Source code for sympy.geometry.util

"""Utility functions for geometrical entities.

Contains
========
intersection
convex_hull
are_coplanar
are_similar

"""
from __future__ import print_function, division

from sympy import Dummy, S, Symbol, Function, solve
from sympy.core.compatibility import string_types, is_sequence


def idiff(eq, y, x, n=1):
    """Return ``dy/dx`` assuming that ``eq == 0``.

    Parameters
    ==========

    y : the dependent variable or a list of dependent variables (with y first)
    x : the variable that the derivative is being taken with respect to
    n : the order of the derivative (default is 1)

    Examples
    ========

    >>> from sympy.abc import x, y, a
    >>> from sympy.geometry.util import idiff

    >>> circ = x**2 + y**2 - 4
    >>> idiff(circ, y, x)
    -x/y
    >>> idiff(circ, y, x, 2).simplify()
    -(x**2 + y**2)/y**3

    Here, ``a`` is assumed to be independent of ``x``:

    >>> idiff(x + a + y, y, x)
    -1

    Now the x-dependence of ``a`` is made explicit by listing ``a`` after
    ``y`` in a list.

    >>> idiff(x + a + y, [y, a], x)
    -Derivative(a, x) - 1

    See Also
    ========

    sympy.core.function.Derivative: represents unevaluated derivatives
    sympy.core.function.diff: explicitly differentiates wrt symbols

    """
    if is_sequence(y):
        dep = set(y)
        y = y[0]
    elif isinstance(y, Symbol):
        dep = set([y])
    else:
        raise ValueError("expecting x-dependent symbol(s) but got: %s" % y)

    f = dict([(s, Function(
        s.name)(x)) for s in eq.free_symbols if s != x and s in dep])
    dydx = Function(y.name)(x).diff(x)
    eq = eq.subs(f)
    derivs = {}
    for i in range(n):
        yp = solve(eq.diff(x), dydx)[0].subs(derivs)
        if i == n - 1:
            return yp.subs([(v, k) for k, v in f.items()])
        derivs[dydx] = yp
        eq = dydx - yp
        dydx = dydx.diff(x)


def _symbol(s, matching_symbol=None):
    """Return s if s is a Symbol, else return either a new Symbol (real=True)
    with the same name s or the matching_symbol if s is a string and it matches
    the name of the matching_symbol.

    >>> from sympy import Symbol
    >>> from sympy.geometry.util import _symbol
    >>> x = Symbol('x')
    >>> _symbol('y')
    y
    >>> _.is_real
    True
    >>> _symbol(x)
    x
    >>> _.is_real is None
    True
    >>> arb = Symbol('foo')
    >>> _symbol('arb', arb) # arb's name is foo so foo will not be returned
    arb
    >>> _symbol('foo', arb) # now it will
    foo

    NB: the symbol here may not be the same as a symbol with the same
    name defined elsewhere as a result of different assumptions.

    See Also
    ========

    sympy.core.symbol.Symbol

    """
    if isinstance(s, string_types):
        if matching_symbol and matching_symbol.name == s:
            return matching_symbol
        return Symbol(s, real=True)
    elif isinstance(s, Symbol):
        return s
    else:
        raise ValueError('symbol must be string for symbol name or Symbol')


def _uniquely_named_symbol(xname, *exprs):
    """Return a symbol which, when printed, will have a name unique
    from any other already in the expressions given. The name is made
    unique by prepending underscores.
    """
    prefix = '%s'
    x = prefix % xname
    syms = set.union(*[e.free_symbols for e in exprs])
    while any(x == str(s) for s in syms):
        prefix = '_' + prefix
        x = prefix % xname
    return _symbol(x)


[docs]def intersection(*entities): """The intersection of a collection of GeometryEntity instances. Parameters ========== entities : sequence of GeometryEntity Returns ======= intersection : list of GeometryEntity Raises ====== NotImplementedError When unable to calculate intersection. Notes ===== The intersection of any geometrical entity with itself should return a list with one item: the entity in question. An intersection requires two or more entities. If only a single entity is given then the function will return an empty list. It is possible for `intersection` to miss intersections that one knows exists because the required quantities were not fully simplified internally. Reals should be converted to Rationals, e.g. Rational(str(real_num)) or else failures due to floating point issues may result. See Also ======== sympy.geometry.entity.GeometryEntity.intersection Examples ======== >>> from sympy.geometry import Point, Line, Circle, intersection >>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(-1, 5) >>> l1, l2 = Line(p1, p2), Line(p3, p2) >>> c = Circle(p2, 1) >>> intersection(l1, p2) [Point(1, 1)] >>> intersection(l1, l2) [Point(1, 1)] >>> intersection(c, p2) [] >>> intersection(c, Point(1, 0)) [Point(1, 0)] >>> intersection(c, l2) [Point(-sqrt(5)/5 + 1, 2*sqrt(5)/5 + 1), Point(sqrt(5)/5 + 1, -2*sqrt(5)/5 + 1)] """ from .entity import GeometryEntity from .point import Point if len(entities) <= 1: return [] for i, e in enumerate(entities): if not isinstance(e, GeometryEntity): try: entities[i] = Point(e) except NotImplementedError: raise ValueError('%s is not a GeometryEntity and cannot be made into Point' % str(e)) res = entities[0].intersection(entities[1]) for entity in entities[2:]: newres = [] for x in res: newres.extend(x.intersection(entity)) res = newres return res
[docs]def convex_hull(*args): """The convex hull surrounding the Points contained in the list of entities. Parameters ========== args : a collection of Points, Segments and/or Polygons Returns ======= convex_hull : Polygon Notes ===== This can only be performed on a set of non-symbolic points. References ========== [1] http://en.wikipedia.org/wiki/Graham_scan [2] Andrew's Monotone Chain Algorithm (A.M. Andrew, "Another Efficient Algorithm for Convex Hulls in Two Dimensions", 1979) http://geomalgorithms.com/a10-_hull-1.html See Also ======== sympy.geometry.point.Point, sympy.geometry.polygon.Polygon Examples ======== >>> from sympy.geometry import Point, convex_hull >>> points = [(1,1), (1,2), (3,1), (-5,2), (15,4)] >>> convex_hull(*points) Polygon(Point(-5, 2), Point(1, 1), Point(3, 1), Point(15, 4)) """ from .entity import GeometryEntity from .point import Point from .line import Segment from .polygon import Polygon p = set() for e in args: if not isinstance(e, GeometryEntity): try: e = Point(e) except NotImplementedError: raise ValueError('%s is not a GeometryEntity and cannot be made into Point' % str(e)) if isinstance(e, Point): p.add(e) elif isinstance(e, Segment): p.update(e.points) elif isinstance(e, Polygon): p.update(e.vertices) else: raise NotImplementedError( 'Convex hull for %s not implemented.' % type(e)) p = list(p) if len(p) == 1: return p[0] elif len(p) == 2: return Segment(p[0], p[1]) def _orientation(p, q, r): '''Return positive if p-q-r are clockwise, neg if ccw, zero if collinear.''' return (q.y - p.y)*(r.x - p.x) - (q.x - p.x)*(r.y - p.y) # scan to find upper and lower convex hulls of a set of 2d points. U = [] L = [] p.sort(key=lambda x: x.args) for p_i in p: while len(U) > 1 and _orientation(U[-2], U[-1], p_i) <= 0: U.pop() while len(L) > 1 and _orientation(L[-2], L[-1], p_i) >= 0: L.pop() U.append(p_i) L.append(p_i) U.reverse() convexHull = tuple(L + U[1:-1]) if len(convexHull) == 2: return Segment(convexHull[0], convexHull[1]) return Polygon(*convexHull)
def are_coplanar(*e): """ Returns True if the given entities are coplanar otherwise False Parameters ========== e: entities to be checked for being coplanar Returns ======= Boolean Examples ======== >>> from sympy import Point3D, Line3D >>> from sympy.geometry.util import are_coplanar >>> a = Line3D(Point3D(5, 0, 0), Point3D(1, -1, 1)) >>> b = Line3D(Point3D(0, -2, 0), Point3D(3, 1, 1)) >>> c = Line3D(Point3D(0, -1, 0), Point3D(5, -1, 9)) >>> are_coplanar(a, b, c) False """ from sympy.geometry.line3d import LinearEntity3D from sympy.geometry.point3d import Point3D from sympy.geometry.plane import Plane # XXX update tests for coverage e = set(e) # first work with a Plane if present for i in list(e): if isinstance(i, Plane): e.remove(i) return all(p.is_coplanar(i) for p in e) if all(isinstance(i, Point3D) for i in e): if len(e) < 3: return False # remove pts that are collinear with 2 pts a, b = e.pop(), e.pop() for i in list(e): if Point3D.are_collinear(a, b, i): e.remove(i) if not e: return False else: # define a plane p = Plane(a, b, e.pop()) for i in e: if i not in p: return False return True else: pt3d = [] for i in e: if isinstance(i, Point3D): pt3d.append(i) elif isinstance(i, LinearEntity3D): pt3d.extend(i.args) elif isinstance(i, GeometryEntity): # XXX we should have a GeometryEntity3D class so we can tell the difference between 2D and 3D -- here we just want to deal with 2D objects; if new 3D objects are encountered that we didn't hanlde above, an error should be raised # all 2D objects have some Point that defines them; so convert those points to 3D pts by making z=0 for p in i.args: if isinstance(p, Point): pt3d.append(Point3D(*(p.args + (0,)))) return are_coplanar(*pt3d)
[docs]def are_similar(e1, e2): """Are two geometrical entities similar. Can one geometrical entity be uniformly scaled to the other? Parameters ========== e1 : GeometryEntity e2 : GeometryEntity Returns ======= are_similar : boolean Raises ====== GeometryError When `e1` and `e2` cannot be compared. Notes ===== If the two objects are equal then they are similar. See Also ======== sympy.geometry.entity.GeometryEntity.is_similar Examples ======== >>> from sympy import Point, Circle, Triangle, are_similar >>> c1, c2 = Circle(Point(0, 0), 4), Circle(Point(1, 4), 3) >>> t1 = Triangle(Point(0, 0), Point(1, 0), Point(0, 1)) >>> t2 = Triangle(Point(0, 0), Point(2, 0), Point(0, 2)) >>> t3 = Triangle(Point(0, 0), Point(3, 0), Point(0, 1)) >>> are_similar(t1, t2) True >>> are_similar(t1, t3) False """ from .exceptions import GeometryError if e1 == e2: return True try: return e1.is_similar(e2) except AttributeError: try: return e2.is_similar(e1) except AttributeError: n1 = e1.__class__.__name__ n2 = e2.__class__.__name__ raise GeometryError( "Cannot test similarity between %s and %s" % (n1, n2))
[docs]def centroid(*args): """Find the centroid (center of mass) of the collection containing only Points, Segments or Polygons. The centroid is the weighted average of the individual centroid where the weights are the lengths (of segments) or areas (of polygons). Overlapping regions will add to the weight of that region. If there are no objects (or a mixture of objects) then None is returned. See Also ======== sympy.geometry.point.Point, sympy.geometry.line.Segment, sympy.geometry.polygon.Polygon Examples ======== >>> from sympy import Point, Segment, Polygon >>> from sympy.geometry.util import centroid >>> p = Polygon((0, 0), (10, 0), (10, 10)) >>> q = p.translate(0, 20) >>> p.centroid, q.centroid (Point(20/3, 10/3), Point(20/3, 70/3)) >>> centroid(p, q) Point(20/3, 40/3) >>> p, q = Segment((0, 0), (2, 0)), Segment((0, 0), (2, 2)) >>> centroid(p, q) Point(1, -sqrt(2) + 2) >>> centroid(Point(0, 0), Point(2, 0)) Point(1, 0) Stacking 3 polygons on top of each other effectively triples the weight of that polygon: >>> p = Polygon((0, 0), (1, 0), (1, 1), (0, 1)) >>> q = Polygon((1, 0), (3, 0), (3, 1), (1, 1)) >>> centroid(p, q) Point(3/2, 1/2) >>> centroid(p, p, p, q) # centroid x-coord shifts left Point(11/10, 1/2) Stacking the squares vertically above and below p has the same effect: >>> centroid(p, p.translate(0, 1), p.translate(0, -1), q) Point(11/10, 1/2) """ from sympy.geometry import Polygon, Segment, Point if args: if all(isinstance(g, Point) for g in args): c = Point(0, 0) for g in args: c += g den = len(args) elif all(isinstance(g, Segment) for g in args): c = Point(0, 0) L = 0 for g in args: l = g.length c += g.midpoint*l L += l den = L elif all(isinstance(g, Polygon) for g in args): c = Point(0, 0) A = 0 for g in args: a = g.area c += g.centroid*a A += a den = A c /= den return c.func(*[i.simplify() for i in c.args])