+ i'Rt^RIHt^RIHt^RIHt^RIHt^RI H t H t ^RI H t ^RIHt^RIHtHt^R IHt^R IHt!R R ] 4tR #)zCCurves in 2-dimensional Euclidean space. Contains ======== Curve )sqrt)diff)Tuple)_symbol)GeometryEntity GeometrySet)Point) integrate)Matrix rot_axis3) is_sequence) prec_to_dpsca]tRt^toRtRtRtRtRRltRRlt ] R4t ] R4t ] R 4t ] R 4t] R 4t] R 4tRR ltRRltRRltRRltRtVtR#)CurveaA 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 Curve, sin, cos, interpolate >>> from sympy.abc import t, a >>> 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 c>\V4'd\V4^8wd\R\V4,4h\V4'd\V4^8wd\R\V4,4h\P !V\ V!\ V!4#)z3Function argument should be (x(t), y(t)) but got %sz3Limit argument should be (t, tmin, tmax) but got %s)r len ValueErrorstrr__new__r)clsfunctionlimitss&&&t/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/geometry/curve.pyr Curve.__new__Ls8$$H (:"8}-. .6""c&kQ&6"6{+, ,%%c5(+;UF^LLc:VPVPV4#N)subs parameter)selffs&&r__call__Curve.__call__Vsyy++rc WP8Xd2\VPUu.uFq3PW4NK up!#R#uupir)rr functionsr)r oldnewr!s&&& r _eval_subsCurve._eval_subsYs9 .. T^^D^66#+^DE E !DsAc VPwpwrEp\V4p\VUu.uFqP!RRV/VBNK up4pWV3Uu.uFqP!RRV/VBNK upwrVVP W4WV34#uupiuupi)n)argsr tupleevalffunc) r precoptionsr!tabdpsis &&, r _eval_evalfCurve._eval_evalf]syy 9A!$ a8a77,S,G,a8 9456:6a)#))6:yyI&&9:s BBc  Vf\VP!#\WPRR7pVPpVPVP8wd?VPRVP 49d\ RVP,4h\VPUu.uFqDPW24NK up!#uupi)aA 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 Curve, Symbol >>> from sympy.abc import s >>> 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 Trealc38"TFqPxK R#5ir)name).0r!s& r (Curve.arbitrary_point..s@.?ff.?szFSymbol %s already appears in object and cannot be used as a parameter.)rr%rrr> free_symbolsrr)r rtnewr3ws&& rarbitrary_pointCurve.arbitrary_pointdsX  $..) )y..t< NN II  @d.?.?@@57;yyAB B?1vva?@@?s'Cc \4pVPVPR,,FpWP,pK VP VP 04pV#)aTReturn a set of symbols other than the bound symbols used to parametrically define the Curve. Returns ======= set : Set of all non-parameterized symbols. Examples ======== >>> from sympy.abc import t, a >>> from sympy import Curve >>> Curve((t, t**2), (t, 0, 2)).free_symbols set() >>> Curve((t, t**2), (t, a, 2)).free_symbols {a} NN)setr%rrB differencer)r freer4s& rrBCurve.free_symbolssO,u$++b/11A NN "D2/0 rc :\VP^,4#)zThe dimension of the curve. Returns ======= int : the dimension of curve. Examples ======== >>> from sympy.abc import t >>> from sympy import Curve >>> C = Curve((t, t**2), (t, 0, 2)) >>> C.ambient_dimension 2 )rr-r s&rambient_dimensionCurve.ambient_dimensions*499Q<  rc (VP^,#)aThe functions specifying the curve. Returns ======= functions : list of parameterized coordinate functions. Examples ======== >>> from sympy.abc import t >>> from sympy import Curve >>> C = Curve((t, t**2), (t, 0, 2)) >>> C.functions (t, t**2) See Also ======== parameter r-rOs&rr%Curve.functions2yy|rc (VP^,#)aThe limits for the curve. Returns ======= limits : tuple Contains parameter and lower and upper limits. Examples ======== >>> from sympy.abc import t >>> from sympy import Curve >>> C = Curve([t, t**3], (t, -2, 2)) >>> C.limits (t, -2, 2) See Also ======== plot_interval rSrOs&rr Curve.limitsrUrc 6VP^,^,#)zThe curve function variable. Returns ======= Symbol : returns a bound symbol. Examples ======== >>> from sympy.abc import t >>> from sympy import Curve >>> C = Curve([t, t**2], (t, 0, 2)) >>> C.parameter t See Also ======== functions rSrOs&rrCurve.parameters2yy|Arc a\\V3RlSP444p\VSP4#)zThe curve length. Examples ======== >>> from sympy import Curve >>> from sympy.abc import t >>> Curve((t, t), (t, 0, 1)).length sqrt(2) c3l<"TF)p\VSP^,4^,xK+ R#5iN)rr)r?r0r s& rr@Curve.length..,s(V~tT$ A7::~s14)rsumr%r r)r integrandsf rlength Curve.lengths/Vt~~VVW DKK00rc x\WPRR7pV.\VPR,4,#)a;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 ======= 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;rH)rrlistr)r rr3s&& r plot_intervalCurve.plot_interval/s/B I~~D 9sT$++b/***rNc V'd\V^R7)pM \^^4pVP!VP!p\VP4pVP ^4\ ^^V4pV\V4,pVPVR,P4^,VP4pV)pVP!VP!#)afThis function is used to rotate a curve along given point ``pt`` at given angle(in radian). Parameters ========== angle : the angle at which the curve will be rotated(in radian) in counterclockwise direction. default value of angle is 0. pt : Point the point along which the curve will be rotated. If no point given, the curve will be rotated around origin. Returns ======= Curve : returns a curve rotated at given angle along given point. Examples ======== >>> from sympy import Curve, pi >>> from sympy.abc import x >>> Curve((x, x), (x, 0, 1)).rotate(pi/2) Curve((-x, x), (x, 0, 1)) dim)r]:NrN) r translater-rdr%appendr r r0tolistr)r angleptrvr!s&&& rrotate Curve.rotateSs: ""BqB ^^RWW %    1aO Yu  YYqx(+T[[ 9S||RWW%%rc V'dP\V^R7pVP!V)P!PW4P!VP!#VPwrEVP WA,WR,3VP 4#)aOverride GeometryEntity.scale since Curve is not made up of Points. Returns ======= Curve : returns scaled curve. Examples ======== >>> from sympy import Curve >>> from sympy.abc import x >>> Curve((x, x), (x, 0, 1)).scale(2) Curve((2*x, x), (x, 0, 1)) rh)rrjr-scaler%r0r)r xyrnfxfys&&&& rrs Curve.scale}sj$ rq!B>>RC::.44Q:DDbggN Nyy"$t{{33rc rVPwr4VPW1,WB,3VP4#)zTranslate the Curve by (x, y). Returns ======= Curve : returns a translated curve. Examples ======== >>> from sympy import Curve >>> from sympy.abc import x >>> Curve((x, x), (x, 0, 1)).translate(1, 2) Curve((x + 1, x + 2), (x, 0, 1)) )r%r0r)r rtrurvrws&&& rrjCurve.translates-$yy"&"&)4;;77rr,))r3r\)rIrIN)r]r])__name__ __module__ __qualname____firstlineno____doc__rr"r(r8rEpropertyrBrPr%rrrarerprsrj__static_attributes____classdictcell__) __classdict__s@rrrs3jM,F'5An6!!,444 1 1"+H(&T4088rrN)r(sympy.functions.elementary.miscellaneousr sympy.corersympy.core.containersrsympy.core.symbolrsympy.geometry.entityrrsympy.geometry.pointrsympy.integralsr sympy.matricesr r sympy.utilities.iterablesr mpmath.libmp.libmpfr rr,rrrs8:'%=&%,1+R8KR8r