# -*- coding: ISO-8859-1 -*- # # # Copyright (C) 2002-2006 Jörg Lehmann # Copyright (C) 2003-2006 Michael Schindler # Copyright (C) 2002-2006 André Wobst # # This file is part of PyX (http://pyx.sourceforge.net/). # # PyX is free software; you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation; either version 2 of the License, or # (at your option) any later version. # # PyX is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # You should have received a copy of the GNU General Public License # along with PyX; if not, write to the Free Software # Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA from __future__ import nested_scopes import math try: from math import radians, degrees except ImportError: # fallback implementation for Python 2.1 def radians(x): return x*math.pi/180 def degrees(x): return x*180/math.pi import mathutils, path, trafo, unit import bbox as bboxmodule try: sum([]) except NameError: # fallback implementation for Python 2.2 and below def sum(list): return reduce(lambda x, y: x+y, list, 0) try: enumerate([]) except NameError: # fallback implementation for Python 2.2 and below def enumerate(list): return zip(xrange(len(list)), list) # use new style classes when possible __metaclass__ = type class _marker: pass ################################################################################ # specific exception for normpath-related problems class NormpathException(Exception): pass # invalid result marker class _invalid: """invalid result marker class The following norm(sub)path(item) methods: - trafo - rotation - tangent_pt - tangent - curvature_pt - curvradius_pt return list of result values, which might contain the invalid instance defined below to signal points, where the result is undefined due to properties of the norm(sub)path(item). Accessing invalid leads to an NormpathException, but you can test the result values by "is invalid". """ def invalid1(self): raise NormpathException("invalid result (the requested value is undefined due to path properties)") __str__ = __repr__ = __neg__ = invalid1 def invalid2(self, other): self.invalid1() __cmp__ = __add__ = __iadd__ = __sub__ = __isub__ = __mul__ = __imul__ = __div__ = __truediv__ = __idiv__ = invalid2 invalid = _invalid() ################################################################################ # global epsilon (default precision of normsubpaths) _epsilon = 1e-5 # minimal relative speed (abort condition for tangent information) _minrelspeed = 1e-5 def set(epsilon=None, minrelspeed=None): global _epsilon global _minrelspeed if epsilon is not None: _epsilon = epsilon if minrelspeed is not None: _minrelspeed = minrelspeed ################################################################################ # normsubpathitems ################################################################################ class normsubpathitem: """element of a normalized sub path Various operations on normsubpathitems might be subject of approximitions. Those methods get the finite precision epsilon, which is the accuracy needed expressed as a length in pts. normsubpathitems should never be modified inplace, since references might be shared between several normsubpaths. """ def arclen_pt(self, epsilon): """return arc length in pts""" pass def _arclentoparam_pt(self, lengths_pt, epsilon): """return a tuple of params and the total length arc length in pts""" pass def arclentoparam_pt(self, lengths_pt, epsilon): """return a tuple of params""" pass def at_pt(self, params): """return coordinates at params in pts""" pass def atbegin_pt(self): """return coordinates of first point in pts""" pass def atend_pt(self): """return coordinates of last point in pts""" pass def bbox(self): """return bounding box of normsubpathitem""" pass def cbox(self): """return control box of normsubpathitem The control box also fully encloses the normsubpathitem but in the case of a Bezier curve it is not the minimal box doing so. On the other hand, it is much faster to calculate. """ pass def curvature_pt(self, params): """return the curvature at params in 1/pts The result contains the invalid instance at positions, where the curvature is undefined.""" pass def curveradius_pt(self, params): """return the curvature radius at params in pts The curvature radius is the inverse of the curvature. Where the curvature is undefined, the invalid instance is returned. Note that this radius can be negative or positive, depending on the sign of the curvature.""" pass def intersect(self, other, epsilon): """intersect self with other normsubpathitem""" pass def modifiedbegin_pt(self, x_pt, y_pt): """return a normsubpathitem with a modified beginning point""" pass def modifiedend_pt(self, x_pt, y_pt): """return a normsubpathitem with a modified end point""" pass def _paramtoarclen_pt(self, param, epsilon): """return a tuple of arc lengths and the total arc length in pts""" pass def pathitem(self): """return pathitem corresponding to normsubpathitem""" def reversed(self): """return reversed normsubpathitem""" pass def rotation(self, params): """return rotation trafos (i.e. trafos without translations) at params""" pass def segments(self, params): """return segments of the normsubpathitem The returned list of normsubpathitems for the segments between the params. params needs to contain at least two values. """ pass def trafo(self, params): """return transformations at params""" def transformed(self, trafo): """return transformed normsubpathitem according to trafo""" pass def outputPS(self, file, writer): """write PS code corresponding to normsubpathitem to file""" pass def outputPDF(self, file, writer): """write PDF code corresponding to normsubpathitem to file""" pass class normline_pt(normsubpathitem): """Straight line from (x0_pt, y0_pt) to (x1_pt, y1_pt) (coordinates in pts)""" __slots__ = "x0_pt", "y0_pt", "x1_pt", "y1_pt" def __init__(self, x0_pt, y0_pt, x1_pt, y1_pt): self.x0_pt = x0_pt self.y0_pt = y0_pt self.x1_pt = x1_pt self.y1_pt = y1_pt def __str__(self): return "normline_pt(%g, %g, %g, %g)" % (self.x0_pt, self.y0_pt, self.x1_pt, self.y1_pt) def _arclentoparam_pt(self, lengths_pt, epsilon): # do self.arclen_pt inplace for performance reasons l_pt = math.hypot(self.x0_pt-self.x1_pt, self.y0_pt-self.y1_pt) return [length_pt/l_pt for length_pt in lengths_pt], l_pt def arclentoparam_pt(self, lengths_pt, epsilon): """return a tuple of params""" return self._arclentoparam_pt(lengths_pt, epsilon)[0] def arclen_pt(self, epsilon): return math.hypot(self.x0_pt-self.x1_pt, self.y0_pt-self.y1_pt) def at_pt(self, params): return [(self.x0_pt+(self.x1_pt-self.x0_pt)*t, self.y0_pt+(self.y1_pt-self.y0_pt)*t) for t in params] def atbegin_pt(self): return self.x0_pt, self.y0_pt def atend_pt(self): return self.x1_pt, self.y1_pt def bbox(self): return bboxmodule.bbox_pt(min(self.x0_pt, self.x1_pt), min(self.y0_pt, self.y1_pt), max(self.x0_pt, self.x1_pt), max(self.y0_pt, self.y1_pt)) cbox = bbox def curvature_pt(self, params): return [0] * len(params) def curveradius_pt(self, params): return [invalid] * len(params) def intersect(self, other, epsilon): if isinstance(other, normline_pt): a_deltax_pt = self.x1_pt - self.x0_pt a_deltay_pt = self.y1_pt - self.y0_pt b_deltax_pt = other.x1_pt - other.x0_pt b_deltay_pt = other.y1_pt - other.y0_pt try: det = 1.0 / (b_deltax_pt * a_deltay_pt - b_deltay_pt * a_deltax_pt) except ArithmeticError: return [] ba_deltax0_pt = other.x0_pt - self.x0_pt ba_deltay0_pt = other.y0_pt - self.y0_pt a_t = (b_deltax_pt * ba_deltay0_pt - b_deltay_pt * ba_deltax0_pt) * det b_t = (a_deltax_pt * ba_deltay0_pt - a_deltay_pt * ba_deltax0_pt) * det # check for intersections out of bound # TODO: we might allow for a small out of bound errors. if not (0<=a_t<=1 and 0<=b_t<=1): return [] # return parameters of intersection return [(a_t, b_t)] else: return [(s_t, o_t) for o_t, s_t in other.intersect(self, epsilon)] def modifiedbegin_pt(self, x_pt, y_pt): return normline_pt(x_pt, y_pt, self.x1_pt, self.y1_pt) def modifiedend_pt(self, x_pt, y_pt): return normline_pt(self.x0_pt, self.y0_pt, x_pt, y_pt) def _paramtoarclen_pt(self, params, epsilon): totalarclen_pt = self.arclen_pt(epsilon) arclens_pt = [totalarclen_pt * param for param in params + [1]] return arclens_pt[:-1], arclens_pt[-1] def pathitem(self): return path.lineto_pt(self.x1_pt, self.y1_pt) def reversed(self): return normline_pt(self.x1_pt, self.y1_pt, self.x0_pt, self.y0_pt) def rotation(self, params): return [trafo.rotate(degrees(math.atan2(self.y1_pt-self.y0_pt, self.x1_pt-self.x0_pt)))]*len(params) def segments(self, params): if len(params) < 2: raise ValueError("at least two parameters needed in segments") result = [] xl_pt = yl_pt = None for t in params: xr_pt = self.x0_pt + (self.x1_pt-self.x0_pt)*t yr_pt = self.y0_pt + (self.y1_pt-self.y0_pt)*t if xl_pt is not None: result.append(normline_pt(xl_pt, yl_pt, xr_pt, yr_pt)) xl_pt = xr_pt yl_pt = yr_pt return result def trafo(self, params): rotate = trafo.rotate(degrees(math.atan2(self.y1_pt-self.y0_pt, self.x1_pt-self.x0_pt))) return [trafo.translate_pt(*at_pt) * rotate for param, at_pt in zip(params, self.at_pt(params))] def transformed(self, trafo): return normline_pt(*(trafo.apply_pt(self.x0_pt, self.y0_pt) + trafo.apply_pt(self.x1_pt, self.y1_pt))) def outputPS(self, file, writer): file.write("%g %g lineto\n" % (self.x1_pt, self.y1_pt)) def outputPDF(self, file, writer): file.write("%f %f l\n" % (self.x1_pt, self.y1_pt)) class normcurve_pt(normsubpathitem): """Bezier curve with control points x0_pt, y0_pt, x1_pt, y1_pt, x2_pt, y2_pt, x3_pt, y3_pt (coordinates in pts)""" __slots__ = "x0_pt", "y0_pt", "x1_pt", "y1_pt", "x2_pt", "y2_pt", "x3_pt", "y3_pt" def __init__(self, x0_pt, y0_pt, x1_pt, y1_pt, x2_pt, y2_pt, x3_pt, y3_pt): self.x0_pt = x0_pt self.y0_pt = y0_pt self.x1_pt = x1_pt self.y1_pt = y1_pt self.x2_pt = x2_pt self.y2_pt = y2_pt self.x3_pt = x3_pt self.y3_pt = y3_pt def __str__(self): return "normcurve_pt(%g, %g, %g, %g, %g, %g, %g, %g)" % (self.x0_pt, self.y0_pt, self.x1_pt, self.y1_pt, self.x2_pt, self.y2_pt, self.x3_pt, self.y3_pt) def _midpointsplit(self, epsilon): """split curve into two parts Helper method to reduce the complexity of a problem by turning a normcurve_pt into several normline_pt segments. This method returns normcurve_pt instances only, when they are not yet straight enough to be replaceable by normcurve_pt instances. Thus a recursive midpointsplitting will turn a curve into line segments with the given precision epsilon. """ # first, we have to calculate the midpoints between adjacent # control points x01_pt = 0.5*(self.x0_pt + self.x1_pt) y01_pt = 0.5*(self.y0_pt + self.y1_pt) x12_pt = 0.5*(self.x1_pt + self.x2_pt) y12_pt = 0.5*(self.y1_pt + self.y2_pt) x23_pt = 0.5*(self.x2_pt + self.x3_pt) y23_pt = 0.5*(self.y2_pt + self.y3_pt) # In the next iterative step, we need the midpoints between 01 and 12 # and between 12 and 23 x01_12_pt = 0.5*(x01_pt + x12_pt) y01_12_pt = 0.5*(y01_pt + y12_pt) x12_23_pt = 0.5*(x12_pt + x23_pt) y12_23_pt = 0.5*(y12_pt + y23_pt) # Finally the midpoint is given by xmidpoint_pt = 0.5*(x01_12_pt + x12_23_pt) ymidpoint_pt = 0.5*(y01_12_pt + y12_23_pt) # Before returning the normcurves we check whether we can # replace them by normlines within an error of epsilon pts. # The maximal error value is given by the modulus of the # difference between the length of the control polygon # (i.e. |P1-P0|+|P2-P1|+|P3-P2|), which consitutes an upper # bound for the length, and the length of the straight line # between start and end point of the normcurve (i.e. |P3-P1|), # which represents a lower bound. l0_pt = math.hypot(xmidpoint_pt - self.x0_pt, ymidpoint_pt - self.y0_pt) l1_pt = math.hypot(x01_pt - self.x0_pt, y01_pt - self.y0_pt) l2_pt = math.hypot(x01_12_pt - x01_pt, y01_12_pt - y01_pt) l3_pt = math.hypot(xmidpoint_pt - x01_12_pt, ymidpoint_pt - y01_12_pt) if l1_pt+l2_pt+l3_pt-l0_pt < epsilon: a = _leftnormline_pt(self.x0_pt, self.y0_pt, xmidpoint_pt, ymidpoint_pt, l1_pt, l2_pt, l3_pt) else: a = _leftnormcurve_pt(self.x0_pt, self.y0_pt, x01_pt, y01_pt, x01_12_pt, y01_12_pt, xmidpoint_pt, ymidpoint_pt) l0_pt = math.hypot(self.x3_pt - xmidpoint_pt, self.y3_pt - ymidpoint_pt) l1_pt = math.hypot(x12_23_pt - xmidpoint_pt, y12_23_pt - ymidpoint_pt) l2_pt = math.hypot(x23_pt - x12_23_pt, y23_pt - y12_23_pt) l3_pt = math.hypot(self.x3_pt - x23_pt, self.y3_pt - y23_pt) if l1_pt+l2_pt+l3_pt-l0_pt < epsilon: b = _rightnormline_pt(xmidpoint_pt, ymidpoint_pt, self.x3_pt, self.y3_pt, l1_pt, l2_pt, l3_pt) else: b = _rightnormcurve_pt(xmidpoint_pt, ymidpoint_pt, x12_23_pt, y12_23_pt, x23_pt, y23_pt, self.x3_pt, self.y3_pt) return a, b def _arclentoparam_pt(self, lengths_pt, epsilon): a, b = self._midpointsplit(epsilon) params_a, arclen_a_pt = a._arclentoparam_pt(lengths_pt, epsilon) params_b, arclen_b_pt = b._arclentoparam_pt([length_pt - arclen_a_pt for length_pt in lengths_pt], epsilon) params = [] for param_a, param_b, length_pt in zip(params_a, params_b, lengths_pt): if length_pt > arclen_a_pt: params.append(b.subparamtoparam(param_b)) else: params.append(a.subparamtoparam(param_a)) return params, arclen_a_pt + arclen_b_pt def arclentoparam_pt(self, lengths_pt, epsilon): """return a tuple of params""" return self._arclentoparam_pt(lengths_pt, epsilon)[0] def arclen_pt(self, epsilon): a, b = self._midpointsplit(epsilon) return a.arclen_pt(epsilon) + b.arclen_pt(epsilon) def at_pt(self, params): return [( (-self.x0_pt+3*self.x1_pt-3*self.x2_pt+self.x3_pt)*t*t*t + (3*self.x0_pt-6*self.x1_pt+3*self.x2_pt )*t*t + (-3*self.x0_pt+3*self.x1_pt )*t + self.x0_pt, (-self.y0_pt+3*self.y1_pt-3*self.y2_pt+self.y3_pt)*t*t*t + (3*self.y0_pt-6*self.y1_pt+3*self.y2_pt )*t*t + (-3*self.y0_pt+3*self.y1_pt )*t + self.y0_pt ) for t in params] def atbegin_pt(self): return self.x0_pt, self.y0_pt def atend_pt(self): return self.x3_pt, self.y3_pt def bbox(self): xmin_pt, xmax_pt = path._bezierpolyrange(self.x0_pt, self.x1_pt, self.x2_pt, self.x3_pt) ymin_pt, ymax_pt = path._bezierpolyrange(self.y0_pt, self.y1_pt, self.y2_pt, self.y3_pt) return bboxmodule.bbox_pt(xmin_pt, ymin_pt, xmax_pt, ymax_pt) def cbox(self): return bboxmodule.bbox_pt(min(self.x0_pt, self.x1_pt, self.x2_pt, self.x3_pt), min(self.y0_pt, self.y1_pt, self.y2_pt, self.y3_pt), max(self.x0_pt, self.x1_pt, self.x2_pt, self.x3_pt), max(self.y0_pt, self.y1_pt, self.y2_pt, self.y3_pt)) def curvature_pt(self, params): result = [] # see notes in rotation approxarclen = (math.hypot(self.x1_pt-self.x0_pt, self.y1_pt-self.y0_pt) + math.hypot(self.x2_pt-self.x1_pt, self.y2_pt-self.y1_pt) + math.hypot(self.x3_pt-self.x2_pt, self.y3_pt-self.y2_pt)) for param in params: xdot = ( 3 * (1-param)*(1-param) * (-self.x0_pt + self.x1_pt) + 6 * (1-param)*param * (-self.x1_pt + self.x2_pt) + 3 * param*param * (-self.x2_pt + self.x3_pt) ) ydot = ( 3 * (1-param)*(1-param) * (-self.y0_pt + self.y1_pt) + 6 * (1-param)*param * (-self.y1_pt + self.y2_pt) + 3 * param*param * (-self.y2_pt + self.y3_pt) ) xddot = ( 6 * (1-param) * (self.x0_pt - 2*self.x1_pt + self.x2_pt) + 6 * param * (self.x1_pt - 2*self.x2_pt + self.x3_pt) ) yddot = ( 6 * (1-param) * (self.y0_pt - 2*self.y1_pt + self.y2_pt) + 6 * param * (self.y1_pt - 2*self.y2_pt + self.y3_pt) ) hypot = math.hypot(xdot, ydot) if hypot/approxarclen > _minrelspeed: result.append((xdot*yddot - ydot*xddot) / hypot**3) else: result.append(invalid) return result def curveradius_pt(self, params): result = [] # see notes in rotation approxarclen = (math.hypot(self.x1_pt-self.x0_pt, self.y1_pt-self.y0_pt) + math.hypot(self.x2_pt-self.x1_pt, self.y2_pt-self.y1_pt) + math.hypot(self.x3_pt-self.x2_pt, self.y3_pt-self.y2_pt)) for param in params: xdot = ( 3 * (1-param)*(1-param) * (-self.x0_pt + self.x1_pt) + 6 * (1-param)*param * (-self.x1_pt + self.x2_pt) + 3 * param*param * (-self.x2_pt + self.x3_pt) ) ydot = ( 3 * (1-param)*(1-param) * (-self.y0_pt + self.y1_pt) + 6 * (1-param)*param * (-self.y1_pt + self.y2_pt) + 3 * param*param * (-self.y2_pt + self.y3_pt) ) xddot = ( 6 * (1-param) * (self.x0_pt - 2*self.x1_pt + self.x2_pt) + 6 * param * (self.x1_pt - 2*self.x2_pt + self.x3_pt) ) yddot = ( 6 * (1-param) * (self.y0_pt - 2*self.y1_pt + self.y2_pt) + 6 * param * (self.y1_pt - 2*self.y2_pt + self.y3_pt) ) hypot = math.hypot(xdot, ydot) if hypot/approxarclen > _minrelspeed: result.append(hypot**3 / (xdot*yddot - ydot*xddot)) else: result.append(invalid) return result def intersect(self, other, epsilon): # There can be no intersection point, when the control boxes are not # overlapping. Note that we use the control box instead of the bounding # box here, because the former can be calculated more efficiently for # Bezier curves. if not self.cbox().intersects(other.cbox()): return [] a, b = self._midpointsplit(epsilon) # To improve the performance in the general case we alternate the # splitting process between the two normsubpathitems return ( [(a.subparamtoparam(a_t), o_t) for o_t, a_t in other.intersect(a, epsilon)] + [(b.subparamtoparam(b_t), o_t) for o_t, b_t in other.intersect(b, epsilon)] ) def modifiedbegin_pt(self, x_pt, y_pt): return normcurve_pt(x_pt, y_pt, self.x1_pt, self.y1_pt, self.x2_pt, self.y2_pt, self.x3_pt, self.y3_pt) def modifiedend_pt(self, x_pt, y_pt): return normcurve_pt(self.x0_pt, self.y0_pt, self.x1_pt, self.y1_pt, self.x2_pt, self.y2_pt, x_pt, y_pt) def _paramtoarclen_pt(self, params, epsilon): arclens_pt = [segment.arclen_pt(epsilon) for segment in self.segments([0] + list(params) + [1])] for i in range(1, len(arclens_pt)): arclens_pt[i] += arclens_pt[i-1] return arclens_pt[:-1], arclens_pt[-1] def pathitem(self): return path.curveto_pt(self.x1_pt, self.y1_pt, self.x2_pt, self.y2_pt, self.x3_pt, self.y3_pt) def reversed(self): return normcurve_pt(self.x3_pt, self.y3_pt, self.x2_pt, self.y2_pt, self.x1_pt, self.y1_pt, self.x0_pt, self.y0_pt) def rotation(self, params): result = [] # We need to take care of the case of tdx_pt and tdy_pt close to zero. # We should not compare those values to epsilon (which is a length) directly. # Furthermore we want this "speed" in general and it's abort condition in # particular to be invariant on the actual size of the normcurve. Hence we # first calculate a crude approximation for the arclen. approxarclen = (math.hypot(self.x1_pt-self.x0_pt, self.y1_pt-self.y0_pt) + math.hypot(self.x2_pt-self.x1_pt, self.y2_pt-self.y1_pt) + math.hypot(self.x3_pt-self.x2_pt, self.y3_pt-self.y2_pt)) for param in params: tdx_pt = (3*( -self.x0_pt+3*self.x1_pt-3*self.x2_pt+self.x3_pt)*param*param + 2*( 3*self.x0_pt-6*self.x1_pt+3*self.x2_pt )*param + (-3*self.x0_pt+3*self.x1_pt )) tdy_pt = (3*( -self.y0_pt+3*self.y1_pt-3*self.y2_pt+self.y3_pt)*param*param + 2*( 3*self.y0_pt-6*self.y1_pt+3*self.y2_pt )*param + (-3*self.y0_pt+3*self.y1_pt )) # We scale the speed such the "relative speed" of a line is 1 independend of # the length of the line. For curves we want this "relative speed" to be higher than # _minrelspeed: if math.hypot(tdx_pt, tdy_pt)/approxarclen > _minrelspeed: result.append(trafo.rotate(degrees(math.atan2(tdy_pt, tdx_pt)))) else: # Note that we can't use the rule of l'Hopital here, since it would # not provide us with a sign for the tangent. Hence we wouldn't # notice whether the sign changes (which is a typical case at cusps). result.append(invalid) return result def segments(self, params): if len(params) < 2: raise ValueError("at least two parameters needed in segments") # first, we calculate the coefficients corresponding to our # original bezier curve. These represent a useful starting # point for the following change of the polynomial parameter a0x_pt = self.x0_pt a0y_pt = self.y0_pt a1x_pt = 3*(-self.x0_pt+self.x1_pt) a1y_pt = 3*(-self.y0_pt+self.y1_pt) a2x_pt = 3*(self.x0_pt-2*self.x1_pt+self.x2_pt) a2y_pt = 3*(self.y0_pt-2*self.y1_pt+self.y2_pt) a3x_pt = -self.x0_pt+3*(self.x1_pt-self.x2_pt)+self.x3_pt a3y_pt = -self.y0_pt+3*(self.y1_pt-self.y2_pt)+self.y3_pt result = [] for i in range(len(params)-1): t1 = params[i] dt = params[i+1]-t1 # [t1,t2] part # # the new coefficients of the [t1,t1+dt] part of the bezier curve # are then given by expanding # a0 + a1*(t1+dt*u) + a2*(t1+dt*u)**2 + # a3*(t1+dt*u)**3 in u, yielding # # a0 + a1*t1 + a2*t1**2 + a3*t1**3 + # ( a1 + 2*a2 + 3*a3*t1**2 )*dt * u + # ( a2 + 3*a3*t1 )*dt**2 * u**2 + # a3*dt**3 * u**3 # # from this values we obtain the new control points by inversion # # TODO: we could do this more efficiently by reusing for # (x0_pt, y0_pt) the control point (x3_pt, y3_pt) from the previous # Bezier curve x0_pt = a0x_pt + a1x_pt*t1 + a2x_pt*t1*t1 + a3x_pt*t1*t1*t1 y0_pt = a0y_pt + a1y_pt*t1 + a2y_pt*t1*t1 + a3y_pt*t1*t1*t1 x1_pt = (a1x_pt+2*a2x_pt*t1+3*a3x_pt*t1*t1)*dt/3.0 + x0_pt y1_pt = (a1y_pt+2*a2y_pt*t1+3*a3y_pt*t1*t1)*dt/3.0 + y0_pt x2_pt = (a2x_pt+3*a3x_pt*t1)*dt*dt/3.0 - x0_pt + 2*x1_pt y2_pt = (a2y_pt+3*a3y_pt*t1)*dt*dt/3.0 - y0_pt + 2*y1_pt x3_pt = a3x_pt*dt*dt*dt + x0_pt - 3*x1_pt + 3*x2_pt y3_pt = a3y_pt*dt*dt*dt + y0_pt - 3*y1_pt + 3*y2_pt result.append(normcurve_pt(x0_pt, y0_pt, x1_pt, y1_pt, x2_pt, y2_pt, x3_pt, y3_pt)) return result def trafo(self, params): result = [] for rotation, at_pt in zip(self.rotation(params), self.at_pt(params)): if rotation is invalid: result.append(rotation) else: result.append(trafo.translate_pt(*at_pt) * rotation) return result def transformed(self, trafo): x0_pt, y0_pt = trafo.apply_pt(self.x0_pt, self.y0_pt) x1_pt, y1_pt = trafo.apply_pt(self.x1_pt, self.y1_pt) x2_pt, y2_pt = trafo.apply_pt(self.x2_pt, self.y2_pt) x3_pt, y3_pt = trafo.apply_pt(self.x3_pt, self.y3_pt) return normcurve_pt(x0_pt, y0_pt, x1_pt, y1_pt, x2_pt, y2_pt, x3_pt, y3_pt) def outputPS(self, file, writer): file.write("%g %g %g %g %g %g curveto\n" % (self.x1_pt, self.y1_pt, self.x2_pt, self.y2_pt, self.x3_pt, self.y3_pt)) def outputPDF(self, file, writer): file.write("%f %f %f %f %f %f c\n" % (self.x1_pt, self.y1_pt, self.x2_pt, self.y2_pt, self.x3_pt, self.y3_pt)) def x_pt(self, t): return ((( self.x3_pt-3*self.x2_pt+3*self.x1_pt-self.x0_pt)*t + 3*self.x0_pt-6*self.x1_pt+3*self.x2_pt)*t + 3*self.x1_pt-3*self.x0_pt)*t + self.x0_pt def xdot_pt(self, t): return ((3*self.x3_pt-9*self.x2_pt+9*self.x1_pt-3*self.x0_pt)*t + 6*self.x0_pt-12*self.x1_pt+6*self.x2_pt)*t + 3*self.x1_pt - 3*self.x0_pt def xddot_pt(self, t): return (6*self.x3_pt-18*self.x2_pt+18*self.x1_pt-6*self.x0_pt)*t + 6*self.x0_pt - 12*self.x1_pt + 6*self.x2_pt def xdddot_pt(self, t): return 6*self.x3_pt-18*self.x2_pt+18*self.x1_pt-6*self.x0_pt def y_pt(self, t): return ((( self.y3_pt-3*self.y2_pt+3*self.y1_pt-self.y0_pt)*t + 3*self.y0_pt-6*self.y1_pt+3*self.y2_pt)*t + 3*self.y1_pt-3*self.y0_pt)*t + self.y0_pt def ydot_pt(self, t): return ((3*self.y3_pt-9*self.y2_pt+9*self.y1_pt-3*self.y0_pt)*t + 6*self.y0_pt-12*self.y1_pt+6*self.y2_pt)*t + 3*self.y1_pt - 3*self.y0_pt def yddot_pt(self, t): return (6*self.y3_pt-18*self.y2_pt+18*self.y1_pt-6*self.y0_pt)*t + 6*self.y0_pt - 12*self.y1_pt + 6*self.y2_pt def ydddot_pt(self, t): return 6*self.y3_pt-18*self.y2_pt+18*self.y1_pt-6*self.y0_pt # curve replacements used by midpointsplit: # The replacements are normline_pt and normcurve_pt instances with an # additional subparamtoparam function for proper conversion of the # parametrization. Note that we only one direction (when a parameter # gets calculated), since the other way around direction midpointsplit # is not needed at all class _leftnormline_pt(normline_pt): __slots__ = "x0_pt", "y0_pt", "x1_pt", "y1_pt", "l1_pt", "l2_pt", "l3_pt" def __init__(self, x0_pt, y0_pt, x1_pt, y1_pt, l1_pt, l2_pt, l3_pt): normline_pt.__init__(self, x0_pt, y0_pt, x1_pt, y1_pt) self.l1_pt = l1_pt self.l2_pt = l2_pt self.l3_pt = l3_pt def subparamtoparam(self, param): if 0 <= param <= 1: params = mathutils.realpolyroots(self.l1_pt-2*self.l2_pt+self.l3_pt, -3*self.l1_pt+3*self.l2_pt, 3*self.l1_pt, -param*(self.l1_pt+self.l2_pt+self.l3_pt)) # we might get several solutions and choose the one closest to 0.5 # (we want the solution to be in the range 0 <= param <= 1; in case # we get several solutions in this range, they all will be close to # each other since l1_pt+l2_pt+l3_pt-l0_pt < epsilon) params.sort(lambda t1, t2: cmp(abs(t1-0.5), abs(t2-0.5))) return 0.5*params[0] else: # when we are outside the proper parameter range, we skip the non-linear # transformation, since it becomes slow and it might even start to be # numerically instable return 0.5*param class _rightnormline_pt(_leftnormline_pt): __slots__ = "x0_pt", "y0_pt", "x1_pt", "y1_pt", "l1_pt", "l2_pt", "l3_pt" def subparamtoparam(self, param): return 0.5+_leftnormline_pt.subparamtoparam(self, param) class _leftnormcurve_pt(normcurve_pt): __slots__ = "x0_pt", "y0_pt", "x1_pt", "y1_pt", "x2_pt", "y2_pt", "x3_pt", "y3_pt" def subparamtoparam(self, param): return 0.5*param class _rightnormcurve_pt(normcurve_pt): __slots__ = "x0_pt", "y0_pt", "x1_pt", "y1_pt", "x2_pt", "y2_pt", "x3_pt", "y3_pt" def subparamtoparam(self, param): return 0.5+0.5*param ################################################################################ # normsubpath ################################################################################ class normsubpath: """sub path of a normalized path A subpath consists of a list of normsubpathitems, i.e., normlines_pt and normcurves_pt and can either be closed or not. Some invariants, which have to be obeyed: - All normsubpathitems have to be longer than epsilon pts. - At the end there may be a normline (stored in self.skippedline) whose length is shorter than epsilon -- it has to be taken into account when adding further normsubpathitems - The last point of a normsubpathitem and the first point of the next element have to be equal. - When the path is closed, the last point of last normsubpathitem has to be equal to the first point of the first normsubpathitem. - epsilon might be none, disallowing any numerics, but allowing for arbitrary short paths. This is used in pdf output, where all paths need to be transformed to normpaths. """ __slots__ = "normsubpathitems", "closed", "epsilon", "skippedline" def __init__(self, normsubpathitems=[], closed=0, epsilon=_marker): """construct a normsubpath""" if epsilon is _marker: epsilon = _epsilon self.epsilon = epsilon # If one or more items appended to the normsubpath have been # skipped (because their total length was shorter than epsilon), # we remember this fact by a line because we have to take it # properly into account when appending further normsubpathitems self.skippedline = None self.normsubpathitems = [] self.closed = 0 # a test (might be temporary) for anormsubpathitem in normsubpathitems: assert isinstance(anormsubpathitem, normsubpathitem), "only list of normsubpathitem instances allowed" self.extend(normsubpathitems) if closed: self.close() def __getitem__(self, i): """return normsubpathitem i""" return self.normsubpathitems[i] def __len__(self): """return number of normsubpathitems""" return len(self.normsubpathitems) def __str__(self): l = ", ".join(map(str, self.normsubpathitems)) if self.closed: return "normsubpath([%s], closed=1)" % l else: return "normsubpath([%s])" % l def _distributeparams(self, params): """return a dictionary mapping normsubpathitemindices to a tuple of a paramindices and normsubpathitemparams. normsubpathitemindex specifies a normsubpathitem containing one or several positions. paramindex specify the index of the param in the original list and normsubpathitemparam is the parameter value in the normsubpathitem. """ result = {} for i, param in enumerate(params): if param > 0: index = int(param) if index > len(self.normsubpathitems) - 1: index = len(self.normsubpathitems) - 1 else: index = 0 result.setdefault(index, ([], [])) result[index][0].append(i) result[index][1].append(param - index) return result def append(self, anormsubpathitem): """append normsubpathitem Fails on closed normsubpath. """ if self.epsilon is None: self.normsubpathitems.append(anormsubpathitem) else: # consitency tests (might be temporary) assert isinstance(anormsubpathitem, normsubpathitem), "only normsubpathitem instances allowed" if self.skippedline: assert math.hypot(*[x-y for x, y in zip(self.skippedline.atend_pt(), anormsubpathitem.atbegin_pt())]) < self.epsilon, "normsubpathitems do not match" elif self.normsubpathitems: assert math.hypot(*[x-y for x, y in zip(self.normsubpathitems[-1].atend_pt(), anormsubpathitem.atbegin_pt())]) < self.epsilon, "normsubpathitems do not match" if self.closed: raise NormpathException("Cannot append to closed normsubpath") if self.skippedline: xs_pt, ys_pt = self.skippedline.atbegin_pt() else: xs_pt, ys_pt = anormsubpathitem.atbegin_pt() xe_pt, ye_pt = anormsubpathitem.atend_pt() if (math.hypot(xe_pt-xs_pt, ye_pt-ys_pt) >= self.epsilon or anormsubpathitem.arclen_pt(self.epsilon) >= self.epsilon): if self.skippedline: anormsubpathitem = anormsubpathitem.modifiedbegin_pt(xs_pt, ys_pt) self.normsubpathitems.append(anormsubpathitem) self.skippedline = None else: self.skippedline = normline_pt(xs_pt, ys_pt, xe_pt, ye_pt) def arclen_pt(self): """return arc length in pts""" return sum([npitem.arclen_pt(self.epsilon) for npitem in self.normsubpathitems]) def _arclentoparam_pt(self, lengths_pt): """return a tuple of params and the total length arc length in pts""" # work on a copy which is counted down to negative values lengths_pt = lengths_pt[:] results = [None] * len(lengths_pt) totalarclen = 0 for normsubpathindex, normsubpathitem in enumerate(self.normsubpathitems): params, arclen = normsubpathitem._arclentoparam_pt(lengths_pt, self.epsilon) for i in range(len(results)): if results[i] is None: lengths_pt[i] -= arclen if lengths_pt[i] < 0 or normsubpathindex == len(self.normsubpathitems) - 1: # overwrite the results until the length has become negative results[i] = normsubpathindex + params[i] totalarclen += arclen return results, totalarclen def arclentoparam_pt(self, lengths_pt): """return a tuple of params""" return self._arclentoparam_pt(lengths_pt)[0] def at_pt(self, params): """return coordinates at params in pts""" if not self.normsubpathitems and self.skippedline: return [self.skippedline.atbegin_pt()]*len(params) result = [None] * len(params) for normsubpathitemindex, (indices, params) in self._distributeparams(params).items(): for index, point_pt in zip(indices, self.normsubpathitems[normsubpathitemindex].at_pt(params)): result[index] = point_pt return result def atbegin_pt(self): """return coordinates of first point in pts""" if not self.normsubpathitems and self.skippedline: return self.skippedline.atbegin_pt() return self.normsubpathitems[0].atbegin_pt() def atend_pt(self): """return coordinates of last point in pts""" if self.skippedline: return self.skippedline.atend_pt() return self.normsubpathitems[-1].atend_pt() def bbox(self): """return bounding box of normsubpath""" if self.normsubpathitems: abbox = self.normsubpathitems[0].bbox() for anormpathitem in self.normsubpathitems[1:]: abbox += anormpathitem.bbox() return abbox else: return bboxmodule.empty() def close(self): """close subnormpath Fails on closed normsubpath. """ if self.closed: raise NormpathException("Cannot close already closed normsubpath") if not self.normsubpathitems: if self.skippedline is None: raise NormpathException("Cannot close empty normsubpath") else: raise NormpathException("Normsubpath too short, cannot be closed") xs_pt, ys_pt = self.normsubpathitems[-1].atend_pt() xe_pt, ye_pt = self.normsubpathitems[0].atbegin_pt() self.append(normline_pt(xs_pt, ys_pt, xe_pt, ye_pt)) self.flushskippedline() self.closed = 1 def copy(self): """return copy of normsubpath""" # Since normsubpathitems are never modified inplace, we just # need to copy the normsubpathitems list. We do not pass the # normsubpathitems to the constructor to not repeat the checks # for minimal length of each normsubpathitem. result = normsubpath(epsilon=self.epsilon) result.normsubpathitems = self.normsubpathitems[:] result.closed = self.closed # We can share the reference to skippedline, since it is a # normsubpathitem as well and thus not modified in place either. result.skippedline = self.skippedline return result def curvature_pt(self, params): """return the curvature at params in 1/pts The result contain the invalid instance at positions, where the curvature is undefined.""" result = [None] * len(params) for normsubpathitemindex, (indices, params) in self._distributeparams(params).items(): for index, curvature_pt in zip(indices, self.normsubpathitems[normsubpathitemindex].curvature_pt(params)): result[index] = curvature_pt return result def curveradius_pt(self, params): """return the curvature radius at params in pts The curvature radius is the inverse of the curvature. When the curvature is 0, the invalid instance is returned. Note that this radius can be negative or positive, depending on the sign of the curvature.""" result = [None] * len(params) for normsubpathitemindex, (indices, params) in self._distributeparams(params).items(): for index, radius_pt in zip(indices, self.normsubpathitems[normsubpathitemindex].curveradius_pt(params)): result[index] = radius_pt return result def extend(self, normsubpathitems): """extend path by normsubpathitems Fails on closed normsubpath. """ for normsubpathitem in normsubpathitems: self.append(normsubpathitem) def flushskippedline(self): """flush the skippedline, i.e. apply it to the normsubpath remove the skippedline by modifying the end point of the existing normsubpath """ while self.skippedline: try: lastnormsubpathitem = self.normsubpathitems.pop() except IndexError: raise ValueError("normsubpath too short to flush the skippedline") lastnormsubpathitem = lastnormsubpathitem.modifiedend_pt(*self.skippedline.atend_pt()) self.skippedline = None self.append(lastnormsubpathitem) def intersect(self, other): """intersect self with other normsubpath Returns a tuple of lists consisting of the parameter values of the intersection points of the corresponding normsubpath. """ intersections_a = [] intersections_b = [] epsilon = min(self.epsilon, other.epsilon) # Intersect all subpaths of self with the subpaths of other, possibly including # one intersection point several times for t_a, pitem_a in enumerate(self.normsubpathitems): for t_b, pitem_b in enumerate(other.normsubpathitems): for intersection_a, intersection_b in pitem_a.intersect(pitem_b, epsilon): intersections_a.append(intersection_a + t_a) intersections_b.append(intersection_b + t_b) # although intersectipns_a are sorted for the different normsubpathitems, # within a normsubpathitem, the ordering has to be ensured separately: intersections = zip(intersections_a, intersections_b) intersections.sort() intersections_a = [a for a, b in intersections] intersections_b = [b for a, b in intersections] # for symmetry reasons we enumerate intersections_a as well, although # they are already sorted (note we do not need to sort intersections_a) intersections_a = zip(intersections_a, range(len(intersections_a))) intersections_b = zip(intersections_b, range(len(intersections_b))) intersections_b.sort() # now we search for intersections points which are closer together than epsilon # This task is handled by the following function def closepoints(normsubpath, intersections): split = normsubpath.segments([0] + [intersection for intersection, index in intersections] + [len(normsubpath)]) result = [] if normsubpath.closed: # note that the number of segments of a closed path is off by one # compared to an open path i = 0 while i < len(split): splitnormsubpath = split[i] j = i while not splitnormsubpath.normsubpathitems: # i.e. while "is short" ip1, ip2 = intersections[i-1][1], intersections[j][1] if ip1 0: index = int(param) if index > len(self.normsubpathitems) - 1: index = len(self.normsubpathitems) - 1 param -= index else: index = 0 if index != collectindex: if collectindex is not None: # append end point depening on the forthcoming index if index > collectindex: collectparams.append(1) else: collectparams.append(0) # get segments of the normsubpathitem and add them to the result segments = self.normsubpathitems[collectindex].segments(collectparams) result[-1].append(segments[0]) result.extend([normsubpath([segment], epsilon=self.epsilon) for segment in segments[1:]]) # add normsubpathitems and first segment parameter to close the # gap to the forthcoming index if index > collectindex: for i in range(collectindex+1, index): result[-1].append(self.normsubpathitems[i]) collectparams = [0] else: for i in range(collectindex-1, index, -1): result[-1].append(self.normsubpathitems[i].reversed()) collectparams = [1] collectindex = index collectparams.append(param) # add remaining collectparams to the result segments = self.normsubpathitems[collectindex].segments(collectparams) result[-1].append(segments[0]) result.extend([normsubpath([segment], epsilon=self.epsilon) for segment in segments[1:]]) if self.closed: # join last and first segment together if the normsubpath was # originally closed and first and the last parameters are the # beginning and end points of the normsubpath if ( ( params[0] == 0 and params[-1] == len(self.normsubpathitems) ) or ( params[-1] == 0 and params[0] == len(self.normsubpathitems) ) ): result[-1].normsubpathitems.extend(result[0].normsubpathitems) result = result[-1:] + result[1:-1] return result def trafo(self, params): """return transformations at params""" result = [None] * len(params) for normsubpathitemindex, (indices, params) in self._distributeparams(params).items(): for index, trafo in zip(indices, self.normsubpathitems[normsubpathitemindex].trafo(params)): result[index] = trafo return result def transformed(self, trafo): """return transformed path""" nnormsubpath = normsubpath(epsilon=self.epsilon) for pitem in self.normsubpathitems: nnormsubpath.append(pitem.transformed(trafo)) if self.closed: nnormsubpath.close() elif self.skippedline is not None: nnormsubpath.append(self.skippedline.transformed(trafo)) return nnormsubpath def outputPS(self, file, writer): # if the normsubpath is closed, we must not output a normline at # the end if not self.normsubpathitems: return if self.closed and isinstance(self.normsubpathitems[-1], normline_pt): assert len(self.normsubpathitems) > 1, "a closed normsubpath should contain more than a single normline_pt" normsubpathitems = self.normsubpathitems[:-1] else: normsubpathitems = self.normsubpathitems file.write("%g %g moveto\n" % self.atbegin_pt()) for anormsubpathitem in normsubpathitems: anormsubpathitem.outputPS(file, writer) if self.closed: file.write("closepath\n") def outputPDF(self, file, writer): # if the normsubpath is closed, we must not output a normline at # the end if not self.normsubpathitems: return if self.closed and isinstance(self.normsubpathitems[-1], normline_pt): assert len(self.normsubpathitems) > 1, "a closed normsubpath should contain more than a single normline_pt" normsubpathitems = self.normsubpathitems[:-1] else: normsubpathitems = self.normsubpathitems file.write("%f %f m\n" % self.atbegin_pt()) for anormsubpathitem in normsubpathitems: anormsubpathitem.outputPDF(file, writer) if self.closed: file.write("h\n") ################################################################################ # normpath ################################################################################ class normpathparam: """parameter of a certain point along a normpath""" __slots__ = "normpath", "normsubpathindex", "normsubpathparam" def __init__(self, normpath, normsubpathindex, normsubpathparam): self.normpath = normpath self.normsubpathindex = normsubpathindex self.normsubpathparam = normsubpathparam float(normsubpathparam) def __str__(self): return "normpathparam(%s, %s, %s)" % (self.normpath, self.normsubpathindex, self.normsubpathparam) def __add__(self, other): if isinstance(other, normpathparam): assert self.normpath is other.normpath, "normpathparams have to belong to the same normpath" return self.normpath.arclentoparam_pt(self.normpath.paramtoarclen_pt(self) + other.normpath.paramtoarclen_pt(other)) else: return self.normpath.arclentoparam_pt(self.normpath.paramtoarclen_pt(self) + unit.topt(other)) __radd__ = __add__ def __sub__(self, other): if isinstance(other, normpathparam): assert self.normpath is other.normpath, "normpathparams have to belong to the same normpath" return self.normpath.arclentoparam_pt(self.normpath.paramtoarclen_pt(self) - other.normpath.paramtoarclen_pt(other)) else: return self.normpath.arclentoparam_pt(self.normpath.paramtoarclen_pt(self) - unit.topt(other)) def __rsub__(self, other): # other has to be a length in this case return self.normpath.arclentoparam_pt(-self.normpath.paramtoarclen_pt(self) + unit.topt(other)) def __mul__(self, factor): return self.normpath.arclentoparam_pt(self.normpath.paramtoarclen_pt(self) * factor) __rmul__ = __mul__ def __div__(self, divisor): return self.normpath.arclentoparam_pt(self.normpath.paramtoarclen_pt(self) / divisor) def __neg__(self): return self.normpath.arclentoparam_pt(-self.normpath.paramtoarclen_pt(self)) def __cmp__(self, other): if isinstance(other, normpathparam): assert self.normpath is other.normpath, "normpathparams have to belong to the same normpath" return cmp((self.normsubpathindex, self.normsubpathparam), (other.normsubpathindex, other.normsubpathparam)) else: return cmp(self.normpath.paramtoarclen_pt(self), unit.topt(other)) def arclen_pt(self): """return arc length in pts corresponding to the normpathparam """ return self.normpath.paramtoarclen_pt(self) def arclen(self): """return arc length corresponding to the normpathparam """ return self.normpath.paramtoarclen(self) def _valueorlistmethod(method): """Creates a method which takes a single argument or a list and returns a single value or a list out of method, which always works on lists.""" def wrappedmethod(self, valueorlist, *args, **kwargs): try: for item in valueorlist: break except: return method(self, [valueorlist], *args, **kwargs)[0] return method(self, valueorlist, *args, **kwargs) return wrappedmethod class normpath: """normalized path A normalized path consists of a list of normsubpaths. """ def __init__(self, normsubpaths=None): """construct a normpath from a list of normsubpaths""" if normsubpaths is None: self.normsubpaths = [] # make a fresh list else: self.normsubpaths = normsubpaths for subpath in normsubpaths: assert isinstance(subpath, normsubpath), "only list of normsubpath instances allowed" def __add__(self, other): """create new normpath out of self and other""" result = self.copy() result += other return result def __iadd__(self, other): """add other inplace""" for normsubpath in other.normpath().normsubpaths: self.normsubpaths.append(normsubpath.copy()) return self def __getitem__(self, i): """return normsubpath i""" return self.normsubpaths[i] def __len__(self): """return the number of normsubpaths""" return len(self.normsubpaths) def __str__(self): return "normpath([%s])" % ", ".join(map(str, self.normsubpaths)) def _convertparams(self, params, convertmethod): """return params with all non-normpathparam arguments converted by convertmethod usecases: - self._convertparams(params, self.arclentoparam_pt) - self._convertparams(params, self.arclentoparam) """ converttoparams = [] convertparamindices = [] for i, param in enumerate(params): if not isinstance(param, normpathparam): converttoparams.append(param) convertparamindices.append(i) if converttoparams: params = params[:] for i, param in zip(convertparamindices, convertmethod(converttoparams)): params[i] = param return params def _distributeparams(self, params): """return a dictionary mapping subpathindices to a tuple of a paramindices and subpathparams subpathindex specifies a subpath containing one or several positions. paramindex specify the index of the normpathparam in the original list and subpathparam is the parameter value in the subpath. """ result = {} for i, param in enumerate(params): assert param.normpath is self, "normpathparam has to belong to this path" result.setdefault(param.normsubpathindex, ([], [])) result[param.normsubpathindex][0].append(i) result[param.normsubpathindex][1].append(param.normsubpathparam) return result def append(self, item): """append a normpath by a normsubpath or a pathitem""" if isinstance(item, normsubpath): # the normsubpaths list can be appended by a normsubpath only self.normsubpaths.append(item) elif isinstance(item, path.pathitem): # ... but we are kind and allow for regular path items as well # in order to make a normpath to behave more like a regular path if self.normsubpaths: context = path.context(*(self.normsubpaths[-1].atend_pt() + self.normsubpaths[-1].atbegin_pt())) item.updatenormpath(self, context) else: self.normsubpaths = item.createnormpath(self).normsubpaths def arclen_pt(self): """return arc length in pts""" return sum([normsubpath.arclen_pt() for normsubpath in self.normsubpaths]) def arclen(self): """return arc length""" return self.arclen_pt() * unit.t_pt def _arclentoparam_pt(self, lengths_pt): """return the params matching the given lengths_pt""" # work on a copy which is counted down to negative values lengths_pt = lengths_pt[:] results = [None] * len(lengths_pt) for normsubpathindex, normsubpath in enumerate(self.normsubpaths): params, arclen = normsubpath._arclentoparam_pt(lengths_pt) done = 1 for i, result in enumerate(results): if results[i] is None: lengths_pt[i] -= arclen if lengths_pt[i] < 0 or normsubpathindex == len(self.normsubpaths) - 1: # overwrite the results until the length has become negative results[i] = normpathparam(self, normsubpathindex, params[i]) done = 0 if done: break return results def arclentoparam_pt(self, lengths_pt): """return the param(s) matching the given length(s)_pt in pts""" pass arclentoparam_pt = _valueorlistmethod(_arclentoparam_pt) def arclentoparam(self, lengths): """return the param(s) matching the given length(s)""" return self._arclentoparam_pt([unit.topt(l) for l in lengths]) arclentoparam = _valueorlistmethod(arclentoparam) def _at_pt(self, params): """return coordinates of normpath in pts at params""" result = [None] * len(params) for normsubpathindex, (indices, params) in self._distributeparams(params).items(): for index, point_pt in zip(indices, self.normsubpaths[normsubpathindex].at_pt(params)): result[index] = point_pt return result def at_pt(self, params): """return coordinates of normpath in pts at param(s) or lengths in pts""" return self._at_pt(self._convertparams(params, self.arclentoparam_pt)) at_pt = _valueorlistmethod(at_pt) def at(self, params): """return coordinates of normpath at param(s) or arc lengths""" return [(x_pt * unit.t_pt, y_pt * unit.t_pt) for x_pt, y_pt in self._at_pt(self._convertparams(params, self.arclentoparam))] at = _valueorlistmethod(at) def atbegin_pt(self): """return coordinates of the beginning of first subpath in normpath in pts""" if self.normsubpaths: return self.normsubpaths[0].atbegin_pt() else: raise NormpathException("cannot return first point of empty path") def atbegin(self): """return coordinates of the beginning of first subpath in normpath""" x, y = self.atbegin_pt() return x * unit.t_pt, y * unit.t_pt def atend_pt(self): """return coordinates of the end of last subpath in normpath in pts""" if self.normsubpaths: return self.normsubpaths[-1].atend_pt() else: raise NormpathException("cannot return last point of empty path") def atend(self): """return coordinates of the end of last subpath in normpath""" x, y = self.atend_pt() return x * unit.t_pt, y * unit.t_pt def bbox(self): """return bbox of normpath""" abbox = bboxmodule.empty() for normsubpath in self.normsubpaths: abbox += normsubpath.bbox() return abbox def begin(self): """return param corresponding of the beginning of the normpath""" if self.normsubpaths: return normpathparam(self, 0, 0) else: raise NormpathException("empty path") def copy(self): """return copy of normpath""" result = normpath() for normsubpath in self.normsubpaths: result.append(normsubpath.copy()) return result def _curvature_pt(self, params): """return the curvature in 1/pts at params When the curvature is undefined, the invalid instance is returned.""" result = [None] * len(params) for normsubpathindex, (indices, params) in self._distributeparams(params).items(): for index, curvature_pt in zip(indices, self.normsubpaths[normsubpathindex].curvature_pt(params)): result[index] = curvature_pt return result def curvature_pt(self, params): """return the curvature in 1/pt at params The curvature radius is the inverse of the curvature. When the curvature is undefined, the invalid instance is returned. Note that this radius can be negative or positive, depending on the sign of the curvature.""" result = [None] * len(params) for normsubpathindex, (indices, params) in self._distributeparams(params).items(): for index, curv_pt in zip(indices, self.normsubpaths[normsubpathindex].curvature_pt(params)): result[index] = curv_pt return result curvature_pt = _valueorlistmethod(curvature_pt) def _curveradius_pt(self, params): """return the curvature radius at params in pts The curvature radius is the inverse of the curvature. When the curvature is 0, None is returned. Note that this radius can be negative or positive, depending on the sign of the curvature.""" result = [None] * len(params) for normsubpathindex, (indices, params) in self._distributeparams(params).items(): for index, radius_pt in zip(indices, self.normsubpaths[normsubpathindex].curveradius_pt(params)): result[index] = radius_pt return result def curveradius_pt(self, params): """return the curvature radius in pts at param(s) or arc length(s) in pts The curvature radius is the inverse of the curvature. When the curvature is 0, None is returned. Note that this radius can be negative or positive, depending on the sign of the curvature.""" return self._curveradius_pt(self._convertparams(params, self.arclentoparam_pt)) curveradius_pt = _valueorlistmethod(curveradius_pt) def curveradius(self, params): """return the curvature radius at param(s) or arc length(s) The curvature radius is the inverse of the curvature. When the curvature is 0, None is returned. Note that this radius can be negative or positive, depending on the sign of the curvature.""" result = [] for radius_pt in self._curveradius_pt(self._convertparams(params, self.arclentoparam)): if radius_pt is not invalid: result.append(radius_pt * unit.t_pt) else: result.append(invalid) return result curveradius = _valueorlistmethod(curveradius) def end(self): """return param corresponding of the end of the path""" if self.normsubpaths: return normpathparam(self, len(self)-1, len(self.normsubpaths[-1])) else: raise NormpathException("empty path") def extend(self, normsubpaths): """extend path by normsubpaths or pathitems""" for anormsubpath in normsubpaths: # use append to properly handle regular path items as well as normsubpaths self.append(anormsubpath) def intersect(self, other): """intersect self with other path Returns a tuple of lists consisting of the parameter values of the intersection points of the corresponding normpath. """ other = other.normpath() # here we build up the result intersections = ([], []) # Intersect all normsubpaths of self with the normsubpaths of # other. for ia, normsubpath_a in enumerate(self.normsubpaths): for ib, normsubpath_b in enumerate(other.normsubpaths): for intersection in zip(*normsubpath_a.intersect(normsubpath_b)): intersections[0].append(normpathparam(self, ia, intersection[0])) intersections[1].append(normpathparam(other, ib, intersection[1])) return intersections def join(self, other): """join other normsubpath inplace Both normpaths must contain at least one normsubpath. The last normsubpath of self will be joined to the first normsubpath of other. """ other = other.normpath() if not self.normsubpaths: raise NormpathException("cannot join to empty path") if not other.normsubpaths: raise PathException("cannot join empty path") self.normsubpaths[-1].join(other.normsubpaths[0]) self.normsubpaths.extend(other.normsubpaths[1:]) def joined(self, other): """return joined self and other Both normpaths must contain at least one normsubpath. The last normsubpath of self will be joined to the first normsubpath of other. """ result = self.copy() result.join(other.normpath()) return result # << operator also designates joining __lshift__ = joined def normpath(self): """return a normpath, i.e. self""" return self def _paramtoarclen_pt(self, params): """return arc lengths in pts matching the given params""" result = [None] * len(params) totalarclen_pt = 0 distributeparams = self._distributeparams(params) for normsubpathindex in range(max(distributeparams.keys()) + 1): if distributeparams.has_key(normsubpathindex): indices, params = distributeparams[normsubpathindex] arclens_pt, normsubpatharclen_pt = self.normsubpaths[normsubpathindex]._paramtoarclen_pt(params) for index, arclen_pt in zip(indices, arclens_pt): result[index] = totalarclen_pt + arclen_pt totalarclen_pt += normsubpatharclen_pt else: totalarclen_pt += self.normsubpaths[normsubpathindex].arclen_pt() return result def paramtoarclen_pt(self, params): """return arc length(s) in pts matching the given param(s)""" paramtoarclen_pt = _valueorlistmethod(_paramtoarclen_pt) def paramtoarclen(self, params): """return arc length(s) matching the given param(s)""" return [arclen_pt * unit.t_pt for arclen_pt in self._paramtoarclen_pt(params)] paramtoarclen = _valueorlistmethod(paramtoarclen) def path(self): """return path corresponding to normpath""" pathitems = [] for normsubpath in self.normsubpaths: pathitems.extend(normsubpath.pathitems()) return path.path(*pathitems) def reversed(self): """return reversed path""" nnormpath = normpath() for i in range(len(self.normsubpaths)): nnormpath.normsubpaths.append(self.normsubpaths[-(i+1)].reversed()) return nnormpath def _rotation(self, params): """return rotation at params""" result = [None] * len(params) for normsubpathindex, (indices, params) in self._distributeparams(params).items(): for index, rotation in zip(indices, self.normsubpaths[normsubpathindex].rotation(params)): result[index] = rotation return result def rotation_pt(self, params): """return rotation at param(s) or arc length(s) in pts""" return self._rotation(self._convertparams(params, self.arclentoparam_pt)) rotation_pt = _valueorlistmethod(rotation_pt) def rotation(self, params): """return rotation at param(s) or arc length(s)""" return self._rotation(self._convertparams(params, self.arclentoparam)) rotation = _valueorlistmethod(rotation) def _split_pt(self, params): """split path at params and return list of normpaths""" if not params: return [self.copy()] # instead of distributing the parameters, we need to keep their # order and collect parameters for splitting of normsubpathitem # with index collectindex collectindex = None for param in params: if param.normsubpathindex != collectindex: if collectindex is not None: # append end point depening on the forthcoming index if param.normsubpathindex > collectindex: collectparams.append(len(self.normsubpaths[collectindex])) else: collectparams.append(0) # get segments of the normsubpath and add them to the result segments = self.normsubpaths[collectindex].segments(collectparams) result[-1].append(segments[0]) result.extend([normpath([segment]) for segment in segments[1:]]) # add normsubpathitems and first segment parameter to close the # gap to the forthcoming index if param.normsubpathindex > collectindex: for i in range(collectindex+1, param.normsubpathindex): result[-1].append(self.normsubpaths[i]) collectparams = [0] else: for i in range(collectindex-1, param.normsubpathindex, -1): result[-1].append(self.normsubpaths[i].reversed()) collectparams = [len(self.normsubpaths[param.normsubpathindex])] else: result = [normpath(self.normsubpaths[:param.normsubpathindex])] collectparams = [0] collectindex = param.normsubpathindex collectparams.append(param.normsubpathparam) # add remaining collectparams to the result collectparams.append(len(self.normsubpaths[collectindex])) segments = self.normsubpaths[collectindex].segments(collectparams) result[-1].append(segments[0]) result.extend([normpath([segment]) for segment in segments[1:]]) result[-1].extend(self.normsubpaths[collectindex+1:]) return result def split_pt(self, params): """split path at param(s) or arc length(s) in pts and return list of normpaths""" try: for param in params: break except: params = [params] return self._split_pt(self._convertparams(params, self.arclentoparam_pt)) def split(self, params): """split path at param(s) or arc length(s) and return list of normpaths""" try: for param in params: break except: params = [params] return self._split_pt(self._convertparams(params, self.arclentoparam)) def _tangent(self, params, length_pt): """return tangent vector of path at params If length_pt in pts is not None, the tangent vector will be scaled to the desired length. """ result = [None] * len(params) tangenttemplate = path.line_pt(0, 0, length_pt, 0).normpath() for normsubpathindex, (indices, params) in self._distributeparams(params).items(): for index, atrafo in zip(indices, self.normsubpaths[normsubpathindex].trafo(params)): if atrafo is invalid: result[index] = invalid else: result[index] = tangenttemplate.transformed(atrafo) return result def tangent_pt(self, params, length_pt): """return tangent vector of path at param(s) or arc length(s) in pts If length in pts is not None, the tangent vector will be scaled to the desired length. """ return self._tangent(self._convertparams(params, self.arclentoparam_pt), length_pt) tangent_pt = _valueorlistmethod(tangent_pt) def tangent(self, params, length): """return tangent vector of path at param(s) or arc length(s) If length is not None, the tangent vector will be scaled to the desired length. """ return self._tangent(self._convertparams(params, self.arclentoparam), unit.topt(length)) tangent = _valueorlistmethod(tangent) def _trafo(self, params): """return transformation at params""" result = [None] * len(params) for normsubpathindex, (indices, params) in self._distributeparams(params).items(): for index, trafo in zip(indices, self.normsubpaths[normsubpathindex].trafo(params)): result[index] = trafo return result def trafo_pt(self, params): """return transformation at param(s) or arc length(s) in pts""" return self._trafo(self._convertparams(params, self.arclentoparam_pt)) trafo_pt = _valueorlistmethod(trafo_pt) def trafo(self, params): """return transformation at param(s) or arc length(s)""" return self._trafo(self._convertparams(params, self.arclentoparam)) trafo = _valueorlistmethod(trafo) def transformed(self, trafo): """return transformed normpath""" return normpath([normsubpath.transformed(trafo) for normsubpath in self.normsubpaths]) def outputPS(self, file, writer): for normsubpath in self.normsubpaths: normsubpath.outputPS(file, writer) def outputPDF(self, file, writer): for normsubpath in self.normsubpaths: normsubpath.outputPDF(file, writer)