B [@s|dZddlmZmZddlmZmZmZddlm Z ddl m Z m Z ddl mZmZddlmZmZmZddlmZmZdd lmZdd lmZmZdd lmZdd lmZdd l m!Z!m"Z"m#Z#m$Z$ddl%m&Z&m'Z'm(Z(ddl)m*Z*m+Z+ddl,m-Z-ddl.m/Z/m0Z0ddl1m2Z2m3Z3ddl4m5Z5m6Z6m7Z7ddl8m9Z9m:Z:ddl;mGddde>Z?ddl@mAZAdS)z?Elliptical geometrical entities. Contains * Ellipse * Circle )divisionprint_function)Spisympify) fuzzy_bool)Rationaloo)rangeordered)Dummy_uniquely_named_symbol_symbol)simplifytrigsimp)sqrt)cossin) elliptic_e) GeometryError)Ray2D Segment2DLine2DLinearEntity3D) DomainErrorPolyPolynomialError) _not_a_coeff_nsort)solve) filldedent func_name)GeometryEntity GeometrySet)PointPoint2DPoint3D)Line LinearEntity)idiffNcs~eZdZdZddZddZfddZdNd d ZdOd dZe ddZ e ddZ dPddZ e ddZ e ddZe ddZe ddZe ddZd d!ZdQd$d%ZdRd&d'Ze d(d)Ze d*d+Ze d,d-Zd.d/Zd0d1Ze d2d3Ze d4d5ZdSd6d7Ze d8d9Ze d:d;ZdTdd?Z d@dAZ!dVfdCdD Z"dWdFdGZ#dHdIZ$e dJdKZ%dXdLdMZ&Z'S)YEllipsea"An elliptical GeometryEntity. Parameters ========== center : Point, optional Default value is Point(0, 0) hradius : number or SymPy expression, optional vradius : number or SymPy expression, optional eccentricity : number or SymPy expression, optional Two of `hradius`, `vradius` and `eccentricity` must be supplied to create an Ellipse. The third is derived from the two supplied. Attributes ========== center hradius vradius area circumference eccentricity periapsis apoapsis focus_distance foci Raises ====== GeometryError When `hradius`, `vradius` and `eccentricity` are incorrectly supplied as parameters. TypeError When `center` is not a Point. See Also ======== Circle Notes ----- Constructed from a center and two radii, the first being the horizontal radius (along the x-axis) and the second being the vertical radius (along the y-axis). When symbolic value for hradius and vradius are used, any calculation that refers to the foci or the major or minor axis will assume that the ellipse has its major radius on the x-axis. If this is not true then a manual rotation is necessary. Examples ======== >>> from sympy import Ellipse, Point, Rational >>> e1 = Ellipse(Point(0, 0), 5, 1) >>> e1.hradius, e1.vradius (5, 1) >>> e2 = Ellipse(Point(3, 1), hradius=3, eccentricity=Rational(4, 5)) >>> e2 Ellipse(Point2D(3, 1), 3, 9/5) cCsht|trRtddd}tddd}|||||j||ji}tt|t j kSt|t rd||kSdS)NxT)realyF) isinstancer%r equationsubsr,r.rrrZZeror+)selfor,r.Zresr45lib/python3.7/site-packages/sympy/geometry/ellipse.py __contains__es    zEllipse.__contains__cCs.t|to,|j|jko,|j|jko,|j|jkS)z5Is the other GeometryEntity the same as this ellipse?)r/r+centerhradiusvradius)r2r3r4r4r5__eq__ps zEllipse.__eq__cstt|S)N)superr+__hash__)r2) __class__r4r5r<vszEllipse.__hash__NcKst|}t|}t|}|dkr,tdd}n t|dd}t|dkrRtd|tttdd|||fdkrztd|dk r|dkr|td|d}n|dkr|td|d}||krt||f|St j ||||f|S) Nr)dimz3The center of "{0}" must be a two dimensional pointcSs|dk S)Nr4)r,r4r4r5sz!Ellipse.__new__..zTExactly two arguments of "hradius", "vradius", and "eccentricity" must not be None."r") rr%len ValueErrorformatlistfilterrCircler#__new__)clsr7r8r9 eccentricitykwargsr4r4r5rGys$    zEllipse.__new__?#66cc99cCsHddlm}||j}||j||j}}dd|||j|j||S)a%Returns SVG ellipse element for the Ellipse. Parameters ========== scale_factor : float Multiplication factor for the SVG stroke-width. Default is 1. fill_color : str, optional Hex string for fill color. Default is "#66cc99". r)Nzkg@)Zsympy.core.evalfrMr7r8r9rCr,r.)r2Z scale_factorZ fill_colorrMchvr4r4r5_svgs  z Ellipse._svgcCsdS)Nr>r4)r2r4r4r5ambient_dimensionszEllipse.ambient_dimensioncCs|jd|jS)aThe apoapsis of the ellipse. The greatest distance between the focus and the contour. Returns ======= apoapsis : number See Also ======== periapsis : Returns shortest distance between foci and contour Examples ======== >>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.apoapsis 2*sqrt(2) + 3 r")majorrI)r2r4r4r5apoapsisszEllipse.apoapsistcCsbt|dd}|jdd|jDkr4ttd|jt|jj|jt ||jj |j t |S)aXA parameterized point on the ellipse. Parameters ========== parameter : str, optional Default value is 't'. Returns ======= arbitrary_point : Point Raises ====== ValueError When `parameter` already appears in the functions. See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import Point, Ellipse >>> e1 = Ellipse(Point(0, 0), 3, 2) >>> e1.arbitrary_point() Point2D(3*cos(t), 2*sin(t)) T)r-css|] }|jVqdS)N)name).0fr4r4r5 sz*Ellipse.arbitrary_point..zFSymbol %s already appears in object and cannot be used as a parameter.) rrV free_symbolsrBr r%r7r,r8rr.r9r)r2 parameterrUr4r4r5arbitrary_points "  zEllipse.arbitrary_pointcCsttj|j|jS)a The area of the ellipse. Returns ======= area : number Examples ======== >>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.area 3*pi )rrPir8r9)r2r4r4r5areasz Ellipse.areacCs:|j|j}}|jj||jj||jj||jj|fS)zwReturn a tuple (xmin, ymin, xmax, ymax) representing the bounding rectangle for the geometric figure. )r8r9r7r,r.)r2rOrPr4r4r5bounds szEllipse.boundscCs |jdS)abThe center of the ellipse. Returns ======= center : number See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.center Point2D(0, 0) r)args)r2r4r4r5r7szEllipse.centercCsH|jdkrd|jS|jdkr,dt|jSd|jt|jdSdS)zThe circumference of the ellipse. Examples ======== >>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.circumference 12*elliptic_e(8/9) r"rr>N)rIrSrr8r)r2r4r4r5 circumference/s    zEllipse.circumferencecCs |j|jS)a5The eccentricity of the ellipse. Returns ======= eccentricity : number Examples ======== >>> from sympy import Point, Ellipse, sqrt >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, sqrt(2)) >>> e1.eccentricity sqrt(7)/3 )focus_distancerS)r2r4r4r5rIFszEllipse.eccentricitycsntdd|krdSt|jdkrRfdd|jD\}}d|j||}n|j|j}t|jS)a Return True if p is enclosed by (is inside of) self. Notes ----- Being on the border of self is considered False. Parameters ========== p : Point Returns ======= encloses_point : True, False or None See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import Ellipse, S >>> from sympy.abc import t >>> e = Ellipse((0, 0), 3, 2) >>> e.encloses_point((0, 0)) True >>> e.encloses_point(e.arbitrary_point(t).subs(t, S.Half)) False >>> e.encloses_point((4, 0)) False r>)r?Fcsg|]}|qSr4)distance)rWrX)pr4r5 sz*Ellipse.encloses_point..) r%rAfocirSradiusr7rdr is_positive)r2reZh1Zh2testr4)rer5encloses_point[s% zEllipse.encloses_pointr,r.cCsPt|dd}t|dd}||jj|jd}||jj|jd}||dS)aXThe equation of the ellipse. Parameters ========== x : str, optional Label for the x-axis. Default value is 'x'. y : str, optional Label for the y-axis. Default value is 'y'. Returns ======= equation : sympy expression See Also ======== arbitrary_point : Returns parameterized point on ellipse Examples ======== >>> from sympy import Point, Ellipse >>> e1 = Ellipse(Point(1, 0), 3, 2) >>> e1.equation() y**2/4 + (x/3 - 1/3)**2 - 1 T)r-r>r")rr7r,r8r.r9)r2r,r.t1t2r4r4r5r0s   zEllipse.equationcCst|jdkrtdt|dd}t|dd}|j||jjtdd}|j||jj tdd}|||jd|jdtddS)aThe equation of evolute of the ellipse. Parameters ========== x : str, optional Label for the x-axis. Default value is 'x'. y : str, optional Label for the y-axis. Default value is 'y'. Returns ======= equation : sympy expression Examples ======== >>> from sympy import Point, Ellipse >>> e1 = Ellipse(Point(1, 0), 3, 2) >>> e1.evolute() 2**(2/3)*y**(2/3) + (3*x - 3)**(2/3) - 5**(2/3) z.Evolute of arbitrary Ellipse is not supported.T)r-r>) rAr`NotImplementedErrorrr8r7r,rr9r.)r2r,r.rlrmr4r4r5evolutes  zEllipse.evolutecCs|j}|j|j}}||kr$||fSt|jd|jd}||jkrd|td| |td|fS||jkr|t| d|t|dfSdS)ahThe foci of the ellipse. Notes ----- The foci can only be calculated if the major/minor axes are known. Raises ====== ValueError When the major and minor axis cannot be determined. See Also ======== sympy.geometry.point.Point focus_distance : Returns the distance between focus and center Examples ======== >>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.foci (Point2D(-2*sqrt(2), 0), Point2D(2*sqrt(2), 0)) r>rN)r7r8r9rrSminorr%)r2rNZhrZvrfdr4r4r5rgs  z Ellipse.focicCst|j|jdS)aThe focal distance of the ellipse. The distance between the center and one focus. Returns ======= focus_distance : number See Also ======== foci Examples ======== >>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.focus_distance 2*sqrt(2) r)r%rdr7rg)r2r4r4r5rcszEllipse.focus_distancecCs |jdS)a^The horizontal radius of the ellipse. Returns ======= hradius : number See Also ======== vradius, major, minor Examples ======== >>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.hradius 3 r")r`)r2r4r4r5r8szEllipse.hradiusc s*tddd}tddd}ttr6|kr0gSgSntttfr|||}t|tjdjd||g||g}t t fdd|DStt r |Stt tfr|kr|S|||}t t d dt|||g||gDSn&ttrtd ntd td S) aIThe intersection of this ellipse and another geometrical entity `o`. Parameters ========== o : GeometryEntity Returns ======= intersection : list of GeometryEntity objects Notes ----- Currently supports intersections with Point, Line, Segment, Ray, Circle and Ellipse types. See Also ======== sympy.geometry.entity.GeometryEntity Examples ======== >>> from sympy import Ellipse, Point, Line, sqrt >>> e = Ellipse(Point(0, 0), 5, 7) >>> e.intersection(Point(0, 0)) [] >>> e.intersection(Point(5, 0)) [Point2D(5, 0)] >>> e.intersection(Line(Point(0,0), Point(0, 1))) [Point2D(0, -7), Point2D(0, 7)] >>> e.intersection(Line(Point(5,0), Point(5, 1))) [Point2D(5, 0)] >>> e.intersection(Line(Point(6,0), Point(6, 1))) [] >>> e = Ellipse(Point(-1, 0), 4, 3) >>> e.intersection(Ellipse(Point(1, 0), 4, 3)) [Point2D(0, -3*sqrt(15)/4), Point2D(0, 3*sqrt(15)/4)] >>> e.intersection(Ellipse(Point(5, 0), 4, 3)) [Point2D(2, -3*sqrt(7)/4), Point2D(2, 3*sqrt(7)/4)] >>> e.intersection(Ellipse(Point(100500, 0), 4, 3)) [] >>> e.intersection(Ellipse(Point(0, 0), 3, 4)) [Point2D(3, 0), Point2D(-363/175, -48*sqrt(111)/175), Point2D(-363/175, 48*sqrt(111)/175)] >>> e.intersection(Ellipse(Point(-1, 0), 3, 4)) [Point2D(-17/5, -12/5), Point2D(-17/5, 12/5), Point2D(7/5, -12/5), Point2D(7/5, 12/5)] r,T)r-r.rr"csg|]}|krt|qSr4)r%)rWi)r3r4r5rfwsz(Ellipse.intersection..cSsg|] }t|qSr4)r%)rWrsr4r4r5rfsz5Entity must be two dimensional, not three dimensionalzIntersection not handled for %sN)r r/r%rrr0rr(pointsrDr Polygon intersectionr+rr TypeErrorr!)r2r3r,r.Zellipse_equationresultr4)r3r5rv7s(4    ,   .  zEllipse.intersectioncszttrdSttrTttr0dSrLtfddDSdSn"ttrptdkSttrtdkrdjko j SdSntt t frLd}tt rֈj ng}xj|D]b}|tdkr@t fdd|jDs:tfdd|jDr:d}qndSq|SqW|Stttfrftd ntd td S) aIs `o` tangent to the ellipse? Parameters ========== o : GeometryEntity An Ellipse, LinearEntity or Polygon Raises ====== NotImplementedError When the wrong type of argument is supplied. Returns ======= is_tangent: boolean True if o is tangent to the ellipse, False otherwise. See Also ======== tangent_lines Examples ======== >>> from sympy import Point, Ellipse, Line >>> p0, p1, p2 = Point(0, 0), Point(3, 0), Point(3, 3) >>> e1 = Ellipse(p0, 3, 2) >>> l1 = Line(p1, p2) >>> e1.is_tangent(l1) True FTc3s,|]$}|d|dVqdS)rN) tangent_linesZequals)rWrs)r3r2r4r5rYsz%Ellipse.is_tangent..r"rc3s|]}d|kVqdS)rNr4)rWrs) intersectr4r5rYsc3s|]}| VqdS)N)rk)rWrs)r2r4r5rYsz5Entity must be two dimensional, not three dimensionalzIs_tangent not handled for %sN)r/r&r+rvallrrArsourcerkrruZsidesanyrtrr'rwr!)r2r3Z all_tangentsZsegmentsZsegmentr4)rzr3r2r5 is_tangents@%           zEllipse.is_tangentcCsT|jdd}t|dkr"|dS|\}}||dk}|dkrB|S|dkrN|S|jS)aLonger axis of the ellipse (if it can be determined) else hradius. Returns ======= major : number or expression See Also ======== hradius, vradius, minor Examples ======== >>> from sympy import Point, Ellipse, Symbol >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.major 3 >>> a = Symbol('a') >>> b = Symbol('b') >>> Ellipse(p1, a, b).major a >>> Ellipse(p1, b, a).major b >>> m = Symbol('m') >>> M = m + 1 >>> Ellipse(p1, m, M).major m + 1 r"rnrTF)r`rAr8)r2ababr3r4r4r5rSs$  z Ellipse.majorcCsT|jdd}t|dkr"|dS|\}}||dk}|dkrB|S|dkrN|S|jS)aShorter axis of the ellipse (if it can be determined) else vradius. Returns ======= minor : number or expression See Also ======== hradius, vradius, major Examples ======== >>> from sympy import Point, Ellipse, Symbol >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.minor 1 >>> a = Symbol('a') >>> b = Symbol('b') >>> Ellipse(p1, a, b).minor b >>> Ellipse(p1, b, a).minor a >>> m = Symbol('m') >>> M = m + 1 >>> Ellipse(p1, m, M).minor m r"rnrTF)r`rAr9)r2rrrr3r4r4r5rqs$  z Ellipse.minorc st|dd}g}|j|jjkr2|t|jtd|j|jjkrT|t|jdd|r\|Stdddtddd|t }d |t|fj }|}t |d} | d}t|jd krByt|} Wn2tttfk r(tt |dd d} YnXfd d | D} ntdfdd | D} dk rfdd | D} fdd | D} dd t| | DS)aGNormal lines between `p` and the ellipse. Parameters ========== p : Point Returns ======= normal_lines : list with 1, 2 or 4 Lines Examples ======== >>> from sympy import Line, Point, Ellipse >>> e = Ellipse((0, 0), 2, 3) >>> c = e.center >>> e.normal_lines(c + Point(1, 0)) [Line2D(Point2D(0, 0), Point2D(1, 0))] >>> e.normal_lines(c) [Line2D(Point2D(0, 0), Point2D(0, 1)), Line2D(Point2D(0, 0), Point2D(1, 0))] Off-axis points require the solution of a quartic equation. This often leads to very large expressions that may be of little practical use. An approximate solution of `prec` digits can be obtained by passing in the desired value: >>> e.normal_lines((3, 3), prec=2) [Line2D(Point2D(-0.81, -2.7), Point2D(0.19, -1.2)), Line2D(Point2D(1.5, -2.0), Point2D(2.5, -2.7))] Whereas the above solution has an operation count of 12, the exact solution has an operation count of 2020. r>)r?)sloperr,T)r-r.r")Z separatedc s(g|] }t|t|dqS)r)r%rr1)rWrs)eqr,r.r4r5rfwsz(Ellipse.normal_lines..z7intersections for the general ellipse are not supportedcs"g|]}tf|jqSr4)r1zipr`)rWpt)normr,r.r4r5rf{sNcsg|]}|qSr4)n)rWr)precr4r5rf}scs"g|]}t|r|n|qSr4)rr)rWrs)rr4r5rf~scSsg|]\}}t||dqS))r)r()rWrsr4r4r5rfs)r%r,r7appendr(r r.r r0r*rrr1Zas_numer_denomexpandrArZrZ real_rootsrrrorr) r2rerrvdydxrseqZyisZxeqZxsolrtZslopesr4)rrrr,r.r5 normal_lines0s:$    zEllipse.normal_linescCs|jd|jS)aThe periapsis of the ellipse. The shortest distance between the focus and the contour. Returns ======= periapsis : number See Also ======== apoapsis : Returns greatest distance between focus and contour Examples ======== >>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.periapsis -2*sqrt(2) + 3 r")rSrI)r2r4r4r5 periapsisszEllipse.periapsiscCs|jd|jdS)a= Calculates the semi-latus rectum of the Ellipse. Semi-latus rectum is defined as one half of the the chord through a focus parallel to the conic section directrix of a conic section. Returns ======= semilatus_rectum : number See Also ======== apoapsis : Returns greatest distance between focus and contour periapsis : The shortest distance between the focus and the contour Examples ======== >>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.semilatus_rectum 1/3 References ========== [1] http://mathworld.wolfram.com/SemilatusRectum.html [2] https://en.wikipedia.org/wiki/Ellipse#Semi-latus_rectum r"r>)rSrI)r2r4r4r5semilatus_rectums$zEllipse.semilatus_rectumcCst|dd}|tj tjgS)aThe plot interval for the default geometric plot of the Ellipse. Parameters ========== parameter : str, optional Default value is 't'. Returns ======= plot_interval : list [parameter, lower_bound, upper_bound] Examples ======== >>> from sympy import Point, Ellipse >>> e1 = Ellipse(Point(0, 0), 3, 2) >>> e1.plot_interval() [t, -pi, pi] T)r-)rrr])r2r[rUr4r4r5 plot_intervals zEllipse.plot_intervalc Csddlm}m}m}tddd}||j\}}|dk rDt|}nt}||} d| d} t d| d} t | ||| | ||| S) aYA random point on the ellipse. Returns ======= point : Point Examples ======== >>> from sympy import Point, Ellipse, Segment >>> e1 = Ellipse(Point(0, 0), 3, 2) >>> e1.random_point() # gives some random point Point2D(...) >>> p1 = e1.random_point(seed=0); p1.n(2) Point2D(2.1, 1.4) Notes ===== When creating a random point, one may simply replace the parameter with a random number. When doing so, however, the random number should be made a Rational or else the point may not test as being in the ellipse: >>> from sympy.abc import t >>> from sympy import Rational >>> arb = e1.arbitrary_point(t); arb Point2D(3*cos(t), 2*sin(t)) >>> arb.subs(t, .1) in e1 False >>> arb.subs(t, Rational(.1)) in e1 True >>> arb.subs(t, Rational('.1')) in e1 True See Also ======== sympy.geometry.point.Point arbitrary_point : Returns parameterized point on ellipse r)rrrrUT)r-Nr>r") Zsympyrrrrr\r`randomZRandomrr%r1) r2ZseedrrrrUr,r.rngrrNrr4r4r5 random_points*    zEllipse.random_pointcsjdtfkr2j}|}|j jSfdddD\}}||}t||}|j t ||f|j dd}t t ddt|t|t|fd S) aOverride GeometryEntity.reflect since the radius is not a GeometryEntity. Examples ======== >>> from sympy import Circle, Line >>> Circle((0, 1), 1).reflect(Line((0, 0), (1, 1))) Circle(Point2D(1, 0), -1) >>> from sympy import Ellipse, Line, Point >>> Ellipse(Point(3, 4), 1, 3).reflect(Line(Point(0, -4), Point(5, 0))) Traceback (most recent call last): ... NotImplementedError: General Ellipse is not supported but the equation of the reflected Ellipse is given by the zeros of: f(x, y) = (9*x/41 + 40*y/41 + 37/41)**2 + (40*x/123 - 3*y/41 - 364/123)**2 - 1 Notes ===== Until the general ellipse (with no axis parallel to the x-axis) is supported a NotImplemented error is raised and the equation whose zeros define the rotated ellipse is given. rcsg|]}t|fddqS)T)r-)r )rWrV)liner2r4r5rf;sz#Ellipse.reflect..ZxyT)Z simultaneouszeGeneral Ellipse is not supported but the equation of the reflected Ellipse is given by the zeros of: zf(%s, %s) = %sN)rr r7reflectfuncr8r9r0r%r1rr`ror str)r2rrNr,r.exprrerxr4)rr2r5rs    zEllipse.reflectrcsz|j|jkr$||j|||jS|tjjrBtt |||Sd|tjjrn||j|||j|jSt ddS)aRotate ``angle`` radians counterclockwise about Point ``pt``. Note: since the general ellipse is not supported, only rotations that are integer multiples of pi/2 are allowed. Examples ======== >>> from sympy import Ellipse, pi >>> Ellipse((1, 0), 2, 1).rotate(pi/2) Ellipse(Point2D(0, 1), 1, 2) >>> Ellipse((1, 0), 2, 1).rotate(pi) Ellipse(Point2D(-1, 0), 2, 1) r>z;Only rotations of pi/2 are currently supported for Ellipse.N) r8r9rr7rotaterr] is_integerr;r+ro)r2Zangler)r=r4r5rFs  zEllipse.rotater"cCs`|j}|r4t|dd}|j| j||j|jS|j}|j}|j|||||||dS)atOverride GeometryEntity.scale since it is the major and minor axes which must be scaled and they are not GeometryEntities. Examples ======== >>> from sympy import Ellipse >>> Ellipse((0, 0), 2, 1).scale(2, 4) Circle(Point2D(0, 0), 4) >>> Ellipse((0, 0), 2, 1).scale(2) Ellipse(Point2D(0, 0), 4, 1) r>)r?)r8r9)r7r% translater`scaler8r9r)r2r,r.rrNrOrPr4r4r5r_s  z Ellipse.scalecCst|dd}||rgS||krx|j|}|jd|j}|jd |j}tt|j|t|j|}t||gSt |j dkr|j \}}|j}d|t ||t ||} n|j t |j|} | j r| jrgStdtd} } || | } t| | | } t|t| | j}t|| | g| | g}t |dkr|dd|jksh|dd|jkshtt||tddt||tddgSt||dt||dgSdS)a`Tangent lines between `p` and the ellipse. If `p` is on the ellipse, returns the tangent line through point `p`. Otherwise, returns the tangent line(s) from `p` to the ellipse, or None if no tangent line is possible (e.g., `p` inside ellipse). Parameters ========== p : Point Returns ======= tangent_lines : list with 1 or 2 Lines Raises ====== NotImplementedError Can only find tangent lines for a point, `p`, on the ellipse. See Also ======== sympy.geometry.point.Point, sympy.geometry.line.Line Examples ======== >>> from sympy import Point, Ellipse >>> e1 = Ellipse(Point(0, 0), 3, 2) >>> e1.tangent_lines(Point(3, 0)) [Line2D(Point2D(3, 0), Point2D(3, -12))] r>)r?r,r.r"rN)r%rkr7r9r,r8r.rr(rArgrdrhZ is_numberrir r0r*rrAssertionError)r2reZdeltaZriserunZp2f1f2Zmajrjr,r.rrrZtangent_pointsr4r4r5ryus8%        &(zEllipse.tangent_linescCs |jdS)a\The vertical radius of the ellipse. Returns ======= vradius : number See Also ======== hradius, major, minor Examples ======== >>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.vradius 1 r>)r`)r2r4r4r5r9szEllipse.vradiuscCstj|j|jdd}tj|jd|jd}d}|dkrJ|||fS||j|d|jjd}||j|d|jjd}||j|d|jj|d|jj}|||fS)aReturns the second moment and product moment area of an ellipse. Parameters ========== point : Point, two-tuple of sympifiable objects, or None(default=None) point is the point about which second moment of area is to be found. If "point=None" it will be calculated about the axis passing through the centroid of the ellipse. Returns ======= I_xx, I_yy, I_xy : number or sympy expression I_xx, I_yy are second moment of area of an ellise. I_xy is product moment of area of an ellipse. Examples ======== >>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.second_moment_of_area() (3*pi/4, 27*pi/4, 0) References ========== https://en.wikipedia.org/wiki/List_of_second_moments_of_area rnrarNr"r>)rr]r8r9r^r7r.r,)r2pointZI_xxZI_yyZI_xyr4r4r5second_moment_of_areas" *zEllipse.second_moment_of_area)NNNN)rKrL)rU)r,r.)r,r.)N)rU)N)rN)r"r"N)N)(__name__ __module__ __qualname____doc__r6r:r<rGrQpropertyrRrTr\r^r_r7rbrIrkr0rprgrcr8rvr~rSrqrrrrrrrrryr9r __classcell__r4r4)r=r5r+#sF@       )    5 $ -  PK / / R  '  9, O r+c@s`eZdZdZddZeddZddd Zd d Zed d Z ddZ dddZ eddZ dS)rFa%A circle in space. Constructed simply from a center and a radius, or from three non-collinear points. Parameters ========== center : Point radius : number or sympy expression points : sequence of three Points Attributes ========== radius (synonymous with hradius, vradius, major and minor) circumference equation Raises ====== GeometryError When trying to construct circle from three collinear points. When trying to construct circle from incorrect parameters. See Also ======== Ellipse, sympy.geometry.point.Point Examples ======== >>> from sympy.geometry import Point, Circle >>> # a circle constructed from a center and radius >>> c1 = Circle(Point(0, 0), 5) >>> c1.hradius, c1.vradius, c1.radius (5, 5, 5) >>> # a circle constructed from three points >>> c2 = Circle(Point(0, 0), Point(1, 1), Point(1, 0)) >>> c2.hradius, c2.vradius, c2.radius, c2.center (sqrt(2)/2, sqrt(2)/2, sqrt(2)/2, Point2D(1/2, 1/2)) cOsd\}}t|dkrVdd|D}tj|r4tdddlm}||}|j}|j}n(t|dkr~t|d dd }t|d}|dks|dkst j |||f|Std dS) N)NNrncSsg|]}t|ddqS)r>)r?)r%)rWrr4r4r5rfCsz"Circle.__new__..z5Cannot construct a circle from three collinear pointsr")Triangler>r)r?z)Circle.__new__ received unknown arguments) rAr%Z is_collinearrpolygonrZ circumcenterZ circumradiusrr#rG)rHr`rJrNrrrUr4r4r5rG@s      zCircle.__new__cCsdtj|jS)a"The circumference of the circle. Returns ======= circumference : number or SymPy expression Examples ======== >>> from sympy import Point, Circle >>> c1 = Circle(Point(3, 4), 6) >>> c1.circumference 12*pi r>)rr]rh)r2r4r4r5rbUszCircle.circumferencer,r.cCsJt|dd}t|dd}||jjd}||jjd}|||jdS)aThe equation of the circle. Parameters ========== x : str or Symbol, optional Default value is 'x'. y : str or Symbol, optional Default value is 'y'. Returns ======= equation : SymPy expression Examples ======== >>> from sympy import Point, Circle >>> c1 = Circle(Point(0, 0), 5) >>> c1.equation() x**2 + y**2 - 25 T)r-r>)rr7r,r.rS)r2r,r.rlrmr4r4r5r0is   zCircle.equationcCs t||S)aThe intersection of this circle with another geometrical entity. Parameters ========== o : GeometryEntity Returns ======= intersection : list of GeometryEntities Examples ======== >>> from sympy import Point, Circle, Line, Ray >>> p1, p2, p3 = Point(0, 0), Point(5, 5), Point(6, 0) >>> p4 = Point(5, 0) >>> c1 = Circle(p1, 5) >>> c1.intersection(p2) [] >>> c1.intersection(p4) [Point2D(5, 0)] >>> c1.intersection(Ray(p1, p2)) [Point2D(5*sqrt(2)/2, 5*sqrt(2)/2)] >>> c1.intersection(Line(p2, p3)) [] )r+rv)r2r3r4r4r5rvszCircle.intersectioncCs |jdS)atThe radius of the circle. Returns ======= radius : number or sympy expression See Also ======== Ellipse.major, Ellipse.minor, Ellipse.hradius, Ellipse.vradius Examples ======== >>> from sympy import Point, Circle >>> c1 = Circle(Point(3, 4), 6) >>> c1.radius 6 r")r`)r2r4r4r5rhsz Circle.radiuscCs |j}||}|||j S)aOverride GeometryEntity.reflect since the radius is not a GeometryEntity. Examples ======== >>> from sympy import Circle, Line >>> Circle((0, 1), 1).reflect(Line((0, 0), (1, 1))) Circle(Point2D(1, 0), -1) )r7rrrh)r2rrNr4r4r5rs  zCircle.reflectr"NcCs|j}|r4t|dd}|j| j||j|jS|||}dd||fD\}}||krp||||jS|j}}t|||||dS)a;Override GeometryEntity.scale since the radius is not a GeometryEntity. Examples ======== >>> from sympy import Circle >>> Circle((0, 0), 1).scale(2, 2) Circle(Point2D(0, 0), 2) >>> Circle((0, 0), 1).scale(2, 4) Ellipse(Point2D(0, 0), 2, 4) r>)r?cSsg|] }t|qSr4)abs)rWrsr4r4r5rfsz Circle.scale..)r8r9)r7r%rr`rrrhr+)r2r,r.rrNrOrPr4r4r5rs    z Circle.scalecCs t|jS)a This Ellipse property is an alias for the Circle's radius. Whereas hradius, major and minor can use Ellipse's conventions, the vradius does not exist for a circle. It is always a positive value in order that the Circle, like Polygons, will have an area that can be positive or negative as determined by the sign of the hradius. Examples ======== >>> from sympy import Point, Circle >>> c1 = Circle(Point(3, 4), 6) >>> c1.vradius 6 )rrh)r2r4r4r5r9szCircle.vradius)r,r.)r"r"N) rrrrrGrrbr0rvrhrrr9r4r4r4r5rFs.    rF)ru)BrZ __future__rrZ sympy.corerrrZsympy.core.logicrZsympy.core.numbersrr Zsympy.core.compatibilityr r Zsympy.core.symbolr r rZsympy.simplifyrrZ(sympy.functions.elementary.miscellaneousrZ(sympy.functions.elementary.trigonometricrrZ*sympy.functions.special.elliptic_integralsrZsympy.geometry.exceptionsrZsympy.geometry.linerrrrZ sympy.polysrrrZsympy.polys.polyutilsrrZ sympy.solversrZsympy.utilities.miscr r!Zentityr#r$rr%r&r'rr(r)utilr*rr+rFrrur4r4r4r5sD      vo