637 lines
20 KiB
Python
637 lines
20 KiB
Python
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"""The definition of the base geometrical entity with attributes common to
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all derived geometrical entities.
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Contains
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========
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GeometryEntity
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GeometricSet
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Notes
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=====
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A GeometryEntity is any object that has special geometric properties.
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A GeometrySet is a superclass of any GeometryEntity that can also
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be viewed as a sympy.sets.Set. In particular, points are the only
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GeometryEntity not considered a Set.
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Rn is a GeometrySet representing n-dimensional Euclidean space. R2 and
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R3 are currently the only ambient spaces implemented.
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"""
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from sympy.core.basic import Basic
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from sympy.core.compatibility import is_sequence
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from sympy.core.containers import Tuple
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from sympy.core.sympify import sympify
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from sympy.functions import cos, sin
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from sympy.matrices import eye
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from sympy.multipledispatch import dispatch
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from sympy.sets import Set
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from sympy.sets.handlers.intersection import intersection_sets
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from sympy.sets.handlers.union import union_sets
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from sympy.utilities.misc import func_name
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# How entities are ordered; used by __cmp__ in GeometryEntity
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ordering_of_classes = [
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"Point2D",
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"Point3D",
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"Point",
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"Segment2D",
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"Ray2D",
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"Line2D",
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"Segment3D",
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"Line3D",
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"Ray3D",
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"Segment",
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"Ray",
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"Line",
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"Plane",
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"Triangle",
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"RegularPolygon",
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"Polygon",
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"Circle",
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"Ellipse",
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"Curve",
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"Parabola"
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]
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class GeometryEntity(Basic):
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"""The base class for all geometrical entities.
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This class doesn't represent any particular geometric entity, it only
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provides the implementation of some methods common to all subclasses.
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"""
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def __cmp__(self, other):
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"""Comparison of two GeometryEntities."""
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n1 = self.__class__.__name__
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n2 = other.__class__.__name__
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c = (n1 > n2) - (n1 < n2)
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if not c:
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return 0
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i1 = -1
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for cls in self.__class__.__mro__:
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try:
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i1 = ordering_of_classes.index(cls.__name__)
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break
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except ValueError:
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i1 = -1
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if i1 == -1:
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return c
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i2 = -1
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for cls in other.__class__.__mro__:
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try:
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i2 = ordering_of_classes.index(cls.__name__)
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break
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except ValueError:
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i2 = -1
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if i2 == -1:
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return c
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return (i1 > i2) - (i1 < i2)
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def __contains__(self, other):
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"""Subclasses should implement this method for anything more complex than equality."""
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if type(self) == type(other):
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return self == other
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raise NotImplementedError()
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def __getnewargs__(self):
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"""Returns a tuple that will be passed to __new__ on unpickling."""
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return tuple(self.args)
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def __ne__(self, o):
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"""Test inequality of two geometrical entities."""
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return not self == o
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def __new__(cls, *args, **kwargs):
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# Points are sequences, but they should not
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# be converted to Tuples, so use this detection function instead.
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def is_seq_and_not_point(a):
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# we cannot use isinstance(a, Point) since we cannot import Point
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if hasattr(a, 'is_Point') and a.is_Point:
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return False
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return is_sequence(a)
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args = [Tuple(*a) if is_seq_and_not_point(a) else sympify(a) for a in args]
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return Basic.__new__(cls, *args)
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def __radd__(self, a):
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"""Implementation of reverse add method."""
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return a.__add__(self)
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def __rtruediv__(self, a):
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"""Implementation of reverse division method."""
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return a.__truediv__(self)
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def __repr__(self):
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"""String representation of a GeometryEntity that can be evaluated
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by sympy."""
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return type(self).__name__ + repr(self.args)
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def __rmul__(self, a):
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"""Implementation of reverse multiplication method."""
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return a.__mul__(self)
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def __rsub__(self, a):
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"""Implementation of reverse subtraction method."""
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return a.__sub__(self)
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def __str__(self):
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"""String representation of a GeometryEntity."""
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from sympy.printing import sstr
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return type(self).__name__ + sstr(self.args)
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def _eval_subs(self, old, new):
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from sympy.geometry.point import Point, Point3D
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if is_sequence(old) or is_sequence(new):
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if isinstance(self, Point3D):
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old = Point3D(old)
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new = Point3D(new)
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else:
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old = Point(old)
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new = Point(new)
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return self._subs(old, new)
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def _repr_svg_(self):
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"""SVG representation of a GeometryEntity suitable for IPython"""
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from sympy.core.evalf import N
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try:
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bounds = self.bounds
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except (NotImplementedError, TypeError):
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# if we have no SVG representation, return None so IPython
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# will fall back to the next representation
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return None
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if any([not x.is_number or not x.is_finite for x in bounds]):
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return None
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svg_top = '''<svg xmlns="http://www.w3.org/2000/svg"
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xmlns:xlink="http://www.w3.org/1999/xlink"
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width="{1}" height="{2}" viewBox="{0}"
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preserveAspectRatio="xMinYMin meet">
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<defs>
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<marker id="markerCircle" markerWidth="8" markerHeight="8"
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refx="5" refy="5" markerUnits="strokeWidth">
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<circle cx="5" cy="5" r="1.5" style="stroke: none; fill:#000000;"/>
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</marker>
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<marker id="markerArrow" markerWidth="13" markerHeight="13" refx="2" refy="4"
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orient="auto" markerUnits="strokeWidth">
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<path d="M2,2 L2,6 L6,4" style="fill: #000000;" />
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</marker>
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<marker id="markerReverseArrow" markerWidth="13" markerHeight="13" refx="6" refy="4"
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orient="auto" markerUnits="strokeWidth">
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<path d="M6,2 L6,6 L2,4" style="fill: #000000;" />
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</marker>
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</defs>'''
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# Establish SVG canvas that will fit all the data + small space
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xmin, ymin, xmax, ymax = map(N, bounds)
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if xmin == xmax and ymin == ymax:
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# This is a point; buffer using an arbitrary size
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xmin, ymin, xmax, ymax = xmin - .5, ymin -.5, xmax + .5, ymax + .5
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else:
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# Expand bounds by a fraction of the data ranges
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expand = 0.1 # or 10%; this keeps arrowheads in view (R plots use 4%)
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widest_part = max([xmax - xmin, ymax - ymin])
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expand_amount = widest_part * expand
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xmin -= expand_amount
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ymin -= expand_amount
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xmax += expand_amount
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ymax += expand_amount
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dx = xmax - xmin
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dy = ymax - ymin
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width = min([max([100., dx]), 300])
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height = min([max([100., dy]), 300])
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scale_factor = 1. if max(width, height) == 0 else max(dx, dy) / max(width, height)
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try:
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svg = self._svg(scale_factor)
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except (NotImplementedError, TypeError):
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# if we have no SVG representation, return None so IPython
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# will fall back to the next representation
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return None
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view_box = "{} {} {} {}".format(xmin, ymin, dx, dy)
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transform = "matrix(1,0,0,-1,0,{})".format(ymax + ymin)
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svg_top = svg_top.format(view_box, width, height)
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return svg_top + (
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'<g transform="{}">{}</g></svg>'
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).format(transform, svg)
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def _svg(self, scale_factor=1., fill_color="#66cc99"):
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"""Returns SVG path element for the GeometryEntity.
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Parameters
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==========
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scale_factor : float
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Multiplication factor for the SVG stroke-width. Default is 1.
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fill_color : str, optional
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Hex string for fill color. Default is "#66cc99".
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"""
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raise NotImplementedError()
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def _sympy_(self):
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return self
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@property
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def ambient_dimension(self):
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"""What is the dimension of the space that the object is contained in?"""
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raise NotImplementedError()
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@property
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def bounds(self):
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"""Return a tuple (xmin, ymin, xmax, ymax) representing the bounding
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rectangle for the geometric figure.
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"""
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raise NotImplementedError()
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def encloses(self, o):
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"""
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Return True if o is inside (not on or outside) the boundaries of self.
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The object will be decomposed into Points and individual Entities need
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only define an encloses_point method for their class.
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See Also
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========
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sympy.geometry.ellipse.Ellipse.encloses_point
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sympy.geometry.polygon.Polygon.encloses_point
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Examples
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========
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>>> from sympy import RegularPolygon, Point, Polygon
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>>> t = Polygon(*RegularPolygon(Point(0, 0), 1, 3).vertices)
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>>> t2 = Polygon(*RegularPolygon(Point(0, 0), 2, 3).vertices)
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>>> t2.encloses(t)
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True
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>>> t.encloses(t2)
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False
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"""
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from sympy.geometry.point import Point
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from sympy.geometry.line import Segment, Ray, Line
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from sympy.geometry.ellipse import Ellipse
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from sympy.geometry.polygon import Polygon, RegularPolygon
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if isinstance(o, Point):
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return self.encloses_point(o)
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elif isinstance(o, Segment):
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return all(self.encloses_point(x) for x in o.points)
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elif isinstance(o, Ray) or isinstance(o, Line):
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return False
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elif isinstance(o, Ellipse):
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return self.encloses_point(o.center) and \
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self.encloses_point(
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Point(o.center.x + o.hradius, o.center.y)) and \
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not self.intersection(o)
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elif isinstance(o, Polygon):
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if isinstance(o, RegularPolygon):
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if not self.encloses_point(o.center):
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return False
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return all(self.encloses_point(v) for v in o.vertices)
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raise NotImplementedError()
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def equals(self, o):
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return self == o
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def intersection(self, o):
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"""
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Returns a list of all of the intersections of self with o.
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Notes
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=====
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An entity is not required to implement this method.
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If two different types of entities can intersect, the item with
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higher index in ordering_of_classes should implement
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intersections with anything having a lower index.
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See Also
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========
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sympy.geometry.util.intersection
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"""
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raise NotImplementedError()
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def is_similar(self, other):
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"""Is this geometrical entity similar to another geometrical entity?
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Two entities are similar if a uniform scaling (enlarging or
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shrinking) of one of the entities will allow one to obtain the other.
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Notes
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=====
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This method is not intended to be used directly but rather
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through the `are_similar` function found in util.py.
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An entity is not required to implement this method.
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If two different types of entities can be similar, it is only
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required that one of them be able to determine this.
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See Also
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========
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scale
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"""
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raise NotImplementedError()
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def reflect(self, line):
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"""
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Reflects an object across a line.
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Parameters
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==========
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line: Line
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Examples
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========
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>>> from sympy import pi, sqrt, Line, RegularPolygon
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>>> l = Line((0, pi), slope=sqrt(2))
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>>> pent = RegularPolygon((1, 2), 1, 5)
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>>> rpent = pent.reflect(l)
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>>> rpent
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RegularPolygon(Point2D(-2*sqrt(2)*pi/3 - 1/3 + 4*sqrt(2)/3, 2/3 + 2*sqrt(2)/3 + 2*pi/3), -1, 5, -atan(2*sqrt(2)) + 3*pi/5)
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>>> from sympy import pi, Line, Circle, Point
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>>> l = Line((0, pi), slope=1)
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>>> circ = Circle(Point(0, 0), 5)
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>>> rcirc = circ.reflect(l)
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>>> rcirc
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Circle(Point2D(-pi, pi), -5)
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"""
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from sympy import atan, Point, Dummy, oo
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g = self
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l = line
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o = Point(0, 0)
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if l.slope.is_zero:
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y = l.args[0].y
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|
|
if not y: # x-axis
|
||
|
|
return g.scale(y=-1)
|
||
|
|
reps = [(p, p.translate(y=2*(y - p.y))) for p in g.atoms(Point)]
|
||
|
|
elif l.slope is oo:
|
||
|
|
x = l.args[0].x
|
||
|
|
if not x: # y-axis
|
||
|
|
return g.scale(x=-1)
|
||
|
|
reps = [(p, p.translate(x=2*(x - p.x))) for p in g.atoms(Point)]
|
||
|
|
else:
|
||
|
|
if not hasattr(g, 'reflect') and not all(
|
||
|
|
isinstance(arg, Point) for arg in g.args):
|
||
|
|
raise NotImplementedError(
|
||
|
|
'reflect undefined or non-Point args in %s' % g)
|
||
|
|
a = atan(l.slope)
|
||
|
|
c = l.coefficients
|
||
|
|
d = -c[-1]/c[1] # y-intercept
|
||
|
|
# apply the transform to a single point
|
||
|
|
x, y = Dummy(), Dummy()
|
||
|
|
xf = Point(x, y)
|
||
|
|
xf = xf.translate(y=-d).rotate(-a, o).scale(y=-1
|
||
|
|
).rotate(a, o).translate(y=d)
|
||
|
|
# replace every point using that transform
|
||
|
|
reps = [(p, xf.xreplace({x: p.x, y: p.y})) for p in g.atoms(Point)]
|
||
|
|
return g.xreplace(dict(reps))
|
||
|
|
|
||
|
|
def rotate(self, angle, pt=None):
|
||
|
|
"""Rotate ``angle`` radians counterclockwise about Point ``pt``.
|
||
|
|
|
||
|
|
The default pt is the origin, Point(0, 0)
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
scale, translate
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Point, RegularPolygon, Polygon, pi
|
||
|
|
>>> t = Polygon(*RegularPolygon(Point(0, 0), 1, 3).vertices)
|
||
|
|
>>> t # vertex on x axis
|
||
|
|
Triangle(Point2D(1, 0), Point2D(-1/2, sqrt(3)/2), Point2D(-1/2, -sqrt(3)/2))
|
||
|
|
>>> t.rotate(pi/2) # vertex on y axis now
|
||
|
|
Triangle(Point2D(0, 1), Point2D(-sqrt(3)/2, -1/2), Point2D(sqrt(3)/2, -1/2))
|
||
|
|
|
||
|
|
"""
|
||
|
|
newargs = []
|
||
|
|
for a in self.args:
|
||
|
|
if isinstance(a, GeometryEntity):
|
||
|
|
newargs.append(a.rotate(angle, pt))
|
||
|
|
else:
|
||
|
|
newargs.append(a)
|
||
|
|
return type(self)(*newargs)
|
||
|
|
|
||
|
|
def scale(self, x=1, y=1, pt=None):
|
||
|
|
"""Scale the object by multiplying the x,y-coordinates by x and y.
|
||
|
|
|
||
|
|
If pt is given, the scaling is done relative to that point; the
|
||
|
|
object is shifted by -pt, scaled, and shifted by pt.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
rotate, translate
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import RegularPolygon, Point, Polygon
|
||
|
|
>>> t = Polygon(*RegularPolygon(Point(0, 0), 1, 3).vertices)
|
||
|
|
>>> t
|
||
|
|
Triangle(Point2D(1, 0), Point2D(-1/2, sqrt(3)/2), Point2D(-1/2, -sqrt(3)/2))
|
||
|
|
>>> t.scale(2)
|
||
|
|
Triangle(Point2D(2, 0), Point2D(-1, sqrt(3)/2), Point2D(-1, -sqrt(3)/2))
|
||
|
|
>>> t.scale(2, 2)
|
||
|
|
Triangle(Point2D(2, 0), Point2D(-1, sqrt(3)), Point2D(-1, -sqrt(3)))
|
||
|
|
|
||
|
|
"""
|
||
|
|
from sympy.geometry.point import Point
|
||
|
|
if pt:
|
||
|
|
pt = Point(pt, dim=2)
|
||
|
|
return self.translate(*(-pt).args).scale(x, y).translate(*pt.args)
|
||
|
|
return type(self)(*[a.scale(x, y) for a in self.args]) # if this fails, override this class
|
||
|
|
|
||
|
|
def translate(self, x=0, y=0):
|
||
|
|
"""Shift the object by adding to the x,y-coordinates the values x and y.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
rotate, scale
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import RegularPolygon, Point, Polygon
|
||
|
|
>>> t = Polygon(*RegularPolygon(Point(0, 0), 1, 3).vertices)
|
||
|
|
>>> t
|
||
|
|
Triangle(Point2D(1, 0), Point2D(-1/2, sqrt(3)/2), Point2D(-1/2, -sqrt(3)/2))
|
||
|
|
>>> t.translate(2)
|
||
|
|
Triangle(Point2D(3, 0), Point2D(3/2, sqrt(3)/2), Point2D(3/2, -sqrt(3)/2))
|
||
|
|
>>> t.translate(2, 2)
|
||
|
|
Triangle(Point2D(3, 2), Point2D(3/2, sqrt(3)/2 + 2), Point2D(3/2, 2 - sqrt(3)/2))
|
||
|
|
|
||
|
|
"""
|
||
|
|
newargs = []
|
||
|
|
for a in self.args:
|
||
|
|
if isinstance(a, GeometryEntity):
|
||
|
|
newargs.append(a.translate(x, y))
|
||
|
|
else:
|
||
|
|
newargs.append(a)
|
||
|
|
return self.func(*newargs)
|
||
|
|
|
||
|
|
def parameter_value(self, other, t):
|
||
|
|
"""Return the parameter corresponding to the given point.
|
||
|
|
Evaluating an arbitrary point of the entity at this parameter
|
||
|
|
value will return the given point.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Line, Point
|
||
|
|
>>> from sympy.abc import t
|
||
|
|
>>> a = Point(0, 0)
|
||
|
|
>>> b = Point(2, 2)
|
||
|
|
>>> Line(a, b).parameter_value((1, 1), t)
|
||
|
|
{t: 1/2}
|
||
|
|
>>> Line(a, b).arbitrary_point(t).subs(_)
|
||
|
|
Point2D(1, 1)
|
||
|
|
"""
|
||
|
|
from sympy.geometry.point import Point
|
||
|
|
from sympy.core.symbol import Dummy
|
||
|
|
from sympy.solvers.solvers import solve
|
||
|
|
if not isinstance(other, GeometryEntity):
|
||
|
|
other = Point(other, dim=self.ambient_dimension)
|
||
|
|
if not isinstance(other, Point):
|
||
|
|
raise ValueError("other must be a point")
|
||
|
|
T = Dummy('t', real=True)
|
||
|
|
sol = solve(self.arbitrary_point(T) - other, T, dict=True)
|
||
|
|
if not sol:
|
||
|
|
raise ValueError("Given point is not on %s" % func_name(self))
|
||
|
|
return {t: sol[0][T]}
|
||
|
|
|
||
|
|
|
||
|
|
class GeometrySet(GeometryEntity, Set):
|
||
|
|
"""Parent class of all GeometryEntity that are also Sets
|
||
|
|
(compatible with sympy.sets)
|
||
|
|
"""
|
||
|
|
def _contains(self, other):
|
||
|
|
"""sympy.sets uses the _contains method, so include it for compatibility."""
|
||
|
|
|
||
|
|
if isinstance(other, Set) and other.is_FiniteSet:
|
||
|
|
return all(self.__contains__(i) for i in other)
|
||
|
|
|
||
|
|
return self.__contains__(other)
|
||
|
|
|
||
|
|
@dispatch(GeometrySet, Set) # type:ignore # noqa:F811
|
||
|
|
def union_sets(self, o): # noqa:F811
|
||
|
|
""" Returns the union of self and o
|
||
|
|
for use with sympy.sets.Set, if possible. """
|
||
|
|
|
||
|
|
from sympy.sets import Union, FiniteSet
|
||
|
|
|
||
|
|
# if its a FiniteSet, merge any points
|
||
|
|
# we contain and return a union with the rest
|
||
|
|
if o.is_FiniteSet:
|
||
|
|
other_points = [p for p in o if not self._contains(p)]
|
||
|
|
if len(other_points) == len(o):
|
||
|
|
return None
|
||
|
|
return Union(self, FiniteSet(*other_points))
|
||
|
|
if self._contains(o):
|
||
|
|
return self
|
||
|
|
return None
|
||
|
|
|
||
|
|
|
||
|
|
@dispatch(GeometrySet, Set) # type: ignore # noqa:F811
|
||
|
|
def intersection_sets(self, o): # noqa:F811
|
||
|
|
""" Returns a sympy.sets.Set of intersection objects,
|
||
|
|
if possible. """
|
||
|
|
|
||
|
|
from sympy.sets import FiniteSet, Union
|
||
|
|
from sympy.geometry import Point
|
||
|
|
|
||
|
|
try:
|
||
|
|
# if o is a FiniteSet, find the intersection directly
|
||
|
|
# to avoid infinite recursion
|
||
|
|
if o.is_FiniteSet:
|
||
|
|
inter = FiniteSet(*(p for p in o if self.contains(p)))
|
||
|
|
else:
|
||
|
|
inter = self.intersection(o)
|
||
|
|
except NotImplementedError:
|
||
|
|
# sympy.sets.Set.reduce expects None if an object
|
||
|
|
# doesn't know how to simplify
|
||
|
|
return None
|
||
|
|
|
||
|
|
# put the points in a FiniteSet
|
||
|
|
points = FiniteSet(*[p for p in inter if isinstance(p, Point)])
|
||
|
|
non_points = [p for p in inter if not isinstance(p, Point)]
|
||
|
|
|
||
|
|
return Union(*(non_points + [points]))
|
||
|
|
|
||
|
|
def translate(x, y):
|
||
|
|
"""Return the matrix to translate a 2-D point by x and y."""
|
||
|
|
rv = eye(3)
|
||
|
|
rv[2, 0] = x
|
||
|
|
rv[2, 1] = y
|
||
|
|
return rv
|
||
|
|
|
||
|
|
|
||
|
|
def scale(x, y, pt=None):
|
||
|
|
"""Return the matrix to multiply a 2-D point's coordinates by x and y.
|
||
|
|
|
||
|
|
If pt is given, the scaling is done relative to that point."""
|
||
|
|
rv = eye(3)
|
||
|
|
rv[0, 0] = x
|
||
|
|
rv[1, 1] = y
|
||
|
|
if pt:
|
||
|
|
from sympy.geometry.point import Point
|
||
|
|
pt = Point(pt, dim=2)
|
||
|
|
tr1 = translate(*(-pt).args)
|
||
|
|
tr2 = translate(*pt.args)
|
||
|
|
return tr1*rv*tr2
|
||
|
|
return rv
|
||
|
|
|
||
|
|
|
||
|
|
def rotate(th):
|
||
|
|
"""Return the matrix to rotate a 2-D point about the origin by ``angle``.
|
||
|
|
|
||
|
|
The angle is measured in radians. To Point a point about a point other
|
||
|
|
then the origin, translate the Point, do the rotation, and
|
||
|
|
translate it back:
|
||
|
|
|
||
|
|
>>> from sympy.geometry.entity import rotate, translate
|
||
|
|
>>> from sympy import Point, pi
|
||
|
|
>>> rot_about_11 = translate(-1, -1)*rotate(pi/2)*translate(1, 1)
|
||
|
|
>>> Point(1, 1).transform(rot_about_11)
|
||
|
|
Point2D(1, 1)
|
||
|
|
>>> Point(0, 0).transform(rot_about_11)
|
||
|
|
Point2D(2, 0)
|
||
|
|
"""
|
||
|
|
s = sin(th)
|
||
|
|
rv = eye(3)*cos(th)
|
||
|
|
rv[0, 1] = s
|
||
|
|
rv[1, 0] = -s
|
||
|
|
rv[2, 2] = 1
|
||
|
|
return rv
|