/usr/share/pyshared/sympy/geometry/line.py is in python-sympy 0.7.1.rc1-3.
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1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 | """Line-like geometrical entities.
Contains
--------
LinearEntity
Line
Ray
Segment
"""
from sympy.core import S, C, sympify, Dummy
from sympy.functions.elementary.trigonometric import _pi_coeff as pi_coeff
from sympy.core.numbers import Float, Rational
from sympy.simplify import simplify
from sympy.solvers import solve
from sympy.geometry.exceptions import GeometryError
from entity import GeometryEntity
from point import Point
from util import _symbol
class LinearEntity(GeometryEntity):
"""An abstract base class for all linear entities (line, ray and segment)
in a 2-dimensional Euclidean space.
Attributes
----------
p1
p2
coefficients
slope
points
Notes
-----
This is an abstract class and is not meant to be instantiated.
Subclasses should implement the following methods:
__eq__
__contains__
"""
def __new__(cls, p1, p2, **kwargs):
p1 = Point(p1)
p2 = Point(p2)
if p1 == p2:
# Rolygon returns lower priority classes...should LinearEntity, too?
return p1 # raise ValueError("%s.__new__ requires two unique Points." % cls.__name__)
return GeometryEntity.__new__(cls, p1, p2, **kwargs)
@property
def p1(self):
"""The first defining point of a linear entity.
See Also
--------
Point
Examples
--------
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l = Line(p1, p2)
>>> l.p1
Point(0, 0)
"""
return self.__getitem__(0)
@property
def p2(self):
"""The second defining point of a linear entity.
See Also
--------
Point
Examples
--------
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l = Line(p1, p2)
>>> l.p2
Point(5, 3)
"""
return self.__getitem__(1)
@property
def coefficients(self):
"""The coefficients (a, b, c) for the linear equation
ax + by + c = 0.
Examples
--------
>>> from sympy import Point, Line
>>> from sympy.abc import x, y
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l = Line(p1, p2)
>>> l.coefficients
(-3, 5, 0)
>>> p3 = Point(x, y)
>>> l2 = Line(p1, p3)
>>> l2.coefficients
(-y, x, 0)
"""
p1, p2 = self.points
if p1[0] == p2[0]:
return (S.One, S.Zero, -p1[0])
elif p1[1] == p2[1]:
return (S.Zero, S.One, -p1[1])
return (self.p1[1]-self.p2[1],
self.p2[0]-self.p1[0],
self.p1[0]*self.p2[1] - self.p1[1]*self.p2[0])
def is_concurrent(*lines):
"""Is a sequence of linear entities concurrent?
Two or more linear entities are concurrent if they all
intersect at a single point.
Parameters
----------
lines : a sequence of linear entities.
Returns
-------
True if the set of linear entities are concurrent, False
otherwise.
Notes
-----
Simply take the first two lines and find their intersection.
If there is no intersection, then the first two lines were
parallel and had no intersection so concurrency is impossible
amongst the whole set. Otherwise, check to see if the
intersection point of the first two lines is a member on
the rest of the lines. If so, the lines are concurrent.
Examples
--------
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(3, 5)
>>> p3, p4 = Point(-2, -2), Point(0, 2)
>>> l1, l2, l3 = Line(p1, p2), Line(p1, p3), Line(p1, p4)
>>> l1.is_concurrent(l2, l3)
True
>>> l4 = Line(p2, p3)
>>> l4.is_concurrent(l2, l3)
False
"""
# Concurrency requires intersection at a single point; One linear
# entity cannot be concurrent.
if len(lines) <= 1:
return False
try:
# Get the intersection (if parallel)
p = lines[0].intersection(lines[1])
if len(p) == 0: return False
# Make sure the intersection is on every linear entity
for line in lines[2:]:
if p[0] not in line:
return False
return True
except AttributeError:
return False
def is_parallel(l1, l2):
"""Are two linear entities parallel?
Parameters
----------
l1 : LinearEntity
l2 : LinearEntity
Returns
-------
True if l1 and l2 are parallel, False otherwise.
Examples
--------
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(1, 1)
>>> p3, p4 = Point(3, 4), Point(6, 7)
>>> l1, l2 = Line(p1, p2), Line(p3, p4)
>>> Line.is_parallel(l1, l2)
True
>>> p5 = Point(6, 6)
>>> l3 = Line(p3, p5)
>>> Line.is_parallel(l1, l3)
False
"""
try:
a1, b1, c1 = l1.coefficients
a2, b2, c2 = l2.coefficients
return bool(simplify(a1*b2 - b1*a2) == 0)
except AttributeError:
return False
def is_perpendicular(l1, l2):
"""Are two linear entities parallel?
Parameters
----------
l1 : LinearEntity
l2 : LinearEntity
Returns
-------
True if l1 and l2 are perpendicular, False otherwise.
Examples
--------
>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(-1, 1)
>>> l1, l2 = Line(p1, p2), Line(p1, p3)
>>> l1.is_perpendicular(l2)
True
>>> p4 = Point(5, 3)
>>> l3 = Line(p1, p4)
>>> l1.is_perpendicular(l3)
False
"""
try:
a1, b1, c1 = l1.coefficients
a2, b2, c2 = l2.coefficients
return bool(simplify(a1*a2 + b1*b2) == 0)
except AttributeError:
return False
def angle_between(l1, l2):
"""The angle formed between the two linear entities.
Parameters
----------
l1 : LinearEntity
l2 : LinearEntity
Returns
-------
angle : angle in radians
Notes
-----
From the dot product of vectors v1 and v2 it is known that:
dot(v1, v2) = |v1|*|v2|*cos(A)
where A is the angle formed between the two vectors. We can
get the directional vectors of the two lines and readily
find the angle between the two using the above formula.
Examples
--------
>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(0, 4), Point(2, 0)
>>> l1, l2 = Line(p1, p2), Line(p1, p3)
>>> l1.angle_between(l2)
pi/2
"""
v1 = l1.p2 - l1.p1
v2 = l2.p2 - l2.p1
return C.acos((v1[0]*v2[0] + v1[1]*v2[1]) / (abs(v1)*abs(v2)))
def parallel_line(self, p):
"""Create a new Line parallel to this linear entity which passes
through the point `p`.
Parameters
----------
p : Point
Returns
-------
line : Line
Examples
--------
>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(2, 3), Point(-2, 2)
>>> l1 = Line(p1, p2)
>>> l2 = l1.parallel_line(p3)
>>> p3 in l2
True
>>> l1.is_parallel(l2)
True
"""
d = self.p1 - self.p2
return Line(p, p + d)
def perpendicular_line(self, p):
"""Create a new Line perpendicular to this linear entity which passes
through the point `p`.
Parameters
----------
p : Point
Returns
-------
line : Line
Examples
--------
>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(2, 3), Point(-2, 2)
>>> l1 = Line(p1, p2)
>>> l2 = l1.perpendicular_line(p3)
>>> p3 in l2
True
>>> l1.is_perpendicular(l2)
True
"""
d1, d2 = self.p1 - self.p2
if d2 == 0: # If an horizontal line
if p[1] == self.p1[1]: # if p is on this linear entity
p2 = Point(p[0], p[1] + 1)
return Line(p, p2)
else:
p2 = Point(p[0], self.p1[1])
return Line(p, p2)
else:
p2 = Point(p[0] - d2, p[1] + d1)
return Line(p, p2)
def perpendicular_segment(self, p):
"""Create a perpendicular line segment from `p` to this line.
Parameters
----------
p : Point
Returns
-------
segment : Segment
Notes
-----
Returns `p` itself if `p` is on this linear entity.
Examples
--------
>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, 2)
>>> l1 = Line(p1, p2)
>>> s1 = l1.perpendicular_segment(p3)
>>> l1.is_perpendicular(s1)
True
>>> p3 in s1
True
"""
if p in self:
return p
pl = self.perpendicular_line(p)
p2 = self.intersection(pl)[0]
return Segment(p, p2)
@property
def length(self):
return S.Infinity
@property
def slope(self):
"""The slope of this linear entity, or infinity if vertical.
Returns
-------
slope : number or sympy expression
Examples
--------
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(3, 5)
>>> l1 = Line(p1, p2)
>>> l1.slope
5/3
>>> p3 = Point(0, 4)
>>> l2 = Line(p1, p3)
>>> l2.slope
oo
"""
d1, d2 = self.p1 - self.p2
if d1 == 0:
return S.Infinity
return simplify(d2/d1)
@property
def points(self):
"""The two points used to define this linear entity.
Returns
-------
points : tuple of Points
See Also
--------
Point
Examples
--------
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 11)
>>> l1 = Line(p1, p2)
>>> l1.points
(Point(0, 0), Point(5, 11))
"""
return (self.p1, self.p2)
def projection(self, o):
"""Project a point, line, ray, or segment onto this linear entity.
Parameters
----------
other : Point or LinearEntity (Line, Ray, Segment)
Returns
-------
projection : Point or LinearEntity (Line, Ray, Segment)
The return type matches the type of the parameter `other`.
Raises
------
GeometryError
When method is unable to perform projection.
See Also
--------
Point
Notes
-----
A projection involves taking the two points that define
the linear entity and projecting those points onto a
Line and then reforming the linear entity using these
projections.
A point P is projected onto a line L by finding the point
on L that is closest to P. This is done by creating a
perpendicular line through P and L and finding its
intersection with L.
Examples
--------
>>> from sympy import Point, Line, Segment, Rational
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(Rational(1, 2), 0)
>>> l1 = Line(p1, p2)
>>> l1.projection(p3)
Point(1/4, 1/4)
>>> p4, p5 = Point(10, 0), Point(12, 1)
>>> s1 = Segment(p4, p5)
>>> l1.projection(s1)
Segment(Point(5, 5), Point(13/2, 13/2))
"""
tline = Line(self.p1, self.p2)
def project(p):
"""Project a point onto the line representing self."""
if p in tline:
return p
l1 = tline.perpendicular_line(p)
return tline.intersection(l1)[0]
projected = None
if isinstance(o, Point):
return project(o)
elif isinstance(o, LinearEntity):
n_p1 = project(o.p1)
n_p2 = project(o.p2)
if n_p1 == n_p2:
projected = n_p1
else:
projected = o.__class__(n_p1, n_p2)
# Didn't know how to project so raise an error
if projected is None:
n1 = self.__class__.__name__
n2 = o.__class__.__name__
raise GeometryError("Do not know how to project %s onto %s" % (n2, n1))
return self.intersection(projected)[0]
def intersection(self, o):
"""The intersection with another geometrical entity.
Parameters
----------
o : Point or LinearEntity
Returns
-------
intersection : list of geometrical entities
Examples
--------
>>> from sympy import Point, Line, Segment
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(7, 7)
>>> l1 = Line(p1, p2)
>>> l1.intersection(p3)
[Point(7, 7)]
>>> p4, p5 = Point(5, 0), Point(0, 3)
>>> l2 = Line(p4, p5)
>>> l1.intersection(l2)
[Point(15/8, 15/8)]
>>> p6, p7 = Point(0, 5), Point(2, 6)
>>> s1 = Segment(p6, p7)
>>> l1.intersection(s1)
[]
"""
if isinstance(o, Point):
if o in self:
return [o]
else:
return []
elif isinstance(o, LinearEntity):
a1, b1, c1 = self.coefficients
a2, b2, c2 = o.coefficients
t = simplify(a1*b2 - a2*b1)
if t == 0: # are parallel?
if isinstance(self, Line):
if o.p1 in self:
return [o]
return []
elif isinstance(o, Line):
if self.p1 in o:
return [self]
return []
elif isinstance(self, Ray):
if isinstance(o, Ray):
# case 1, rays in the same direction
if self.xdirection == o.xdirection:
if self.source[0] < o.source[0]:
return [o]
return [self]
# case 2, rays in the opposite directions
else:
if o.source in self:
if self.source == o.source:
return [self.source]
return [Segment(o.source, self.source)]
return []
elif isinstance(o, Segment):
if o.p1 in self:
if o.p2 in self:
return [o]
return [Segment(o.p1, self.source)]
elif o.p2 in self:
return [Segment(o.p2, self.source)]
return []
elif isinstance(self, Segment):
if isinstance(o, Ray):
return o.intersection(self)
elif isinstance(o, Segment):
# A reminder that the points of Segments are ordered
# in such a way that the following works. See
# Segment.__new__ for details on the ordering.
if self.p1 not in o:
if self.p2 not in o:
# Neither of the endpoints are in o so either
# o is contained in this segment or it isn't
if o in self:
return [self]
return []
else:
# p1 not in o but p2 is. Either there is a
# segment as an intersection, or they only
# intersect at an endpoint
if self.p2 == o.p1:
return [o.p1]
return [Segment(o.p1, self.p2)]
elif self.p2 not in o:
# p2 not in o but p1 is. Either there is a
# segment as an intersection, or they only
# intersect at an endpoint
if self.p1 == o.p2:
return [o.p2]
return [Segment(o.p2, self.p1)]
# Both points of self in o so the whole segment
# is in o
return [self]
# Unknown linear entity
return []
# Not parallel, so find the point of intersection
px = simplify((b1*c2 - c1*b2) / t)
py = simplify((a2*c1 - a1*c2) / t)
inter = Point(px, py)
if inter in self and inter in o:
return [inter]
return []
return o.intersection(self)
def random_point(self):
"""A random point on a LinearEntity.
Returns
-------
point : Point
See Also
--------
Point
Examples
--------
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l1 = Line(p1, p2)
>>> p3 = l1.random_point()
>>> # random point - don't know its coords in advance
>>> p3 # doctest: +ELLIPSIS
Point(...)
>>> # point should belong to the line
>>> p3 in l1
True
"""
from random import randint
# The lower and upper
lower, upper = -2**32 - 1, 2**32
if self.slope is S.Infinity:
if isinstance(self, Ray):
if self.ydirection is S.Infinity:
lower = self.p1[1]
else:
upper = self.p1[1]
elif isinstance(self, Segment):
lower = self.p1[1]
upper = self.p2[1]
x = self.p1[0]
y = randint(lower, upper)
else:
if isinstance(self, Ray):
if self.xdirection is S.Infinity:
lower = self.p1[0]
else:
upper = self.p1[0]
elif isinstance(self, Segment):
lower = self.p1[0]
upper = self.p2[0]
a, b, c = self.coefficients
x = randint(lower, upper)
y = simplify((-c - a*x) / b)
return Point(x, y)
def is_similar(self, other):
"""Return True if self and other are contained in the same line."""
def norm(a, b, c):
if a != 0:
return 1, b/a, c/a
elif b != 0:
return a/b, 1, c/b
else:
return c
return norm(*self.coefficients) == norm(*other.coefficients)
def __eq__(self, other):
"""Subclasses should implement this method."""
raise NotImplementedError()
def __hash__(self):
return super(LinearEntity, self).__hash__()
class Line(LinearEntity):
"""An infinite line in space.
A line is declared with two distinct points or a point and slope
as defined using keyword `slope`.
Note
----
At the moment only lines in a 2D space can be declared, because
Points can be defined only for 2D spaces.
Parameters
----------
p1 : Point
pt : Point
slope: sympy expression
See Also
--------
Point
Examples
--------
>>> import sympy
>>> from sympy import Point
>>> from sympy.abc import L
>>> from sympy.geometry import Line
>>> L = Line(Point(2,3), Point(3,5))
>>> L
Line(Point(2, 3), Point(3, 5))
>>> L.points
(Point(2, 3), Point(3, 5))
>>> L.equation()
-2*x + y + 1
>>> L.coefficients
(-2, 1, 1)
Instantiate with keyword `slope`:
>>> Line(Point(0, 0), slope=2)
Line(Point(0, 0), Point(1, 2))
"""
def __new__(cls, p1, pt=None, slope=None, **kwargs):
p1 = Point(p1)
if pt and slope is None:
try:
p2 = Point(pt)
except NotImplementedError:
raise ValueError('The 2nd argument was not a valid Point; if it was meant to be a slope it should be given with keyword "slope".')
if p1 == p2:
raise ValueError('A line requires two distinct points.')
elif slope and pt is None:
slope = sympify(slope)
if slope.is_bounded is False:
# when unbounded slope, don't change x
p2 = p1 + Point(0, 1)
else:
# go over 1 up slope
p2 = p1 + Point(1, slope)
else:
raise ValueError('A 2nd Point or keyword "slope" must be used.')
return LinearEntity.__new__(cls, p1, p2, **kwargs)
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'.
Returns
-------
point : Point
Raises
------
ValueError
When `parameter` already appears in the Line's definition.
See Also
--------
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[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 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 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 __contains__(self, o):
"""Return True if o is on this Line, or False otherwise."""
if isinstance(o, Point):
return Point.is_collinear(self.p1, self.p2, o)
elif not isinstance(o, LinearEntity):
return False
elif isinstance(o, Line):
return self.__eq__(o)
elif not self.is_similar(o):
return False
else:
return o[0] in self and o[1] in self
def __eq__(self, other):
"""Return True if other is equal to this Line, or False otherwise."""
if not isinstance(other, Line):
return False
return Point.is_collinear(self.p1, self.p2, other.p1, other.p2)
def __hash__(self):
return super(Line, self).__hash__()
class Ray(LinearEntity):
"""A Ray is a semi-line in the space with a source point and a direction.
Paramaters
----------
p1 : Point
The source of the Ray
p2 : Point or radian value
This point determines the direction in which the Ray propagates.
If given as an angle it is interpreted in radians with the positive
direction being ccw.
Attributes
----------
source
xdirection
ydirection
See Also
--------
Point
Notes
-----
At the moment only rays in a 2D space can be declared, because
Points can be defined only for 2D spaces.
Examples
--------
>>> import sympy
>>> from sympy import Point, pi
>>> from sympy.abc import r
>>> from sympy.geometry import Ray
>>> r = Ray(Point(2, 3), Point(3, 5))
>>> r = Ray(Point(2, 3), Point(3, 5))
>>> r
Ray(Point(2, 3), Point(3, 5))
>>> r.points
(Point(2, 3), Point(3, 5))
>>> r.source
Point(2, 3)
>>> r.xdirection
oo
>>> r.ydirection
oo
>>> r.slope
2
>>> Ray(Point(0, 0), angle=pi/4).slope
1
"""
def __new__(cls, p1, pt=None, angle=None, **kwargs):
p1 = Point(p1)
if pt and angle is None:
try:
p2 = Point(pt)
except NotImplementedError:
raise ValueError('The 2nd argument was not a valid Point;\nif it was meant to be an angle it should be given with keyword "angle".')
if p1 == p2:
raise ValueError('A Ray requires two distinct points.')
elif angle is not None and pt is None:
# we need to know if the angle is an odd multiple of pi/2
c = pi_coeff(sympify(angle))
p2 = None
if c is not None:
if c.is_Rational:
if c.q == 2:
if c.p == 1:
p2 = p1 + Point(0, 1)
elif c.p == 3:
p2 = p1 + Point(0, -1)
elif c.q == 1:
if c.p == 0:
p2 = p1 + Point(1, 0)
elif c.p == 1:
p2 = p1 + Point(-1, 0)
if p2 is None:
c *= S.Pi
else:
c = angle
if not p2:
p2 = p1 + Point(1, C.tan(c))
else:
raise ValueError('A 2nd point or keyword "angle" must be used.')
return LinearEntity.__new__(cls, p1, p2, **kwargs)
@property
def source(self):
"""The point from which the ray emanates.
Examples
--------
>>> from sympy import Point, Ray
>>> p1, p2 = Point(0, 0), Point(4, 1)
>>> r1 = Ray(p1, p2)
>>> r1.source
Point(0, 0)
"""
return self.p1
@property
def xdirection(self):
"""The x direction of the ray.
Positive infinity if the ray points in the positive x direction,
negative infinity if the ray points in the negative x direction,
or 0 if the ray is vertical.
Examples
--------
>>> from sympy import Point, Ray
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, -1)
>>> r1, r2 = Ray(p1, p2), Ray(p1, p3)
>>> r1.xdirection
oo
>>> r2.xdirection
0
"""
if self.p1[0] < self.p2[0]:
return S.Infinity
elif self.p1[0] == self.p2[0]:
return S.Zero
else:
return S.NegativeInfinity
@property
def ydirection(self):
"""The y direction of the ray.
Positive infinity if the ray points in the positive y direction,
negative infinity if the ray points in the negative y direction,
or 0 if the ray is horizontal.
Examples
--------
>>> from sympy import Point, Ray
>>> p1, p2, p3 = Point(0, 0), Point(-1, -1), Point(-1, 0)
>>> r1, r2 = Ray(p1, p2), Ray(p1, p3)
>>> r1.ydirection
-oo
>>> r2.ydirection
0
"""
if self.p1[1] < self.p2[1]:
return S.Infinity
elif self.p1[1] == self.p2[1]:
return S.Zero
else:
return S.NegativeInfinity
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 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*2**(1/2)/(1 + 5*2**(1/2))]
"""
t = _symbol(parameter)
p = self.arbitrary_point(t)
# get a t corresponding to length of 10
want = 10
u = Segment(self[0], p.subs(t, S.Half)).length # gives unit length
t_need = want/u
return [t, 0, t_need/(1 + t_need)]
def __eq__(self, other):
"""Is the other GeometryEntity equal to this Ray?"""
if not isinstance(other, Ray):
return False
return (self.source == other.source) and (other.p2 in self)
def __hash__(self):
return super(Ray, self).__hash__()
def __contains__(self, o):
"""Is other GeometryEntity contained in this Ray?"""
if isinstance(o, Ray):
d = o.p2 - o.p1
return (Point.is_collinear(self.p1, self.p2, o.p1, o.p2)
and (self.xdirection == o.xdirection)
and (self.ydirection == o.ydirection))
elif isinstance(o, Segment):
return (o.p1 in self) and (o.p2 in self)
elif isinstance(o, Point):
if Point.is_collinear(self.p1, self.p2, o):
if (not self.p1[0].has(C.Symbol) and not self.p1[1].has(C.Symbol)
and not self.p2[0].has(C.Symbol) and not self.p2[1].has(C.Symbol)):
if self.xdirection is S.Infinity:
return o[0] >= self.source[0]
elif self.xdirection is S.NegativeInfinity:
return o[0] <= self.source[0]
elif self.ydirection is S.Infinity:
return o[1] >= self.source[1]
return o[1] <= self.source[1]
else:
# There are symbols lying around, so assume that o
# is contained in this ray (for now)
return True
else:
# Points are not collinear, so the rays are not parallel
# and hence it isimpossible for self to contain o
return False
# No other known entity can be contained in a Ray
return False
class Segment(LinearEntity):
"""An undirected line segment in space.
Parameters
----------
p1 : Point
p2 : Point
Attributes
----------
length : number or sympy expression
midpoint : Point
See Also
--------
Point
Notes
-----
At the moment only segments in a 2D space can be declared, because
Points can be defined only for 2D spaces.
Examples
--------
>>> import sympy
>>> from sympy import Point
>>> from sympy.abc import s
>>> from sympy.geometry import Segment
>>> Segment((1, 0), (1, 1)) # tuples are interpreted as pts
Segment(Point(1, 0), Point(1, 1))
>>> s = Segment(Point(4, 3), Point(1, 1))
>>> s
Segment(Point(1, 1), Point(4, 3))
>>> s.points
(Point(1, 1), Point(4, 3))
>>> s.slope
2/3
>>> s.length
13**(1/2)
>>> s.midpoint
Point(5/2, 2)
"""
def __new__(cls, p1, p2, **kwargs):
# Reorder the two points under the following ordering:
# if p1[0] != p2[0] then p1[0] < p2[0]
# if p1[0] == p2[0] then p1[1] < p2[1]
p1 = Point(p1)
p2 = Point(p2)
if p1 == p2:
return Point(p1)
if p1[0] > p2[0]:
p1, p2 = p2, p1
elif p1[0] == p2[0] and p1[1] > p2[0]:
p1, p2 = p2, p1
return LinearEntity.__new__(cls, p1, p2, **kwargs)
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(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[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 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 perpendicular_bisector(self, p=None):
"""The perpendicular bisector of this segment.
If no point is specified or the point specified is not on the
bisector then the bisector is returned as a Line. Otherwise a
Segment is returned that joins the point specified and the
intersection of the bisector and the segment.
Parameters
----------
p : Point
Returns
-------
bisector : Line or Segment
Examples
--------
>>> from sympy import Point, Segment
>>> p1, p2, p3 = Point(0, 0), Point(6, 6), Point(5, 1)
>>> s1 = Segment(p1, p2)
>>> s1.perpendicular_bisector()
Line(Point(3, 3), Point(9, -3))
>>> s1.perpendicular_bisector(p3)
Segment(Point(3, 3), Point(5, 1))
"""
l = LinearEntity.perpendicular_line(self, self.midpoint)
if p is None or p not in l:
return l
else:
return Segment(self.midpoint, p)
@property
def length(self):
"""The length of the line segment.
Examples
--------
>>> from sympy import Point, Segment
>>> p1, p2 = Point(0, 0), Point(4, 3)
>>> s1 = Segment(p1, p2)
>>> s1.length
5
"""
return Point.distance(self.p1, self.p2)
@property
def midpoint(self):
"""The midpoint of the line segment.
Examples
--------
>>> from sympy import Point, Segment
>>> p1, p2 = Point(0, 0), Point(4, 3)
>>> s1 = Segment(p1, p2)
>>> s1.midpoint
Point(2, 3/2)
"""
return Point.midpoint(self.p1, self.p2)
def distance(self, o):
"""Attempts to find the distance of the line segment to an object"""
if isinstance(o, Point):
return self._do_point_distance(o)
raise NotImplementedError()
def _do_point_distance(self, pt):
"""Calculates the distance between a point and a line segment"""
seg_vector = Point(self.p2[0] - self.p1[0], self.p2[1] - self.p1[1])
pt_vector = Point(pt[0] - self.p1[0], pt[1] - self.p1[1])
t = (seg_vector[0]*pt_vector[0] + seg_vector[1]*pt_vector[1])/self.length**2
if t >= 1:
distance = Point.distance(self.p2, pt)
elif t <= 0:
distance = Point.distance(self.p1, pt)
else:
distance = Point.distance(self.p1 + Point(t*seg_vector[0], t*seg_vector[1]), pt)
return distance
def __eq__(self, other):
"""Is the other GeometryEntity equal to this Ray?"""
if not isinstance(other, Segment):
return False
return (self.p1 == other.p1) and (self.p2 == other.p2)
def __hash__(self):
return super(Segment, self).__hash__()
def __contains__(self, o):
"""Is the other GeometryEntity contained within this Ray?"""
if isinstance(o, Segment):
return o.p1 in self and o.p2 in self
elif isinstance(o, Point):
if Point.is_collinear(self.p1, self.p2, o):
t = Dummy('t')
x, y = self.arbitrary_point(t)
if self.p1.x != self.p2.x:
ti = solve(x - o.x, t)[0]
else:
ti = solve(y - o.y, t)[0]
if ti.is_number:
return 0 <= ti <= 1
return None
# No other known entity can be contained in a Ray
return False
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