3 ƽhL@s dZddlmZddlZddlZddlZddljZej eddddZ Gdd d e Z d d Z d d ZddZddZd4ddZGdddZd5ddZd6ddZddZd d!Zd7d#d$Zd%d&Zd'd(Zd8d*d+Zejd,d-d.d/d0Zejd,d1d.d2d3ZdS)9zO A module providing some utility functions regarding Bezier path manipulation. ) lru_cacheN)maxsizecCsF||kr dSt|||}tjd|d}tj|d||jtS)Nr)minnparangeprodZastypeint)nkir7/tmp/pip-build-7iwl8md4/matplotlib/matplotlib/bezier.py_combs rc@s eZdZdS)NonIntersectingPathExceptionN)__name__ __module__ __qualname__rrrrrsrcs||||}||||} || } } || } } | | | | tdkr\td| | }}| | }}fdd||||gD\}}}}|||| }|||| }||fS)z Return the intersection between the line through (*cx1*, *cy1*) at angle *t1* and the line through (*cx2*, *cy2*) at angle *t2*. g-q=zcGiven lines do not intersect. Please verify that the angles are not equal or differ by 180 degrees.csg|] }|qSrr).0r )ad_bcrr 9sz$get_intersection..)abs ValueError)Zcx1Zcy1cos_t1sin_t1Zcx2Zcy2cos_t2sin_t2Z line1_rhsZ line2_rhsabcdZa_Zb_Zc_Zd_xyr)rrget_intersection s      "r$c Csl|dkr||||fS|| }}| |}}||||||} } ||||||} } | | | | fS)z For a line passing through (*cx*, *cy*) and having an angle *t*, return locations of the two points located along its perpendicular line at the distance of *length*. gr) cxcyZcos_tZsin_tlengthrrrrx1y1Zx2y2rrrget_normal_pointsAs   r+cCs(|ddd||dd|}|S)Nrr)betatZ next_betarrr_de_casteljau1Zs$r/cCs`tj|}|g}x&t||}|j|t|dkrPqWdd|D}ddt|D}||fS)z Split a Bezier segment defined by its control points *beta* into two separate segments divided at *t* and return their control points. rcSsg|] }|dqS)rr)rr-rrrrksz&split_de_casteljau..cSsg|] }|dqS)rr,r)rr-rrrrls)rasarrayr/appendlenreversed)r-r.Z beta_listZ left_betaZ right_betarrrsplit_de_casteljau_s    r4?{Gz?c Cs||}||}||}||}||kr8||kr8tdxrtj|d|d|d|d|krj||fSd||} || } || } || Ar| }| }| }q:| }| }| }q:WdS)a Find the intersection of the Bezier curve with a closed path. The intersection point *t* is approximated by two parameters *t0*, *t1* such that *t0* <= *t* <= *t1*. Search starts from *t0* and *t1* and uses a simple bisecting algorithm therefore one of the end points must be inside the path while the other doesn't. The search stops when the distance of the points parametrized by *t0* and *t1* gets smaller than the given *tolerance*. Parameters ---------- bezier_point_at_t : callable A function returning x, y coordinates of the Bezier at parameter *t*. It must have the signature:: bezier_point_at_t(t: float) -> Tuple[float, float] inside_closedpath : callable A function returning True if a given point (x, y) is inside the closed path. It must have the signature:: inside_closedpath(point: Tuple[float, float]) -> bool t0, t1 : float Start parameters for the search. tolerance : float Maximal allowed distance between the final points. Returns ------- t0, t1 : float The Bezier path parameters. z3Both points are on the same side of the closed pathrrg?N)rrhypot) bezier_point_at_tinside_closedpatht0t1 tolerancestartendZ start_insideZ end_insideZmiddle_tZmiddleZ middle_insiderrr*find_bezier_t_intersecting_with_closedpathqs(&( r@c@s`eZdZdZddZddZddZedd Zed d Z ed d Z eddZ ddZ dS) BezierSegmentz A d-dimensional Bezier segment. Parameters ---------- control_points : (N, d) array Location of the *N* control points. csVtj|_jj\__tjj_fddtjD}jj |j _ dS)Ncs:g|]2}tjjdtj|tjjd|qS)r)math factorial_N)rr )selfrrrsz*BezierSegment.__init__..) rr0_cpointsshaperD_dr_ordersrangeT_px)rEcontrol_pointsZcoeffr)rEr__init__s   zBezierSegment.__init__cCs>tj|}tjjd||jdddtjj||j|jS)a9 Evaluate the Bezier curve at point(s) t in [0, 1]. Parameters ---------- t : float (k,), array_like Points at which to evaluate the curve. Returns ------- float (k, d), array_like Value of the curve for each point in *t*. rNr,)rr0powerouterrIrL)rEr.rrr__call__s zBezierSegment.__call__cCs t||S)z?Evaluate curve at a single point *t*. Returns a Tuple[float*d].)tuple)rEr.rrr point_at_tszBezierSegment.point_at_tcCs|jS)z The control points of the curve.)rF)rErrrrMszBezierSegment.control_pointscCs|jS)zThe dimension of the curve.)rH)rErrr dimensionszBezierSegment.dimensioncCs |jdS)z@Degree of the polynomial. One less the number of control points.r)rD)rErrrdegreeszBezierSegment.degreecCs||j}|dkrtjdt|j}tj|ddddf}tj|ddddf}d||t||}t||||S)a The polynomial coefficients of the Bezier curve. .. warning:: Follows opposite convention from `numpy.polyval`. Returns ------- float, (n+1, d) array_like Coefficients after expanding in polynomial basis, where :math:`n` is the degree of the bezier curve and :math:`d` its dimension. These are the numbers (:math:`C_j`) such that the curve can be written :math:`\sum_{j=0}^n C_j t^j`. Notes ----- The coefficients are calculated as .. math:: {n \choose j} \sum_{i=0}^j (-1)^{i+j} {j \choose i} P_i where :math:`P_i` are the control points of the curve. zFPolynomial coefficients formula unstable for high order Bezier curves!rNr,)rUwarningswarnRuntimeWarningrMrrr)rEr Pjr Z prefactorrrrpolynomial_coefficientssz%BezierSegment.polynomial_coefficientsc Cs|j}|dkr"tjgtjgfS|j}tjd|ddddf|dd}g}g}xFt|jD]8\}}tj|ddd}|j||jtj ||qdWtj |}tj |}tj ||dk@|dk@} || tj || fS)a Return the dimension and location of the curve's interior extrema. The extrema are the points along the curve where one of its partial derivatives is zero. Returns ------- dims : int, array_like Index :math:`i` of the partial derivative which is zero at each interior extrema. dzeros : float, array_like Of same size as dims. The :math:`t` such that :math:`d/dx_i B(t) = 0` rNrr,) rUrarrayr\r enumeraterKrootsr1Z full_like concatenateZisrealreal) rEr ZCjZdCjZdimsr_r pirZin_rangerrraxis_aligned_extremas(   z"BezierSegment.axis_aligned_extremaN) rrr__doc__rNrQrSpropertyrMrTrUr\rdrrrrrAs     $rAc Cs>t|}|j}t|||d\}}t|||d\}}||fS)an Split a Bezier curve into two at the intersection with a closed path. Parameters ---------- bezier : array-like(N, 2) Control points of the Bezier segment. See `.BezierSegment`. inside_closedpath : callable A function returning True if a given point (x, y) is inside the closed path. See also `.find_bezier_t_intersecting_with_closedpath`. tolerance : float The tolerance for the intersection. See also `.find_bezier_t_intersecting_with_closedpath`. Returns ------- left, right Lists of control points for the two Bezier segments. )r=g@)rArSr@r4) Zbezierr:r=Zbzr9r;r<_leftZ_rightrrr)split_bezier_intersecting_with_closedpath3s rhFcCs ddlm}|j}t|\}}||d d}|} d} d} x\|D]L\}}| } | t|d7} ||d d|krtj| d d|g} P|} qBWtd| jd} t | ||\}}t|dkr|j g}|j |j g}nht|dkr|j |j g}|j |j |j g}n bool rjcs$|\}}|d|dkS)Nrjr)Zxyr"r#)r%r&r2rr_fszinside_circle.._fr)r%r&rcrzr)r%r&ryr inside_circles r{cCsB||||}}||||d}|dkr2dS||||fS)Ng?r)r|r|r)Zx0Zy0r(r)ZdxZdyr!rrr get_cos_sins r}h㈵>cCsNtj||}tj||}t||}||kr0dSt|tj|krFdSdSdS)a Check if two lines are parallel. Parameters ---------- dx1, dy1, dx2, dy2 : float The gradients *dy*/*dx* of the two lines. tolerance : float The angular tolerance in radians up to which the lines are considered parallel. Returns ------- is_parallel - 1 if two lines are parallel in same direction. - -1 if two lines are parallel in opposite direction. - False otherwise. rFNr,)rZarctan2rrb)Zdx1Zdy1Zdx2Zdy2r=Ztheta1Ztheta2Zdthetarrrcheck_if_parallels   rcCs~|d\}}|d\}}|d\}}t||||||||}|dkrrtjdt||||\} } | | } } n$t||||\} } t||||\} } t||| | |\} }}}t||| | |\}}}}y8t| || | ||| | \}}t||| | ||| | \}}WnJtk rHd| |d||}}d||d||}}YnX| |f||f||fg}||f||f||fg}||fS)z Given the quadratic Bezier control points *bezier2*, returns control points of quadratic Bezier lines roughly parallel to given one separated by *width*. rrrjz8Lines do not intersect. A straight line is used instead.g?r,)rcbookZ_warn_externalr}r+r$r)bezier2widthc1xc1ycmxcmyc2xc2yZ parallel_testrrrrc1x_leftc1y_left c1x_right c1y_rightZc2x_leftZc2y_leftZ c2x_rightZ c2y_rightZcmx_leftZcmy_leftZ cmx_rightZ cmy_right path_left path_rightrrr get_parallelss>        rcCs>dd|||}dd|||}||f||f||fgS)z Find control points of the Bezier curve passing through (*c1x*, *c1y*), (*mmx*, *mmy*), and (*c2x*, *c2y*), at parametric values 0, 0.5, and 1. g?rlr)rrZmmxZmmyrrrrrrrfind_control_pointssr?c%Cs(|d\}}|d\}}|d\} } t||||\} } t||| | \} }t||| | ||\}}}}t| | | |||\}}}}||d||d}}|| d|| d}}||d||d}}t||||\}}t||||||\}} }!}"t|||| ||}#t|||!|"||}$|#|$fS)z Being similar to get_parallels, returns control points of two quadratic Bezier lines having a width roughly parallel to given one separated by *width*. rrrjg?)r}r+r)%rrZw1ZwmZw2rrrrZc3xZc3yrrrrrrrrZc3x_leftZc3y_leftZ c3x_rightZ c3y_rightZc12xZc12yZc23xZc23yZc123xZc123yZcos_t123Zsin_t123Z c123x_leftZ c123y_leftZ c123x_rightZ c123y_rightrrrrrmake_wedged_bezier2!s&   rz3.3z2Path.cleaned() and remove the final STOP if needed) alternativecCsRddlm}|j}|dkrJtjt|j|j|jd}|j |d<||j|S|SdS)z If the ``codes`` attribute of `.Path` *p* is None, return a copy of *p* with ``codes`` set to (MOVETO, LINETO, LINETO, ..., LINETO); otherwise return *p* itself. r)riN)Zdtyper) rnrirsrfullr2rtrpZ code_typerq)prir rrrmake_path_regularTs   rzPath.make_compound_path()cCsddlm}|j|S)z/Concatenate a list of paths into a single path.r)ri)rnriZmake_compound_path)pathsrirrrconcatenate_pathsfs r)r5r6r7)r7)r7F)r~)r6rr5)re functoolsrrBrWnumpyrZmatplotlib.cbookrZ vectorizerrrr$r+r/r4r@rArhrxr{r}rrrrZ deprecatedrrrrrrs2   ! D~ " = J 3