ó ¡¼™\c@ sÂdZddlmZmZddlmZddlmZmZddl m Z ddl m Z ddl mZddlmZmZdd lmZdd lmZd efd „ƒYZd S(sCCurves in 2-dimensional Euclidean space. Contains ======== Curve iÿÿÿÿ(tdivisiontprint_function(tsqrt(tsympifytdiff(t is_sequence(tTuple(t_symbol(tGeometryEntityt GeometrySet(tPoint(t integratetCurvecB sÂeZdZd„Zd„Zdd„Zed„ƒZed„ƒZed„ƒZ ed„ƒZ ed „ƒZ ed „ƒZ dd „Z d dd „Zdddd„Zd d d„ZRS(sUA 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) cC s›t|ƒ}t|ƒ s+t|ƒdkrDtdt|ƒƒ‚nt|ƒ sct|ƒdkr|tdt|ƒƒ‚ntj|t|Œt|ŒƒS(Nis3Function argument should be (x(t), y(t)) but got %sis3Limit argument should be (t, tmin, tmax) but got %s(RRtlent ValueErrortstrRt__new__R(tclstfunctiontlimitstfun((s3lib/python2.7/site-packages/sympy/geometry/curve.pyRKs cC s?||jkr;tg|jD]}|j||ƒ^qŒSdS(N(t parameterR t functionstsubs(tselftoldtnewtf((s3lib/python2.7/site-packages/sympy/geometry/curve.pyt _eval_subsVsttcC sª|dkrt|jŒSt||jdtƒ}|j}|j|jkr~|jd„|jDƒkr~td|jƒ‚ntg|jD]}|j ||ƒ^q‹ŒS(sò 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) trealcs s|]}|jVqdS(N(tname(t.0R((s3lib/python2.7/site-packages/sympy/geometry/curve.pys ssFSymbol %s already appears in object and cannot be used as a parameter.N( tNoneR RRRtTrueRt free_symbolsRR(RRttnewRtw((s3lib/python2.7/site-packages/sympy/geometry/curve.pytarbitrary_pointZs-   cC sNtƒ}x)|j|jdD]}||jO}qW|j|jhƒ}|S(ss 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} i(tsetRRR#t differenceR(Rtfreeta((s3lib/python2.7/site-packages/sympy/geometry/curve.pyR#’s  cC st|jdƒS(Ni(R targs(R((s3lib/python2.7/site-packages/sympy/geometry/curve.pytambient_dimension¨scC s |jdS(sThe 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) i(R+(R((s3lib/python2.7/site-packages/sympy/geometry/curve.pyR¬scC s |jdS(s›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) i(R+(R((s3lib/python2.7/site-packages/sympy/geometry/curve.pyRÆscC s|jddS(sbThe 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 ii(R+(R((s3lib/python2.7/site-packages/sympy/geometry/curve.pyRásc s5tt‡fd†ˆjDƒƒƒ}t|ˆjƒS(søThe 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) c3 s)|]}t|ˆjdƒdVqdS(iiN(RR(R tfunc(R(s3lib/python2.7/site-packages/sympy/geometry/curve.pys s(RtsumRR R(Rt integrand((Rs3lib/python2.7/site-packages/sympy/geometry/curve.pytlengthûs %cC s0t||jdtƒ}|gt|jdƒS(sàThe 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] Ri(RRR"tlistR(RRR((s3lib/python2.7/site-packages/sympy/geometry/curve.pyt plot_interval s icC sçddlm}m}|r2t|ddƒ }ntddƒ}|j|jŒ}t|jƒ}|jdƒ|dd|ƒ}|||ƒ9}|j |ddd…fj ƒd|j ƒ}|dk rã| }|j|jŒS|S( sjRotate ``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)) iÿÿÿÿ(tMatrixt rot_axis3tdimiiiiN( tsympy.matricesR3R4R t translateR+R1RtappendR-ttolistRR!(RtangletptR3R4trvR((s3lib/python2.7/site-packages/sympy/geometry/curve.pytrotate.s / icC sq|rAt|ddƒ}|j| jŒj||ƒj|jŒS|j\}}|j||||f|jƒS(s9Override 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)) R5i(R R7R+tscaleRR-R(RtxtyR;tfxtfy((s3lib/python2.7/site-packages/sympy/geometry/curve.pyR>Ls )cC s0|j\}}|j||||f|jƒS(s!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)) (RR-R(RR?R@RARB((s3lib/python2.7/site-packages/sympy/geometry/curve.pyR7^s N(t__name__t __module__t__doc__RRR&tpropertyR#R,RRRR0R2R!R=R>R7(((s3lib/python2.7/site-packages/sympy/geometry/curve.pyR s4  8 #N(REt __future__RRtsympyRt sympy.coreRRtsympy.core.compatibilityRtsympy.core.containersRtsympy.core.symbolRtsympy.geometry.entityRR tsympy.geometry.pointR tsympy.integralsR R (((s3lib/python2.7/site-packages/sympy/geometry/curve.pyts