"""Curves in 2-dimensional Euclidean space.
Contains
========
Curve
"""
from __future__ import division, print_function
from sympy import sqrt
from sympy.core import sympify, diff
from sympy.core.compatibility import is_sequence
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
[docs]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.
See Also
========
sympy.core.function.Function
sympy.polys.polyfuncs.interpolate
Examples
========
>>> from sympy import sin, cos, Symbol, interpolate
>>> from sympy.abc import t, a
>>> from sympy.geometry import Curve
>>> 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)
"""
def __new__(cls, function, limits):
fun = sympify(function)
if not is_sequence(fun) or len(fun) != 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(*fun), Tuple(*limits))
def _eval_subs(self, old, new):
if old == self.parameter:
return Point(*[f.subs(old, new) for f in self.functions])
[docs] 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()
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)
"""
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.
Examples
========
>>> from sympy.abc import t, a
>>> from sympy.geometry 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):
return len(self.args[0])
@property
def functions(self):
"""The functions specifying the curve.
Returns
=======
functions : list of parameterized coordinate functions.
See Also
========
parameter
Examples
========
>>> from sympy.abc import t
>>> from sympy.geometry import Curve
>>> C = Curve((t, t**2), (t, 0, 2))
>>> C.functions
(t, t**2)
"""
return self.args[0]
@property
def limits(self):
"""The limits for the curve.
Returns
=======
limits : tuple
Contains parameter and lower and upper limits.
See Also
========
plot_interval
Examples
========
>>> from sympy.abc import t
>>> from sympy.geometry import Curve
>>> C = Curve([t, t**3], (t, -2, 2))
>>> C.limits
(t, -2, 2)
"""
return self.args[1]
@property
def parameter(self):
"""The curve function variable.
Returns
=======
parameter : SymPy symbol
See Also
========
functions
Examples
========
>>> from sympy.abc import t
>>> from sympy.geometry import Curve
>>> C = Curve([t, t**2], (t, 0, 2))
>>> C.parameter
t
"""
return self.args[1][0]
@property
def length(self):
"""The curve length.
Examples
========
>>> from sympy.geometry.curve import Curve
>>> from sympy import cos, sin
>>> 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)
[docs] 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]
See Also
========
limits : Returns limits of the parameter interval
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, real=True)
return [t] + list(self.limits[1:])
[docs] def rotate(self, angle=0, pt=None):
"""Rotate ``angle`` radians counterclockwise about Point ``pt``.
The default pt is the origin, Point(0, 0).
Examples
========
>>> from sympy.geometry.curve import Curve
>>> from sympy.abc import x
>>> from sympy import pi
>>> Curve((x, x), (x, 0, 1)).rotate(pi/2)
Curve((-x, x), (x, 0, 1))
"""
from sympy.matrices import Matrix, rot_axis3
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)
if pt is not None:
pt = -pt
return rv.translate(*pt.args)
return rv
[docs] def scale(self, x=1, y=1, pt=None):
"""Override GeometryEntity.scale since Curve is not made up of Points.
Examples
========
>>> from sympy.geometry.curve import Curve
>>> from sympy import pi
>>> 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)
[docs] def translate(self, x=0, y=0):
"""Translate the Curve by (x, y).
Examples
========
>>> from sympy.geometry.curve import Curve
>>> from sympy import pi
>>> 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)
