
t2 = ((y - self.center[1]) / self.vradius) ** 2
return t1 + t2 - 1
def equation(self, x="x", y="y"):
"""The equation of the circle.
Parameters
----------
x : str or Symbol, optional
Default value is 'x'.
y : str or Symbol, optional
Default value is 'y'.
Returns
-------
equation : sympy expression
Examples
--------
>>> from sympy import Point, Circle
>>> c1 = Circle(Point(0, 0), 5)
>>> c1.equation()
x**2 + y**2 - 25
"""
x = _symbol(x)
y = _symbol(y)
t1 = (x - self.center[0]) ** 2
t2 = (y - self.center[1]) ** 2
return t1 + t2 - self.major ** 2
def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of the Ray. Gives
values that will produce a ray that is 10 units long (where a unit is
the distance between the two points that define the ray).
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
plot_interval : list
[parameter, lower_bound, upper_bound]
Examples
========
>>> from sympy import Point, Ray, pi
>>> r = Ray((0, 0), angle=pi/4)
>>> r.plot_interval()
[t, 0, 10]
"""
t = _symbol(parameter)
return [t, 0, 10]
def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of the curve.
Parameters
----------
parameter : str or Symbol, optional
Default value is 't';
otherwise the provided symbol is used.
Returns
-------
plot_interval : list (plot interval)
[parameter, lower_bound, upper_bound]
Examples
--------
>>> from sympy import Curve, sin
>>> from sympy.abc import x, t, s
>>> Curve((x, sin(x)), (x, 1, 2)).plot_interval()
[t, 1, 2]
>>> Curve((x, sin(x)), (x, 1, 2)).plot_interval(s)
[s, 1, 2]
"""
t = _symbol(parameter, self.parameter)
return [t] + list(self.limits[1:])
def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of line.
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
plot_interval : list (plot interval)
[parameter, lower_bound, upper_bound]
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l1 = Line(p1, p2)
>>> l1.plot_interval()
[t, -5, 5]
"""
t = _symbol(parameter)
return [t, -5, 5]
def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of the Segment.
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
plot_interval : list
[parameter, lower_bound, upper_bound]
Examples
========
>>> from sympy import Point, Segment
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> s1 = Segment(p1, p2)
>>> s1.plot_interval()
[t, 0, 1]
"""
t = _symbol(parameter)
return [t, 0, 1]
def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of the Ellipse.
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
plot_interval : list
[parameter, lower_bound, upper_bound]
Examples
========
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.plot_interval()
[t, -pi, pi]
"""
t = _symbol(parameter)
return [t, -S.Pi, S.Pi]
def arbitrary_point(self, parameter="t"):
"""A parameterized point on the ellipse.
Parameters
----------
parameter : str, optional
Default value is 't'.
Returns
-------
arbitrary_point : Point
Raises
------
ValueError
When `parameter` already appears in the functions.
See Also
--------
Point
Examples
--------
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.arbitrary_point()
Point(3*cos(t), 2*sin(t))
"""
t = _symbol(parameter)
if t.name in (f.name for f in self.free_symbols):
raise ValueError("Symbol %s already appears in object and cannot be used as a parameter." % t.name)
return Point(self.center[0] + self.hradius * C.cos(t), self.center[1] + self.vradius * C.sin(t))
def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of the Ray.
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
plot_interval : list
[parameter, lower_bound, upper_bound]
Examples
========
>>> from sympy import Point, Ray, pi
>>> r = Ray((0, 0), angle=pi/4)
>>> r.plot_interval()
[t, 0, 5*sqrt(2)/(1 + 5*sqrt(2))]
"""
t = _symbol(parameter)
p = self.arbitrary_point(t)
# get a t corresponding to length of 10
want = 10
u = Segment(self.p1, p.subs(t, S.Half)).length # gives unit length
t_need = want/u
return [t, 0, t_need/(1 + t_need)]
def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of line. Gives
values that will produce a line that is +/- 5 units long (where a
unit is the distance between the two points that define the line).
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
plot_interval : list (plot interval)
[parameter, lower_bound, upper_bound]
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l1 = Line(p1, p2)
>>> l1.plot_interval()
[t, -5, 5]
"""
t = _symbol(parameter)
return [t, -5, 5]
def arbitrary_point(self, parameter='t'):
"""A parameterized point on the Segment.
Parameters
==========
parameter : str, optional
The name of the parameter which will be used for the parametric
point. The default value is 't'.
Returns
=======
point : Point
Parameters
==========
parameter : str, optional
The name of the parameter which will be used for the parametric
point. The default value is 't'.
Returns
=======
point : Point
Raises
======
ValueError
When ``parameter`` already appears in the Segment's definition.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Segment
>>> p1, p2 = Point(1, 0), Point(5, 3)
>>> s1 = Segment(p1, p2)
>>> s1.arbitrary_point()
Point(4*t + 1, 3*t)
"""
t = _symbol(parameter)
if t.name in (f.name for f in self.free_symbols):
raise ValueError('Symbol %s already appears in object and cannot be used as a parameter.' % t.name)
x = simplify(self.p1.x + t*(self.p2.x - self.p1.x))
y = simplify(self.p1.y + t*(self.p2.y - self.p1.y))
return Point(x, y)
def arbitrary_point(self, parameter='t'):
"""
A parameterized point on the curve.
Parameters
==========
parameter : str or Symbol, optional
Default value is 't';
the Curve's parameter is selected with None or self.parameter
otherwise the provided symbol is used.
Returns
=======
arbitrary_point : Point
Raises
======
ValueError
When `parameter` already appears in the functions.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Symbol
>>> from sympy.abc import s
>>> from sympy.geometry import Curve
>>> C = Curve([2*s, s**2], (s, 0, 2))
>>> C.arbitrary_point()
Point(2*t, t**2)
>>> C.arbitrary_point(C.parameter)
Point(2*s, s**2)
>>> C.arbitrary_point(None)
Point(2*s, s**2)
>>> C.arbitrary_point(Symbol('a'))
Point(2*a, a**2)
"""
if parameter is None:
return Point(*self.functions)
tnew = _symbol(parameter, self.parameter)
t = self.parameter
if (tnew.name != t.name and
tnew.name in (f.name for f in self.free_symbols)):
raise ValueError('Symbol %s already appears in object '
'and cannot be used as a parameter.' % tnew.name)
return Point(*[w.subs(t, tnew) for w in self.functions])
def equation(self, x='x', y='y'):
"""The equation of the line: ax + by + c.
Parameters
==========
x : str, optional
The name to use for the x-axis, default value is 'x'.
y : str, optional
The name to use for the y-axis, default value is 'y'.
Returns
=======
equation : sympy expression
See Also
========
LinearEntity.coefficients
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(1, 0), Point(5, 3)
>>> l1 = Line(p1, p2)
>>> l1.equation()
-3*x + 4*y + 3
"""
x, y = _symbol(x), _symbol(y)
p1, p2 = self.points
if p1.x == p2.x:
return x - p1.x
elif p1.y == p2.y:
return y - p1.y
a, b, c = self.coefficients
return simplify(a*x + b*y + c)
def arbitrary_point(self, parameter='t'):
"""A parameterized point on the polygon.
The parameter, varying from 0 to 1, assigns points to the position on
the perimeter that is that fraction of the total perimeter. So the
point evaluated at t=1/2 would return the point from the first vertex
that is 1/2 way around the polygon.
Parameters
----------
parameter : str, optional
Default value is 't'.
Returns
-------
arbitrary_point : Point
Raises
------
ValueError
When `parameter` already appears in the Polygon's definition.
See Also
--------
Point
Examples
--------
>>> from sympy import Polygon, S, Symbol
>>> t = Symbol('t', real=True)
>>> tri = Polygon((0, 0), (1, 0), (1, 1))
>>> p = tri.arbitrary_point('t')
>>> perimeter = tri.perimeter
>>> s1, s2 = [s.length for s in tri.sides[:2]]
>>> p.subs(t, (s1 + s2/2)/perimeter)
Point(1, 1/2)
"""
t = _symbol(parameter)
if t.name in (f.name for f in self.free_symbols):
raise ValueError('Symbol %s already appears in object and cannot be used as a parameter.' % t.name)
sides = []
perimeter = self.perimeter
perim_fraction_start = 0
for s in self.sides:
side_perim_fraction = s.length/perimeter
perim_fraction_end = perim_fraction_start + side_perim_fraction
pt = s.arbitrary_point(parameter).subs(
t, (t - perim_fraction_start)/side_perim_fraction)
sides.append((pt, (perim_fraction_start <= t < perim_fraction_end)))
perim_fraction_start = perim_fraction_end
return Piecewise(*sides)
def arbitrary_point(self, parameter='t'):
"""A parameterized point on the Segment.
Parameters
----------
parameter : str, optional
The name of the parameter which will be used for the parametric
point. The default value is 't'.
Returns
-------
point : Point
Parameters
----------
parameter : str, optional
The name of the parameter which will be used for the parametric
point. The default value is 't'.
Returns
-------
point : Point
Raises
------
ValueError
When `parameter` already appears in the Segment's definition.
See Also
--------
Point
Examples
--------
>>> from sympy import Point, Segment
>>> p1, p2 = Point(1, 0), Point(5, 3)
>>> s1 = Segment(p1, p2)
>>> s1.arbitrary_point()
Point(1 + 4*t, 3*t)
"""
t = _symbol(parameter)
if t.name in (f.name for f in self.free_symbols):
raise ValueError('Symbol %s already appears in object and cannot be used as a parameter.' % t.name)
x = simplify(self.p1[0] + t*(self.p2[0] - self.p1[0]))
y = simplify(self.p1[1] + t*(self.p2[1] - self.p1[1]))
return Point(x, y)
def arbitrary_point(self, parameter='t'):
"""A parameterized point on the Line.
Parameters
==========
parameter : str, optional
The name of the parameter which will be used for the parametric
point. The default value is 't'. When this parameter is 0, the
first point used to define the line will be returned, and when
it is 1 the second point will be returned.
Returns
=======
point : Point
Raises
======
ValueError
When ``parameter`` already appears in the Line's definition.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(1, 0), Point(5, 3)
>>> l1 = Line(p1, p2)
>>> l1.arbitrary_point()
Point(4*t + 1, 3*t)
"""
t = _symbol(parameter)
if t.name in (f.name for f in self.free_symbols):
raise ValueError('Symbol %s already appears in object '
'and cannot be used as a parameter.' % t.name)
x = simplify(self.p1.x + t*(self.p2.x - self.p1.x))
y = simplify(self.p1.y + t*(self.p2.y - self.p1.y))
return Point(x, y)
def random_point(self):
"""A random point on the ellipse.
Returns
=======
point : Point
See Also
========
sympy.geometry.point.Point
arbitrary_point : Returns parameterized point on ellipse
Notes
-----
A random point may not appear to be on the ellipse, ie, `p in e` may
return False. This is because the coordinates of the point will be
floating point values, and when these values are substituted into the
equation for the ellipse the result may not be zero because of floating
point rounding error.
Examples
========
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> p1 = e1.random_point()
>>> # a random point may not appear to be on the ellipse because of
>>> # floating point rounding error
>>> p1 in e1 # doctest: +SKIP
True
>>> p1 # doctest +ELLIPSIS
Point(...)
"""
from random import random
t = _symbol("t")
x, y = self.arbitrary_point(t).args
# get a random value in [-pi, pi)
subs_val = float(S.Pi) * (2 * random() - 1)
return Point(x.subs(t, subs_val), y.subs(t, subs_val))
def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of line.
Parameters
----------
parameter : str, optional
Default value is 't'.
Returns
-------
plot_interval : list (plot interval)
[parameter, lower_bound, upper_bound]
Examples
--------
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l1 = Line(p1, p2)
>>> l1.plot_interval()
[t, -5, 5]
"""
t = _symbol(parameter)
return [t, -5, 5]
def arbitrary_point(self, parameter='t'):
"""A parameterized point on the ellipse.
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
arbitrary_point : Point
Raises
======
ValueError
When `parameter` already appears in the functions.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.arbitrary_point()
Point(3*cos(t), 2*sin(t))
"""
t = _symbol(parameter)
if t.name in (f.name for f in self.free_symbols):
raise ValueError(
'Symbol %s already appears in object and cannot be used as a parameter.'
% t.name)
return Point(self.center[0] + self.hradius * C.cos(t),
self.center[1] + self.vradius * C.sin(t))
def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of the Segment.
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
plot_interval : list
[parameter, lower_bound, upper_bound]
Examples
========
>>> from sympy import Point, Segment
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> s1 = Segment(p1, p2)
>>> s1.plot_interval()
[t, 0, 1]
"""
t = _symbol(parameter)
return [t, 0, 1]
def __new__(cls, *args, **kwargs):
if kwargs.get('n', 0):
n = kwargs.pop('n')
args = list(args)
# return a virtual polygon with n sides
if len(args) == 2: # center, radius
args.append(n)
elif len(args) == 3: # center, radius, rotation
args.insert(2, n)
return RegularPolygon(*args, **kwargs)
vertices = [Point(a) for a in args]
# remove consecutive duplicates
nodup = []
for p in vertices:
if nodup and p == nodup[-1]:
continue
nodup.append(p)
if len(nodup) > 1 and nodup[-1] == nodup[0]:
nodup.pop() # last point was same as first
# remove collinear points unless they are shared points
got = set()
shared = set()
for p in nodup:
if p in got:
shared.add(p)
else:
got.add(p)
i = -3
while i < len(nodup) - 3 and len(nodup) > 2:
a, b, c = sorted([nodup[i], nodup[i + 1], nodup[i + 2]])
if b not in shared and Point.is_collinear(a, b, c):
nodup[i] = a
nodup[i + 1] = None
nodup.pop(i + 1)
i += 1
vertices = filter(lambda x: x is not None, nodup)
if len(vertices) > 3:
rv = GeometryEntity.__new__(cls, *vertices, **kwargs)
elif len(vertices) == 3:
return Triangle(*vertices, **kwargs)
elif len(vertices) == 2:
return Segment(*vertices, **kwargs)
else:
return Point(*vertices, **kwargs)
# reject polygons that have intersecting sides unless the
# intersection is a shared point or a generalized intersection.
# A self-intersecting polygon is easier to detect than a
# random set of segments since only those sides that are not
# part of the convex hull can possibly intersect with other
# sides of the polygon...but for now we use the n**2 algorithm
# and check all sides with intersection with any preceding sides
hit = _symbol('hit')
if not rv.is_convex:
sides = rv.sides
for i, si in enumerate(sides):
pts = si[0], si[1]
ai = si.arbitrary_point(hit)
for j in xrange(i):
sj = sides[j]
if sj[0] not in pts and sj[1] not in pts:
aj = si.arbitrary_point(hit)
tx = (solve(ai[0] - aj[0]) or [S.Zero])[0]
if tx.is_number and 0 <= tx <= 1:
ty = (solve(ai[1] - aj[1]) or [S.Zero])[0]
if (tx or ty) and ty.is_number and 0 <= ty <= 1:
print ai, aj
raise GeometryError("Polygon has intersecting sides.")
return rv
def random_point(self, seed=None):
"""A random point on the ellipse.
Returns
=======
point : Point
See Also
========
sympy.geometry.point.Point
arbitrary_point : Returns parameterized point on ellipse
Notes
-----
A random point may not appear to be on the ellipse, ie, `p in e` may
return False. This is because the coordinates of the point will be
floating point values, and when these values are substituted into the
equation for the ellipse the result may not be zero because of floating
point rounding error.
Examples
========
>>> from sympy import Point, Ellipse, Segment
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.random_point() # gives some random point
Point(...)
>>> p1 = e1.random_point(seed=0); p1.n(2)
Point(2.1, 1.4)
The random_point method assures that the point will test as being
in the ellipse:
>>> p1 in e1
True
Notes
=====
An arbitrary_point with a random value of t substituted into it may
not test as being on the ellipse because the expression tested that
a point is on the ellipse doesn't simplify to zero and doesn't evaluate
exactly to zero:
>>> from sympy.abc import t
>>> e1.arbitrary_point(t)
Point(3*cos(t), 2*sin(t))
>>> p2 = _.subs(t, 0.1)
>>> p2 in e1
False
Note that arbitrary_point routine does not take this approach. A value for
cos(t) and sin(t) (not t) is substituted into the arbitrary point. There is
a small chance that this will give a point that will not test as being
in the ellipse, so the process is repeated (up to 10 times) until a
valid point is obtained.
"""
from sympy import sin, cos, Rational
t = _symbol('t')
x, y = self.arbitrary_point(t).args
# get a random value in [-1, 1) corresponding to cos(t)
# and confirm that it will test as being in the ellipse
if seed is not None:
rng = random.Random(seed)
else:
rng = random
for i in range(10): # should be enough?
# simplify this now or else the Float will turn s into a Float
c = 2*Rational(rng.random()) - 1
s = sqrt(1 - c**2)
p1 = Point(x.subs(cos(t), c), y.subs(sin(t), s))
if p1 in self:
return p1
raise GeometryError(
'Having problems generating a point in the ellipse.')
def random_point(self, seed=None):
"""A random point on the ellipse.
Returns
=======
point : Point
See Also
========
sympy.geometry.point.Point
arbitrary_point : Returns parameterized point on ellipse
Notes
-----
A random point may not appear to be on the ellipse, ie, `p in e` may
return False. This is because the coordinates of the point will be
floating point values, and when these values are substituted into the
equation for the ellipse the result may not be zero because of floating
point rounding error.
Examples
========
>>> from sympy import Point, Ellipse, Segment
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.random_point() # gives some random point
Point(...)
>>> p1 = e1.random_point(seed=0); p1.n(2)
Point(2.1, 1.4)
The random_point method assures that the point will test as being
in the ellipse:
>>> p1 in e1
True
Notes
=====
An arbitrary_point with a random value of t substituted into it may
not test as being on the ellipse because the expression tested that
a point is on the ellipse doesn't simplify to zero and doesn't evaluate
exactly to zero:
>>> from sympy.abc import t
>>> e1.arbitrary_point(t)
Point(3*cos(t), 2*sin(t))
>>> p2 = _.subs(t, 0.1)
>>> p2 in e1
False
Note that arbitrary_point routine does not take this approach. A value for
cos(t) and sin(t) (not t) is substituted into the arbitrary point. There is
a small chance that this will give a point that will not test as being
in the ellipse, so the process is repeated (up to 10 times) until a
valid point is obtained.
"""
from sympy import sin, cos, Rational
t = _symbol('t')
x, y = self.arbitrary_point(t).args
# get a random value in [-1, 1) corresponding to cos(t)
# and confirm that it will test as being in the ellipse
if seed is not None:
rng = random.Random(seed)
else:
rng = random
for i in range(10): # should be enough?
# simplify this now or else the Float will turn s into a Float
c = 2 * Rational(rng.random()) - 1
s = sqrt(1 - c**2)
p1 = Point(x.subs(cos(t), c), y.subs(sin(t), s))
if p1 in self:
return p1
raise GeometryError(
'Having problems generating a point in the ellipse.')
def __new__(cls, *args, **kwargs):
if kwargs.get('n', 0):
n = kwargs.pop('n')
args = list(args)
# return a virtual polygon with n sides
if len(args) == 2: # center, radius
args.append(n)
elif len(args) == 3: # center, radius, rotation
args.insert(2, n)
return RegularPolygon(*args, **kwargs)
vertices = [Point(a) for a in args]
# remove consecutive duplicates
nodup = []
for p in vertices:
if nodup and p == nodup[-1]:
continue
nodup.append(p)
if len(nodup) > 1 and nodup[-1] == nodup[0]:
nodup.pop() # last point was same as first
# remove collinear points unless they are shared points
got = set()
shared = set()
for p in nodup:
if p in got:
shared.add(p)
else:
got.add(p)
i = -3
while i < len(nodup) - 3 and len(nodup) > 2:
a, b, c = sorted([nodup[i], nodup[i + 1], nodup[i + 2]])
if b not in shared and Point.is_collinear(a, b, c):
nodup[i] = a
nodup[i + 1] = None
nodup.pop(i + 1)
i += 1
vertices = filter(lambda x: x is not None, nodup)
if len(vertices) > 3:
rv = GeometryEntity.__new__(cls, *vertices, **kwargs)
elif len(vertices) == 3:
return Triangle(*vertices, **kwargs)
elif len(vertices) == 2:
return Segment(*vertices, **kwargs)
else:
return Point(*vertices, **kwargs)
# reject polygons that have intersecting sides unless the
# intersection is a shared point or a generalized intersection.
# A self-intersecting polygon is easier to detect than a
# random set of segments since only those sides that are not
# part of the convex hull can possibly intersect with other
# sides of the polygon...but for now we use the n**2 algorithm
# and check all sides with intersection with any preceding sides
hit = _symbol('hit')
if not rv.is_convex:
sides = rv.sides
for i, si in enumerate(sides):
pts = si[0], si[1]
ai = si.arbitrary_point(hit)
for j in xrange(i):
sj = sides[j]
if sj[0] not in pts and sj[1] not in pts:
aj = si.arbitrary_point(hit)
tx = (solve(ai[0] - aj[0]) or [S.Zero])[0]
if tx.is_number and 0 <= tx <= 1:
ty = (solve(ai[1] - aj[1]) or [S.Zero])[0]
if (tx or ty) and ty.is_number and 0 <= ty <= 1:
print ai, aj
raise GeometryError(
"Polygon has intersecting sides.")
return rv
def arbitrary_point(self, parameter='t'):
"""A parameterized point on the Ray.
Parameters
----------
parameter : str, optional
The name of the parameter which will be used for the parametric
point. The default value is 't'.
Returns
-------
point : Point
Raises
------
ValueError
When `parameter` already appears in the Ray's definition.
See Also
--------
Point
Examples
--------
>>> from sympy import Ray, Point, Segment, S, simplify, solve
>>> from sympy.abc import t
>>> r = Ray(Point(0, 0), Point(2, 3))
>>> p = r.arbitrary_point(t)
The parameter `t` used in the arbitrary point maps 0 to the
origin of the ray and 1 to the end of the ray at infinity
(which will show up as NaN).
>>> p.subs(t, 0), p.subs(t, 1)
(Point(0, 0), Point(oo, oo))
The unit that `t` moves you is based on the spacing of the
points used to define the ray.
>>> p.subs(t, 1/(S(1) + 1)) # one unit
Point(2, 3)
>>> p.subs(t, 2/(S(1) + 2)) # two units out
Point(4, 6)
>>> p.subs(t, S.Half/(S(1) + S.Half)) # half a unit out
Point(1, 3/2)
If you want to be located a distance of 1 from the origin of the
ray, what value of `t` is needed?
a) find the unit length and pick t accordingly
>>> u = Segment(r[0], p.subs(t, S.Half)).length # S.Half = 1/(1 + 1)
>>> want = 1
>>> t_need = want/u
>>> p_want = p.subs(t, t_need/(1 + t_need))
>>> simplify(Segment(r[0], p_want).length)
1
b) find the t that makes the length from origin to p equal to 1
>>> l = Segment(r[0], p).length
>>> t_need = solve(l**2 - want**2, t) # use the square to remove abs() if it is there
>>> t_need = [w for w in t_need if w.n() > 0][0] # take positive t
>>> p_want = p.subs(t, t_need)
>>> simplify(Segment(r[0], p_want).length)
1
"""
t = _symbol(parameter)
if t.name in (f.name for f in self.free_symbols):
raise ValueError(
'Symbol %s already appears in object and cannot be used as a parameter.'
% t.name)
m = self.slope
x = simplify(self.p1[0] + t / (1 - t) * (self.p2[0] - self.p1[0]))
y = simplify(self.p1[1] + t / (1 - t) * (self.p2[1] - self.p1[1]))
return Point(x, y)
def arbitrary_point(self, parameter='t'):
"""A parameterized point on the Ray.
Parameters
==========
parameter : str, optional
The name of the parameter which will be used for the parametric
point. The default value is 't'.
Returns
=======
point : Point
Raises
======
ValueError
When ``parameter`` already appears in the Ray's definition.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Ray, Point, Segment, S, simplify, solve
>>> from sympy.abc import t
>>> r = Ray(Point(0, 0), Point(2, 3))
>>> p = r.arbitrary_point(t)
The parameter `t` used in the arbitrary point maps 0 to the
origin of the ray and 1 to the end of the ray at infinity
(which will show up as NaN).
>>> p.subs(t, 0), p.subs(t, 1)
(Point(0, 0), Point(oo, oo))
The unit that `t` moves you is based on the spacing of the
points used to define the ray.
>>> p.subs(t, 1/(S(1) + 1)) # one unit
Point(2, 3)
>>> p.subs(t, 2/(S(1) + 2)) # two units out
Point(4, 6)
>>> p.subs(t, S.Half/(S(1) + S.Half)) # half a unit out
Point(1, 3/2)
If you want to be located a distance of 1 from the origin of the
ray, what value of `t` is needed?
a) Find the unit length and pick `t` accordingly.
>>> u = Segment(r.p1, p.subs(t, S.Half)).length # S.Half = 1/(1 + 1)
>>> want = 1
>>> t_need = want/u
>>> p_want = p.subs(t, t_need/(1 + t_need))
>>> simplify(Segment(r.p1, p_want).length)
1
b) Find the `t` that makes the length from origin to `p` equal to 1.
>>> l = Segment(r.p1, p).length
>>> t_need = solve(l**2 - want**2, t) # use the square to remove abs() if it is there
>>> t_need = [w for w in t_need if w.n() > 0][0] # take positive t
>>> p_want = p.subs(t, t_need)
>>> simplify(Segment(r.p1, p_want).length)
1
"""
t = _symbol(parameter)
if t.name in (f.name for f in self.free_symbols):
raise ValueError('Symbol %s already appears in object and cannot be used as a parameter.' % t.name)
x = simplify(self.p1.x + t/(1 - t)*(self.p2.x - self.p1.x))
y = simplify(self.p1.y + t/(1 - t)*(self.p2.y - self.p1.y))
return Point(x, y)