B [y#@sdZddlmZmZddlmZddlmZmZddl m Z ddl m Z ddl mZddlmZmZdd lmZdd lmZGd d d eZd S)zCCurves in 2-dimensional Euclidean space. Contains ======== Curve )divisionprint_function)sqrt)sympifydiff) is_sequence)Tuple)_symbol)GeometryEntity GeometrySet)Point) integratec@seZdZdZddZddZd ddZed d Zed d Z ed dZ eddZ eddZ eddZ d!ddZd"ddZd#ddZd$ddZdS)%CurveaUA curve in space. A curve is defined by parametric functions for the coordinates, a parameter and the lower and upper bounds for the parameter value. Parameters ========== function : list of functions limits : 3-tuple Function parameter and lower and upper bounds. Attributes ========== functions parameter limits Raises ====== ValueError When `functions` are specified incorrectly. When `limits` are specified incorrectly. See Also ======== sympy.core.function.Function sympy.polys.polyfuncs.interpolate Examples ======== >>> from sympy import sin, cos, Symbol, interpolate >>> from sympy.abc import t, a >>> from sympy.geometry import Curve >>> C = Curve((sin(t), cos(t)), (t, 0, 2)) >>> C.functions (sin(t), cos(t)) >>> C.limits (t, 0, 2) >>> C.parameter t >>> C = Curve((t, interpolate([1, 4, 9, 16], t)), (t, 0, 1)); C Curve((t, t**2), (t, 0, 1)) >>> C.subs(t, 4) Point2D(4, 16) >>> C.arbitrary_point(a) Point2D(a, a**2) cCsft|}t|rt|dkr,tdt|t|r@t|dkrPtdt|t|t|t|S)Nz3Function argument should be (x(t), y(t)) but got %sz3Limit argument should be (t, tmin, tmax) but got %s)rrlen ValueErrorstrr __new__r)clsZfunctionlimitsZfunr3lib/python3.7/site-packages/sympy/geometry/curve.pyrKs  z Curve.__new__cs(|jkr$tfdd|jDSdS)Ncsg|]}|qSr)subs).0f)newoldrr Xsz$Curve._eval_subs..) parameterr functions)selfrrr)rrr _eval_subsVs zCurve._eval_substcsr|dkrt|jSt||jdd|jjjkrXjdd|jDkrXtdjtfdd|jDS) a 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) NT)realcss|] }|jVqdS)N)name)rrrrr sz(Curve.arbitrary_point..zFSymbol %s already appears in object and cannot be used as a parameter.csg|]}|qSr)r)rw)r#tnewrrrsz)Curve.arbitrary_point..)r r r rr% free_symbolsr)r!rr)r#r(rarbitrary_pointZs-   zCurve.arbitrary_pointcCs@t}x&|j|jddD]}||jO}qW||jh}|S)as 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} N)setr rr) differencer)r!Zfreearrrr)s zCurve.free_symbolscCst|jdS)Nr)rargs)r!rrrambient_dimensionszCurve.ambient_dimensioncCs |jdS)aThe 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) r)r/)r!rrrr szCurve.functionscCs |jdS)aThe 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) r+)r/)r!rrrrsz Curve.limitscCs|jddS)abThe 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 r+r)r/)r!rrrrszCurve.parametercs(ttfddjD}t|jS)zThe 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) c3s"|]}t|jddVqdS)rrN)rr)rfunc)r!rrr&szCurve.length..)rsumr r r)r!Z integrandr)r!rlengths z Curve.lengthcCs(t||jdd}|gt|jddS)aThe 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)r$r+N)r rlistr)r!rr#rrr plot_interval s zCurve.plot_intervalrNcCsddlm}m}|r$t|dd }n tdd}|j|j}t|j}|d|dd|}|||9}| |dddf d|j }|dk r| }|j|jS|S)ajRotate ``angle`` radians counterclockwise about Point ``pt``. The default pt is the origin, Point(0, 0). Examples ======== >>> from sympy.geometry.curve import Curve >>> from sympy.abc import x >>> from sympy import pi >>> Curve((x, x), (x, 0, 1)).rotate(pi/2) Curve((-x, x), (x, 0, 1)) r)Matrix rot_axis3r)dimr+rN) Zsympy.matricesr6r7r translater/r4r appendr1tolistr)r!Zangleptr6r7rvrrrrrotate.s      " z Curve.rotater+cCsR|r.t|dd}|j| j||j|jS|j\}}|||||f|jS)a9Override GeometryEntity.scale since Curve is not made up of Points. Examples ======== >>> from sympy.geometry.curve import Curve >>> from sympy import pi >>> from sympy.abc import x >>> Curve((x, x), (x, 0, 1)).scale(2) Curve((2*x, x), (x, 0, 1)) r)r8)r r9r/scaler r1r)r!xyr<fxfyrrrr?Ls   z Curve.scalecCs$|j\}}|||||f|jS)a!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)) )r r1r)r!r@rArBrCrrrr9^s zCurve.translate)r#)r#)rN)r+r+N)rr)__name__ __module__ __qualname____doc__rr"r*propertyr)r0r rrr3r5r>r?r9rrrrrs4  8       #  rN)rGZ __future__rrZsympyrZ sympy.corerrZsympy.core.compatibilityrZsympy.core.containersrZsympy.core.symbolr Zsympy.geometry.entityr r Zsympy.geometry.pointr Zsympy.integralsr rrrrrs