o 8VaL'@sdZddlmZddlmZmZddlmZddlm Z ddl m Z ddl m Z mZddlmZdd lmZGd d d eZd S) zCCurves in 2-dimensional Euclidean space. Contains ======== Curve )sqrt)sympifydiff) is_sequence)Tuple)_symbol)GeometryEntity GeometrySet)Point) integratec@seZdZdZddZddZddZd"d d Zed d Z ed dZ eddZ eddZ eddZ eddZd"ddZd#ddZd$ddZd%d d!ZdS)&CurveaNA curve in space. A curve is defined by parametric functions for the coordinates, a parameter and the lower and upper bounds for the parameter value. Parameters ========== function : list of functions limits : 3-tuple Function parameter and lower and upper bounds. Attributes ========== functions parameter limits Raises ====== ValueError When `functions` are specified incorrectly. When `limits` are specified incorrectly. Examples ======== >>> from sympy import sin, cos, interpolate >>> from sympy.abc import t, a >>> from sympy.geometry import Curve >>> C = Curve((sin(t), cos(t)), (t, 0, 2)) >>> C.functions (sin(t), cos(t)) >>> C.limits (t, 0, 2) >>> C.parameter t >>> C = Curve((t, interpolate([1, 4, 9, 16], t)), (t, 0, 1)); C Curve((t, t**2), (t, 0, 1)) >>> C.subs(t, 4) Point2D(4, 16) >>> C.arbitrary_point(a) Point2D(a, a**2) See Also ======== sympy.core.function.Function sympy.polys.polyfuncs.interpolate cCsft|}t|rt|dkrtdt|t|r t|dkr(tdt|t|t|t|S)Nz3Function argument should be (x(t), y(t)) but got %sz3Limit argument should be (t, tmin, tmax) but got %s)rrlen ValueErrorstrr__new__r)clsfunctionlimitsZfunr6/usr/lib/python3/dist-packages/sympy/geometry/curve.pyrJsz Curve.__new__cCs||j|SN)subs parameter)selffrrr__call__UszCurve.__call__cs(|jkrtfdd|jDSdS)Ncsg|]}|qSrr.0rnewoldrr Zz$Curve._eval_subs..)rr functions)rr#r"rr!r _eval_subsXs zCurve._eval_substcsr|dur t|jSt||jdd|jjjkr,jdd|jDvr,tdjtfdd|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 ======= Point : Returns a point in parametric form. Raises ====== ValueError When `parameter` already appears in the functions. 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) See Also ======== sympy.geometry.point.Point NTrealcss|]}|jVqdSr)namerrrr sz(Curve.arbitrary_point..zFSymbol %s already appears in object and cannot be used as a parameter.csg|]}|qSrr)r wr(Ztnewrrr$r%z)Curve.arbitrary_point..)r r&rrr+ free_symbolsr)rrrr.rarbitrary_point\s-  zCurve.arbitrary_pointcCs<t}|j|jddD]}||jO}q ||jh}|S)aReturn a set of symbols other than the bound symbols used to parametrically define the Curve. Returns ======= set : Set of all non-parameterized symbols. Examples ======== >>> from sympy.abc import t, a >>> from sympy.geometry import Curve >>> Curve((t, t**2), (t, 0, 2)).free_symbols set() >>> Curve((t, t**2), (t, a, 2)).free_symbols {a} N)setr&rr/ differencer)rfreearrrr/s  zCurve.free_symbolscCst|jdS)aDThe dimension of the curve. Returns ======= int : the dimension of curve. Examples ======== >>> from sympy.abc import t >>> from sympy.geometry import Curve >>> C = Curve((t, t**2), (t, 0, 2)) >>> C.ambient_dimension 2 r)rargsrrrrambient_dimensionszCurve.ambient_dimensioncC |jdS)aThe functions specifying the curve. Returns ======= functions : list of parameterized coordinate functions. Examples ======== >>> from sympy.abc import t >>> from sympy.geometry import Curve >>> C = Curve((t, t**2), (t, 0, 2)) >>> C.functions (t, t**2) See Also ======== parameter rr6r7rrrr& zCurve.functionscCr9)aThe limits for the curve. Returns ======= limits : tuple Contains parameter and lower and upper limits. Examples ======== >>> from sympy.abc import t >>> from sympy.geometry import Curve >>> C = Curve([t, t**3], (t, -2, 2)) >>> C.limits (t, -2, 2) See Also ======== plot_interval r1r:r7rrrrr;z Curve.limitscCs|jddS)avThe curve function variable. Returns ======= Symbol : returns a bound symbol. Examples ======== >>> from sympy.abc import t >>> from sympy.geometry import Curve >>> C = Curve([t, t**2], (t, 0, 2)) >>> C.parameter t See Also ======== functions r1rr:r7rrrrszCurve.parametercs(ttfddjD}t|jS)zThe curve length. Examples ======== >>> from sympy.geometry.curve import Curve >>> from sympy.abc import t >>> Curve((t, t), (t, 0, 1)).length sqrt(2) c3s$|] }t|jddVqdS)rr N)rr)r funcr7rrr,%s"zCurve.length..)rsumr&r r)rZ integrandrr7rlengths 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 ======= 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 Tr)r1N)rrlistr)rrr(rrr plot_interval(s!zCurve.plot_intervalrNcCsddlm}m}|rt|dd }ntdd}|j|j}t|j}|d|dd|}|||9}| |dddf d|j }| }|j|jS)a*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.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 Zdimr1rN) Zsympy.matricesrArBr translater6r?r&appendr<tolistr)rZangleptrArBZrvrrrrrotateLs      " z Curve.rotater1cCsR|rt|dd}|j| j||j|jS|j\}}|||||f|jS)amOverride GeometryEntity.scale since Curve is not made up of Points. Returns ======= Curve : returns scaled curve. Examples ======== >>> from sympy.geometry.curve import Curve >>> from sympy.abc import x >>> Curve((x, x), (x, 0, 1)).scale(2) Curve((2*x, x), (x, 0, 1)) r rC)r rDr6scaler&r<r)rxyrGfxfyrrrrIxs   z Curve.scalecCs$|j\}}|||||f|jS)a[Translate the Curve by (x, y). Returns ======= Curve : returns a translated curve. Examples ======== >>> from sympy.geometry.curve import Curve >>> from sympy.abc import x >>> Curve((x, x), (x, 0, 1)).translate(1, 2) Curve((x + 1, x + 2), (x, 0, 1)) )r&r<r)rrJrKrLrMrrrrDs zCurve.translate)r()rN)r1r1N)rr)__name__ __module__ __qualname____doc__rrr'r0propertyr/r8r&rrr>r@rHrIrDrrrrr s,6  8        $ ,r N)rQZsympyrZ sympy.corerrZsympy.core.compatibilityrZsympy.core.containersrZsympy.core.symbolrZsympy.geometry.entityrr Zsympy.geometry.pointr Zsympy.integralsr r rrrrs