Source code for diofant.geometry.entity

"""The definition of the base geometrical entity with attributes common to
all derived geometrical entities.

Contains
========

GeometryEntity
GeometricSet

Notes
=====

A GeometryEntity is any object that has special geometric properties.
A GeometrySet is a superclass of any GeometryEntity that can also
be viewed as a diofant.sets.Set.  In particular, points are the only
GeometryEntity not considered a Set.

Rn is a GeometrySet representing n-dimensional Euclidean space. R2 and
R3 are currently the only ambient spaces implemented.

"""

from ..core import Basic, Dummy, Tuple, oo, sympify
from ..core.compatibility import is_sequence
from ..functions import atan, cos, sin
from ..matrices import eye
from ..sets import Set


[docs]class GeometryEntity(Basic): """The base class for all geometrical entities. This class doesn't represent any particular geometric entity, it only provides the implementation of some methods common to all subclasses. """ def __new__(cls, *args, **kwargs): from .point import Point args = [Tuple(*a) if is_sequence(a) and not isinstance(a, Point) else sympify(a) for a in args] return Basic.__new__(cls, *args) def _diofant_(self): return self def __getnewargs__(self): return tuple(self.args)
[docs] def intersection(self, o): """ Returns a list of all of the intersections of self with o. Notes ===== An entity is not required to implement this method. See Also ======== diofant.geometry.util.intersection """ raise NotImplementedError()
[docs] 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 ======== >>> 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)
[docs] 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 ======== >>> 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 .point import Point if pt: pt = Point(pt) 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
[docs] 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 ======== >>> 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, -sqrt(3)/2 + 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 reflect(self, line): from .point import Point g = self l = line o = Point(0, 0) if l.slope == 0: y = l.args[0].y 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 == 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) xf = xf.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))
[docs] def encloses(self, o): """ Return True if o is inside (not on or outside) the boundaries of self. The object will be decomposed into Points and individual Entities need only define an encloses_point method for their class. See Also ======== diofant.geometry.ellipse.Ellipse.encloses_point diofant.geometry.polygon.Polygon.encloses_point Examples ======== >>> t = Polygon(*RegularPolygon(Point(0, 0), 1, 3).vertices) >>> t2 = Polygon(*RegularPolygon(Point(0, 0), 2, 3).vertices) >>> t2.encloses(t) True >>> t.encloses(t2) False """ from .line import Segment, Ray, Line from .ellipse import Ellipse from .polygon import Polygon, RegularPolygon from .point import Point if isinstance(o, Point): return self.encloses_point(o) elif isinstance(o, Segment): return all(self.encloses_point(x) for x in o.points) elif isinstance(o, Ray) or isinstance(o, Line): return False elif isinstance(o, Ellipse): return (self.encloses_point(o.center) and not self.intersection(o) and self.encloses_point(Point(o.center.x + o.hradius, o.center.y))) elif isinstance(o, Polygon): if isinstance(o, RegularPolygon): if not self.encloses_point(o.center): return False return all(self.encloses_point(v) for v in o.vertices) raise NotImplementedError()
@property def ambient_dimension(self): """What is the dimension of the space that the object is contained in?""" raise NotImplementedError()
[docs] def is_similar(self, other): """Is this geometrical entity similar to another geometrical entity? Two entities are similar if a uniform scaling (enlarging or shrinking) of one of the entities will allow one to obtain the other. Notes ===== This method is not intended to be used directly but rather through the `are_similar` function found in util.py. An entity is not required to implement this method. If two different types of entities can be similar, it is only required that one of them be able to determine this. See Also ======== scale """ raise NotImplementedError()
def equals(self, o): return self == o def __radd__(self, a): return a.__add__(self) def __rsub__(self, a): return a.__sub__(self) def __rmul__(self, a): return a.__mul__(self) def __str__(self): """String representation of a GeometryEntity.""" from ..printing import sstr return type(self).__name__ + sstr(self.args) def __repr__(self): """String representation of a GeometryEntity that can be evaluated by diofant. """ return type(self).__name__ + repr(self.args) def __contains__(self, other): """Subclasses should implement this method for anything more complex than equality.""" if type(self) == type(other): return self == other raise NotImplementedError() def _eval_subs(self, old, new): from .point import Point, Point3D if is_sequence(old) or is_sequence(new): if isinstance(self, Point3D): old = Point3D(old) new = Point3D(new) else: old = Point(old) new = Point(new) return self._subs(old, new)
class GeometrySet(GeometryEntity, Set): """Parent class of all GeometryEntity that are also Sets (compatible with diofant.sets). """ def _contains(self, other): """diofant.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) def _union(self, o): """ Returns the union of self and o for use with diofant.sets.Set, if possible. """ from ..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 return Union(self, FiniteSet(*other_points)) if self._contains(o): return self def _intersection(self, o): """Returns a diofant.sets.Set of intersection objects, if possible. """ from .point import Point from ..sets import FiniteSet, Union try: inter = self.intersection(o) except NotImplementedError: # diofant.sets.Set.reduce expects None if an object # doesn't know how to simplify return # 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 .point import Point pt = Point(pt) 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: >>> 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