277 lines
21 KiB
Plaintext
277 lines
21 KiB
Plaintext
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†R jiJ ã óú — d Z ddlmZ ddlZddlZddlZddlmZ ej ed¬« d„ « « Z
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A module providing some utility functions regarding Bézier path manipulation.
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�7‰7�A˜‘E˜A‘I˜q‘=Ó !× (Ñ (¬Ó -Ð-ó c ó — e Zd Zy)Ú NonIntersectingPathExceptionN)Ú__name__Ú
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}}ˆfd„||||fD « \ }}}}||z || z z }||z || z z }||fS )zŽ
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Return the intersection between the line through (*cx1*, *cy1*) at angle
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*t1* and the line through (*cx2*, *cy2*) at angle *t2*.
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gê-�™—q=zcGiven lines do not intersect. Please verify that the angles are not equal or differ by 180 degrees.c 3 ó( •K — | ] }|‰z –— Œ y w)Nr )Ú.0r Úad_bcs €r ú <genexpr>z#get_intersection.<locals>.<genexpr>9 s øè ø€ Ò: A�a˜%•iÑ:ùs ƒ)ÚabsÚ
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ValueError)Úcx1Úcy1Úcos_t1Úsin_t1Úcx2Úcy2Úcos_t2Úsin_t2Ú line1_rhsÚ line2_rhsÚaÚbÚcÚdÚa_Úb_Úc_Úd_ÚxÚyr s @r Úget_intersectionr6 sÓ ø€ 𠘑 ˜v¨™|Ñ+€Iؘ‘ ˜v¨™|Ñ+€Ið �F�7€q€AØ �F�7€q€Aà
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ˆ5ƒz�EÒÜð Nó Oð Oð ��ˆ€B؈R�ˆ€BÛ:¨"¨b°"°bÐ)9Ô:�N€BˆˆB�à
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ˆY‰˜˜i™ Ñ'€AØ
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ˆY‰˜˜i™ Ñ'€Aà ˆaˆ4€Kr c óx — |dk( r| || |fS || }}| |}}||z | z ||z |z }
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} ||z | z ||z |z } } | |
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For a line passing through (*cx*, *cy*) and having an angle *t*, return
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locations of the two points located along its perpendicular line at the
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distance of *length*.
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ç r )
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ÚcxÚcyÚcos_tÚsin_tÚlengthr$ r% r( r) Úx1Úy1Úx2Úy2s
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r Úget_normal_pointsrB A ss € ð �‚|Ø�2�r˜2ˆ~Ð à˜U˜FˆF€FØ�V˜UˆF€Fà
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ˆr�2�rˆ>Ðr c ó. — | d d d|z
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z | dd |z z }|S )Néÿÿÿÿr r )ÚbetaÚtÚ next_betas r Ú_de_casteljau1rH Z s+ € Ø�S�b� ˜Q ™UÑ# d¨1¨2 h°¡lÑ2€IØ Ðr c ó — t j | « } | g} t | |« } |j | « t | « dk( rnŒ-|D � cg c] } | d ‘Œ }} t |« D � cg c] } | d ‘Œ }} ||fS c c} w c c} w )u•
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Split a Bézier segment defined by its control points *beta* into two
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separate segments divided at *t* and return their control points.
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r r rD )r
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ÚasarrayrH ÚappendÚlenÚreversed)rE rF Ú beta_listÚ left_betaÚ
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right_betas r Úsplit_de_casteljaurQ _ s‘ € ô
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&/Ö/˜T��a“Ð/€IÐ/Ü'/° Ó':Ö;˜t�$�r“(Ð;€JÐ;à �jÐ Ð ùò 0ùÚ;s Á A8Á& A=c óB — | |« } | |« } ||« } ||« }||k( r||k7 r t d« ‚ t j |d |d z
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|d |d z
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« |k r||fS d||z z } | | « }
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« } || z r| }||
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k( r||fS |
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}| }Œj)uŒ
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Find the intersection of the Bézier curve with a closed path.
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The intersection point *t* is approximated by two parameters *t0*, *t1*
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such that *t0* <= *t* <= *t1*.
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Search starts from *t0* and *t1* and uses a simple bisecting algorithm
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therefore one of the end points must be inside the path while the other
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doesn't. The search stops when the distance of the points parametrized by
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*t0* and *t1* gets smaller than the given *tolerance*.
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Parameters
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----------
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bezier_point_at_t : callable
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A function returning x, y coordinates of the Bézier at parameter *t*.
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It must have the signature::
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bezier_point_at_t(t: float) -> tuple[float, float]
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inside_closedpath : callable
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A function returning True if a given point (x, y) is inside the
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closed path. It must have the signature::
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inside_closedpath(point: tuple[float, float]) -> bool
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t0, t1 : float
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Start parameters for the search.
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tolerance : float
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Maximal allowed distance between the final points.
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Returns
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-------
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t0, t1 : float
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The Bézier path parameters.
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z3Both points are on the same side of the closed pathr r ç à?)r r
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Úhypot) Úbezier_point_at_tÚinside_closedpathÚt0Út1Ú toleranceÚstartÚendÚ start_insideÚ
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end_insideÚmiddle_tÚmiddleÚ
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middle_insides r Ú*find_bezier_t_intersecting_with_closedpathra q s÷ € ñL
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˜bÓ !€EÙ
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€Cá$ UÓ+€LÙ" 3Ó'€Jà�zÒ! e¨s¢lÜ*Ø AóCð Cð ô �8‰8�E˜!‘H˜s 1™vÑ% u¨Q¡x°#°a±&Ñ'8Ó 9¸IÒ EØ�r�6ˆMð ˜"˜r™'‘?ˆÙ" 8Ó,ˆÙ)¨&Ó1ˆ
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؉CàˆBؘŠð ˜2�v�
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؈EØ(ˆLð3 r c óh — e Zd ZdZd„ Zd„ Zd„ Zed„ « Zed„ « Z ed„ « Z
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ed„ « Z d „ Z y
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) Ú
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BezierSegmentu—
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A d-dimensional Bézier segment.
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Parameters
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----------
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control_points : (N, d) array
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Location of the *N* control points.
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c ó — t j |« | _ | j j \ | _ | _ t j | j « | _ t | j « D �cg c]` }t j | j dz
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« t j |« t j | j dz
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|z
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« z z ‘Œb }}| j j |z j | _ y c c}w )Nr )
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rJ Ú_cpointsÚshapeÚ_NÚ_dr Ú_ordersÚrangeÚmathÚ factorialÚTÚ_px)ÚselfÚcontrol_pointsr Úcoeffs r Ú__init__zBezierSegment.__init__Ç sà € ÜŸ
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>Ó2ˆŒ
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Ø Ÿ=™=×.Ñ.шŒ�”Ü—y‘y §¡Ó)ˆŒ ô §¡›.ö*àô — ‘ §¡¨!¡ Ó,Ü—^‘^ AÓ&¬¯ © ¸¿¹À!¹ Àa¹Ó)HÑHóJð *ˆð *ð —M‘M—O‘O eÑ+×.Ñ.ˆ�ùò*s Á9A%D c ó — t j |« }t j j d|z
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z S )u)
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Evaluate the Bézier curve at point(s) *t* in [0, 1].
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Parameters
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----------
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t : (k,) array-like
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Points at which to evaluate the curve.
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Returns
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-------
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(k, d) array
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Value of the curve for each point in *t*.
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r NrD )r
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rJ ÚpowerÚouterri rn ©ro rF s r Ú__call__zBezierSegment.__call__Ð s` € ô
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�J‰J�q‹MˆÜ—‘—‘˜q 1™u d§l¡l±4°R°4Ñ&8Ó9Ü—(‘(—.‘. D§L¡LÓ1ñ2Ø59·X±Xñ>ð >r c ó$ — t | |« « S )zX
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Evaluate the curve at a single point, returning a tuple of *d* floats.
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)Útuplerv s r Ú
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point_at_tzBezierSegment.point_at_tâ s € ô ‘T˜!“W‹~Ð r c ó — | j S )z The control points of the curve.)re ©ro s r rp z BezierSegment.control_pointsè s € ð �}‰}Ð r c ó — | j S )zThe dimension of the curve.)rh r| s r Ú dimensionzBezierSegment.dimensioní s
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€ ð �w‰wˆr c ó — | j dz
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S )z@Degree of the polynomial. One less the number of control points.r )rg r| s r ÚdegreezBezierSegment.degreeò s € ð �w‰w˜‰{Ðr c ó: — | j }|dkD rt j dt « | j }t j |dz « dd…df }t j |dz « ddd…f }d||z z t ||« z }t ||« |z |z S )uº
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The polynomial coefficients of the Bézier curve.
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.. warning:: Follows opposite convention from `numpy.polyval`.
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Returns
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-------
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(n+1, d) array
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Coefficients after expanding in polynomial basis, where :math:`n`
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is the degree of the Bézier curve and :math:`d` its dimension.
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These are the numbers (:math:`C_j`) such that the curve can be
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written :math:`\sum_{j=0}^n C_j t^j`.
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Notes
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-----
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The coefficients are calculated as
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.. math::
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{n \choose j} \sum_{i=0}^j (-1)^{i+j} {j \choose i} P_i
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where :math:`P_i` are the control points of the curve.
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é
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zFPolynomial coefficients formula unstable for high order Bezier curves!r NrD )r€ ÚwarningsÚwarnÚRuntimeWarningrp r
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r r )ro r ÚPÚjr Ú prefactors r Úpolynomial_coefficientsz%BezierSegment.polynomial_coefficients÷ s– € ð2
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�K‰Kˆà ˆrŠ6Ü �M‰Mð 1Ü2@ô
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Bà × Ñ ˆÜ �I‰I�a˜‘c‹Nš1˜d˜7Ñ #ˆÜ �I‰I�a˜‘c‹N˜4¢˜7Ñ #ˆØ˜1˜q™5‘M¤E¨!¨Q£KÑ/ˆ Ü�Q˜‹{˜YÑ&¨Ñ*Ð*r c ó� — | j }|dk r*t j g « t j g « fS | j }t j d|dz « dd…df |dd z }g }g }t |j « D ]V \ }}t j |ddd… « }|j |« |j t j ||« « ŒX t j |« }t j |« }t j |« |dk\ z |dk z } || t j |« | fS )aã
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Return the dimension and location of the curve's interior extrema.
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The extrema are the points along the curve where one of its partial
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derivatives is zero.
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Returns
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-------
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dims : array of int
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Index :math:`i` of the partial derivative which is zero at each
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interior extrema.
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dzeros : array of float
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Of same size as dims. The :math:`t` such that :math:`d/dx_i B(t) =
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0`
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r NrD r )
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Úarrayr‰ r Ú enumeraterm ÚrootsrK Ú full_likeÚ concatenateÚisrealÚreal)
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r Úaxis_aligned_extremaz"BezierSegment.axis_aligned_extrema s € ð
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�K‰KˆØ �Š6Ü—8‘8˜B“<¤§¡¨"£ Ð-Ð -Ø
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×
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)ˆÜ�i‰i˜˜1˜Q™3Ó ¢ 4 Ñ(¨2¨a¨b¨6Ñ1ˆØˆØˆÜ ˜sŸu™uÓ%ò ,‰EˆAˆrÜ—‘˜™D˜b˜D™Ó"ˆAØ �L‰L˜ŒOØ �K‰KœŸ ™ Q¨Ó*Õ +ð ,ô —‘˜uÓ%ˆÜ�~‰~˜dÓ#ˆÜ—9‘9˜UÓ# u°¡zÑ2°e¸q±jÑAˆØ�H‰~œrŸw™w u›~¨hÑ 7Ð7Ð7r N)
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r r r Ú__doc__rr rw rz Úpropertyrp r~ r€ r‰ r˜ r r r rc rc ½ sl „ ñò/ò>ò$ ð ñ ó ð ð ñó ðð ñó ðð ñ!+ó ð!+óF 8r rc c ó„ — t | « }|j }t |||¬« \ }}t | ||z dz « \ }}||fS )ur
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Split a Bézier curve into two at the intersection with a closed path.
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Parameters
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----------
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bezier : (N, 2) array-like
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Control points of the Bézier segment. See `.BezierSegment`.
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inside_closedpath : callable
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A function returning True if a given point (x, y) is inside the
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closed path. See also `.find_bezier_t_intersecting_with_closedpath`.
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tolerance : float
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The tolerance for the intersection. See also
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`.find_bezier_t_intersecting_with_closedpath`.
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Returns
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-------
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left, right
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Lists of control points for the two Bézier segments.
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r )ÚPathéþÿÿÿNr é z*The path does not intersect with the patch)rD r¤ é é z This should never be reached)Úpathr¢ Ú
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!Ñ!ˆÙ �*˜R˜S�/Ó " lÒ 2Ü Ÿ.™.¨.¸¸Ð*=¸zÐ)JÓKˆKÙ Ø#‰ð
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Gô ÐEÓFÐFà × Ñ ˜WÓ %€BÜ;Ø
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ˆF�Ió �K€Dˆ%ä
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ˆ4ƒy�A‚~Ø—k‘k�]ˆ
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Ø—{‘{ D§K¡KÐ0‰ Ü ˆT‹�aŠ Ø—k‘k 4§;¡;Ð/ˆ
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Ø—{‘{ D§K¡K°·±Ð=‰ Ü ˆT‹�aŠ Ø—k‘k 4§;¡;°· ± Ð<ˆ
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Ø—{‘{ D§K¡K°·±¸d¿k¹kÐJ‰ ä Ð ;Ó<Ð<à�a�b�€JØ™�(€Kà ‡z�zÐÙ”r—~‘~ t§}¡}°R°aÐ'8¸*Ð&EÓFÓGˆÙœŸ™¨ °T·]±]À1À2Ð5FÐ'GÓHÓI‰ñ ”r—~‘~ t§}¡}°U°dÐ';¸ZÐ&HÓIÜ—~‘~ t§z¡z°%°4Ð'8¸*Ð&EÓFóHˆñ œŸ™¨ °T·]±]À1À2Ð5FÐ'GÓHÜŸ™¨ °T·Z±ZÀÀ°^Ð'DÓEóGˆñ ™\Ø $ g�ˆà �HÐ Ð r c ó$ ‡ ‡‡— |dz Šˆ ˆˆfd„}|S )zÎ
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Return a function that checks whether a point is in a circle with center
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(*cx*, *cy*) and radius *r*.
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The returned function has the signature::
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f(xy: tuple[float, float]) -> bool
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r¤ c ó6 •— | \ }}|‰z
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dz |‰z
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dz z ‰k S )Nr¤ r )Úxyr4 r5 r9 r: Úr2s €€€r Ú_fzinside_circle.<locals>._f§ s, ø€ ؉ˆˆ1Ø�B‘˜1‰} B¡¨1™}Ñ,¨rÑ1Ð1r r )r9 r: r– rÉ rÈ s `` @r Ú
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inside_circlerÊ œ s ú€ ð
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ˆa‰€Bö2ð €Ir c óR — || z
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||z
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}}||z ||z z dz }|dk( ry||z ||z fS )NrS r )r8 r8 r )Úx0Úy0r> r? ÚdxÚdyr/ s r Ú get_cos_sinrÐ ¯ sF € Ø
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�"‰W�b˜2‘gˆ€BØ ˆb‰�2˜‘7Ñ ˜rÑ!€AàˆA‚vØØ
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�‰6�2˜‘6ˆ>Ðr c óÄ — t j | |« }t j ||« }t ||z
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« }||k ryt |t j z
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« |k ryy)aË
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Check if two lines are parallel.
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Parameters
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----------
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dx1, dy1, dx2, dy2 : float
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The gradients *dy*/*dx* of the two lines.
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tolerance : float
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The angular tolerance in radians up to which the lines are considered
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parallel.
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Returns
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-------
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is_parallel
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- 1 if two lines are parallel in same direction.
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- -1 if two lines are parallel in opposite direction.
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- False otherwise.
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r rD F)r
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Úarctan2r r• )Údx1Údy1Údx2Údy2rY Útheta1Útheta2Údthetas r Úcheck_if_parallelrÚ ¸ sX € ô& �Z‰Z˜˜SÓ
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