# Source code for sympy.geometry.curve

"""Curves in 2-dimensional Euclidean space.

Contains
========
Curve

"""

from __future__ import print_function, division

from sympy.core import sympify
from sympy.core.compatibility import is_sequence
from sympy.core.containers import Tuple
from sympy.geometry.entity import GeometryEntity
from sympy.geometry.point import Point
from .util import _symbol

[docs]class Curve(GeometryEntity): """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) Point(4, 16) >>> C.arbitrary_point(a) Point(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]) @property
[docs] 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 set([a]) """ free = set() for a in self.functions + self.limits[1:]: free |= a.free_symbols free = free.difference(set([self.parameter])) return free
@property
[docs] 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
[docs] 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
[docs] 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]
[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 pt = -Point(pt or (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) 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)
[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() 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])
[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) return [t] + list(self.limits[1:])