You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
425 lines
9.9 KiB
425 lines
9.9 KiB
5 months ago
|
"""Curves in 2-dimensional Euclidean space.
|
||
|
|
|
||
|
|
Contains
|
||
|
|
========
|
||
|
|
Curve
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
from sympy.functions.elementary.miscellaneous import sqrt
|
||
|
|
from sympy.core import diff
|
||
|
|
from sympy.core.containers import Tuple
|
||
|
|
from sympy.core.symbol import _symbol
|
||
|
|
from sympy.geometry.entity import GeometryEntity, GeometrySet
|
||
|
|
from sympy.geometry.point import Point
|
||
|
|
from sympy.integrals import integrate
|
||
|
|
from sympy.matrices import Matrix, rot_axis3
|
||
|
|
from sympy.utilities.iterables import is_sequence
|
||
|
|
|
||
|
|
from mpmath.libmp.libmpf import prec_to_dps
|
||
|
|
|
||
|
|
|
||
|
|
class Curve(GeometrySet):
|
||
|
|
"""A curve in space.
|
||
|
|
|
||
|
|
A curve is defined by parametric functions for the coordinates, a
|
||
|
|
parameter and the lower and upper bounds for the parameter value.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
function : list of functions
|
||
|
|
limits : 3-tuple
|
||
|
|
Function parameter and lower and upper bounds.
|
||
|
|
|
||
|
|
Attributes
|
||
|
|
==========
|
||
|
|
|
||
|
|
functions
|
||
|
|
parameter
|
||
|
|
limits
|
||
|
|
|
||
|
|
Raises
|
||
|
|
======
|
||
|
|
|
||
|
|
ValueError
|
||
|
|
When `functions` are specified incorrectly.
|
||
|
|
When `limits` are specified incorrectly.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Curve, sin, cos, interpolate
|
||
|
|
>>> from sympy.abc import t, a
|
||
|
|
>>> C = Curve((sin(t), cos(t)), (t, 0, 2))
|
||
|
|
>>> C.functions
|
||
|
|
(sin(t), cos(t))
|
||
|
|
>>> C.limits
|
||
|
|
(t, 0, 2)
|
||
|
|
>>> C.parameter
|
||
|
|
t
|
||
|
|
>>> C = Curve((t, interpolate([1, 4, 9, 16], t)), (t, 0, 1)); C
|
||
|
|
Curve((t, t**2), (t, 0, 1))
|
||
|
|
>>> C.subs(t, 4)
|
||
|
|
Point2D(4, 16)
|
||
|
|
>>> C.arbitrary_point(a)
|
||
|
|
Point2D(a, a**2)
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
sympy.core.function.Function
|
||
|
|
sympy.polys.polyfuncs.interpolate
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
def __new__(cls, function, limits):
|
||
|
|
if not is_sequence(function) or len(function) != 2:
|
||
|
|
raise ValueError("Function argument should be (x(t), y(t)) "
|
||
|
|
"but got %s" % str(function))
|
||
|
|
if not is_sequence(limits) or len(limits) != 3:
|
||
|
|
raise ValueError("Limit argument should be (t, tmin, tmax) "
|
||
|
|
"but got %s" % str(limits))
|
||
|
|
|
||
|
|
return GeometryEntity.__new__(cls, Tuple(*function), Tuple(*limits))
|
||
|
|
|
||
|
|
def __call__(self, f):
|
||
|
|
return self.subs(self.parameter, f)
|
||
|
|
|
||
|
|
def _eval_subs(self, old, new):
|
||
|
|
if old == self.parameter:
|
||
|
|
return Point(*[f.subs(old, new) for f in self.functions])
|
||
|
|
|
||
|
|
def _eval_evalf(self, prec=15, **options):
|
||
|
|
f, (t, a, b) = self.args
|
||
|
|
dps = prec_to_dps(prec)
|
||
|
|
f = tuple([i.evalf(n=dps, **options) for i in f])
|
||
|
|
a, b = [i.evalf(n=dps, **options) for i in (a, b)]
|
||
|
|
return self.func(f, (t, a, b))
|
||
|
|
|
||
|
|
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
|
||
|
|
=======
|
||
|
|
|
||
|
|
Point :
|
||
|
|
Returns a point in parametric form.
|
||
|
|
|
||
|
|
Raises
|
||
|
|
======
|
||
|
|
|
||
|
|
ValueError
|
||
|
|
When `parameter` already appears in the functions.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Curve, Symbol
|
||
|
|
>>> from sympy.abc import s
|
||
|
|
>>> C = Curve([2*s, s**2], (s, 0, 2))
|
||
|
|
>>> C.arbitrary_point()
|
||
|
|
Point2D(2*t, t**2)
|
||
|
|
>>> C.arbitrary_point(C.parameter)
|
||
|
|
Point2D(2*s, s**2)
|
||
|
|
>>> C.arbitrary_point(None)
|
||
|
|
Point2D(2*s, s**2)
|
||
|
|
>>> C.arbitrary_point(Symbol('a'))
|
||
|
|
Point2D(2*a, a**2)
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
sympy.geometry.point.Point
|
||
|
|
|
||
|
|
"""
|
||
|
|
if parameter is None:
|
||
|
|
return Point(*self.functions)
|
||
|
|
|
||
|
|
tnew = _symbol(parameter, self.parameter, real=True)
|
||
|
|
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])
|
||
|
|
|
||
|
|
@property
|
||
|
|
def free_symbols(self):
|
||
|
|
"""Return a set of symbols other than the bound symbols used to
|
||
|
|
parametrically define the Curve.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
set :
|
||
|
|
Set of all non-parameterized symbols.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.abc import t, a
|
||
|
|
>>> from sympy import Curve
|
||
|
|
>>> Curve((t, t**2), (t, 0, 2)).free_symbols
|
||
|
|
set()
|
||
|
|
>>> Curve((t, t**2), (t, a, 2)).free_symbols
|
||
|
|
{a}
|
||
|
|
|
||
|
|
"""
|
||
|
|
free = set()
|
||
|
|
for a in self.functions + self.limits[1:]:
|
||
|
|
free |= a.free_symbols
|
||
|
|
free = free.difference({self.parameter})
|
||
|
|
return free
|
||
|
|
|
||
|
|
@property
|
||
|
|
def ambient_dimension(self):
|
||
|
|
"""The dimension of the curve.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
int :
|
||
|
|
the dimension of curve.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.abc import t
|
||
|
|
>>> from sympy import Curve
|
||
|
|
>>> C = Curve((t, t**2), (t, 0, 2))
|
||
|
|
>>> C.ambient_dimension
|
||
|
|
2
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
return len(self.args[0])
|
||
|
|
|
||
|
|
@property
|
||
|
|
def functions(self):
|
||
|
|
"""The functions specifying the curve.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
functions :
|
||
|
|
list of parameterized coordinate functions.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.abc import t
|
||
|
|
>>> from sympy import Curve
|
||
|
|
>>> C = Curve((t, t**2), (t, 0, 2))
|
||
|
|
>>> C.functions
|
||
|
|
(t, t**2)
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
parameter
|
||
|
|
|
||
|
|
"""
|
||
|
|
return self.args[0]
|
||
|
|
|
||
|
|
@property
|
||
|
|
def limits(self):
|
||
|
|
"""The limits for the curve.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
limits : tuple
|
||
|
|
Contains parameter and lower and upper limits.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.abc import t
|
||
|
|
>>> from sympy import Curve
|
||
|
|
>>> C = Curve([t, t**3], (t, -2, 2))
|
||
|
|
>>> C.limits
|
||
|
|
(t, -2, 2)
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
plot_interval
|
||
|
|
|
||
|
|
"""
|
||
|
|
return self.args[1]
|
||
|
|
|
||
|
|
@property
|
||
|
|
def parameter(self):
|
||
|
|
"""The curve function variable.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
Symbol :
|
||
|
|
returns a bound symbol.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.abc import t
|
||
|
|
>>> from sympy import Curve
|
||
|
|
>>> C = Curve([t, t**2], (t, 0, 2))
|
||
|
|
>>> C.parameter
|
||
|
|
t
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
functions
|
||
|
|
|
||
|
|
"""
|
||
|
|
return self.args[1][0]
|
||
|
|
|
||
|
|
@property
|
||
|
|
def length(self):
|
||
|
|
"""The curve length.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Curve
|
||
|
|
>>> from sympy.abc import t
|
||
|
|
>>> Curve((t, t), (t, 0, 1)).length
|
||
|
|
sqrt(2)
|
||
|
|
|
||
|
|
"""
|
||
|
|
integrand = sqrt(sum(diff(func, self.limits[0])**2 for func in self.functions))
|
||
|
|
return integrate(integrand, self.limits)
|
||
|
|
|
||
|
|
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
|
||
|
|
=======
|
||
|
|
|
||
|
|
List :
|
||
|
|
the plot interval as below:
|
||
|
|
[parameter, lower_bound, upper_bound]
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Curve, sin
|
||
|
|
>>> from sympy.abc import x, 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]
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
limits : Returns limits of the parameter interval
|
||
|
|
|
||
|
|
"""
|
||
|
|
t = _symbol(parameter, self.parameter, real=True)
|
||
|
|
return [t] + list(self.limits[1:])
|
||
|
|
|
||
|
|
def rotate(self, angle=0, pt=None):
|
||
|
|
"""This function is used to rotate a curve along given point ``pt`` at given angle(in radian).
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
angle :
|
||
|
|
the angle at which the curve will be rotated(in radian) in counterclockwise direction.
|
||
|
|
default value of angle is 0.
|
||
|
|
|
||
|
|
pt : Point
|
||
|
|
the point along which the curve will be rotated.
|
||
|
|
If no point given, the curve will be rotated around origin.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
Curve :
|
||
|
|
returns a curve rotated at given angle along given point.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Curve, pi
|
||
|
|
>>> from sympy.abc import x
|
||
|
|
>>> Curve((x, x), (x, 0, 1)).rotate(pi/2)
|
||
|
|
Curve((-x, x), (x, 0, 1))
|
||
|
|
|
||
|
|
"""
|
||
|
|
if pt:
|
||
|
|
pt = -Point(pt, dim=2)
|
||
|
|
else:
|
||
|
|
pt = Point(0,0)
|
||
|
|
rv = self.translate(*pt.args)
|
||
|
|
f = list(rv.functions)
|
||
|
|
f.append(0)
|
||
|
|
f = Matrix(1, 3, f)
|
||
|
|
f *= rot_axis3(angle)
|
||
|
|
rv = self.func(f[0, :2].tolist()[0], self.limits)
|
||
|
|
pt = -pt
|
||
|
|
return rv.translate(*pt.args)
|
||
|
|
|
||
|
|
def scale(self, x=1, y=1, pt=None):
|
||
|
|
"""Override GeometryEntity.scale since Curve is not made up of Points.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
Curve :
|
||
|
|
returns scaled curve.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Curve
|
||
|
|
>>> from sympy.abc import x
|
||
|
|
>>> Curve((x, x), (x, 0, 1)).scale(2)
|
||
|
|
Curve((2*x, x), (x, 0, 1))
|
||
|
|
|
||
|
|
"""
|
||
|
|
if pt:
|
||
|
|
pt = Point(pt, dim=2)
|
||
|
|
return self.translate(*(-pt).args).scale(x, y).translate(*pt.args)
|
||
|
|
fx, fy = self.functions
|
||
|
|
return self.func((fx*x, fy*y), self.limits)
|
||
|
|
|
||
|
|
def translate(self, x=0, y=0):
|
||
|
|
"""Translate the Curve by (x, y).
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
Curve :
|
||
|
|
returns a translated curve.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Curve
|
||
|
|
>>> from sympy.abc import x
|
||
|
|
>>> Curve((x, x), (x, 0, 1)).translate(1, 2)
|
||
|
|
Curve((x + 1, x + 2), (x, 0, 1))
|
||
|
|
|
||
|
|
"""
|
||
|
|
fx, fy = self.functions
|
||
|
|
return self.func((fx + x, fy + y), self.limits)
|