Python sympy.Rational类代码示例

def test_point3D():
    p1 = Point3D(x1, x2, x3)
    p2 = Point3D(y1, y2, y3)
    p3 = Point3D(0, 0, 0)
    p4 = Point3D(1, 1, 1)
    p5 = Point3D(0, 1, 2)

    assert p1 in p1
    assert p1 not in p2
    assert p2.y == y2
    assert (p3 + p4) == p4
    assert (p2 - p1) == Point3D(y1 - x1, y2 - x2, y3 - x3)
    assert p4*5 == Point3D(5, 5, 5)
    assert -p2 == Point3D(-y1, -y2, -y3)

    assert Point(34.05, sqrt(3)) == Point(Rational(681, 20), sqrt(3))
    assert Point3D.midpoint(p3, p4) == Point3D(half, half, half)
    assert Point3D.midpoint(p1, p4) == Point3D(half + half*x1, half + half*x2,
                                         half + half*x3)
    assert Point3D.midpoint(p2, p2) == p2
    assert p2.midpoint(p2) == p2

    assert Point3D.distance(p3, p4) == sqrt(3)
    assert Point3D.distance(p1, p1) == 0
    assert Point3D.distance(p3, p2) == sqrt(p2.x**2 + p2.y**2 + p2.z**2)

    p1_1 = Point3D(x1, x1, x1)
    p1_2 = Point3D(y2, y2, y2)
    p1_3 = Point3D(x1 + 1, x1, x1)
    assert Point3D.is_collinear(p3)
    assert Point3D.is_collinear(p3, p4)
    assert Point3D.is_collinear(p3, p4, p1_1, p1_2)
    assert Point3D.is_collinear(p3, p4, p1_1, p1_3) is False
    assert Point3D.is_collinear(p3, p3, p4, p5) is False

    assert p3.intersection(Point3D(0, 0, 0)) == [p3]
    assert p3.intersection(p4) == []


    assert p4 * 5 == Point3D(5, 5, 5)
    assert p4 / 5 == Point3D(0.2, 0.2, 0.2)

    raises(ValueError, lambda: Point3D(0, 0, 0) + 10)

    # Point differences should be simplified
    assert Point3D(x*(x - 1), y, 2) - Point3D(x**2 - x, y + 1, 1) == \
        Point3D(0, -1, 1)

    a, b = Rational(1, 2), Rational(1, 3)
    assert Point(a, b).evalf(2) == \
        Point(a.n(2), b.n(2))
    raises(ValueError, lambda: Point(1, 2) + 1)

    # test transformations
    p = Point3D(1, 1, 1)
    assert p.scale(2, 3) == Point3D(2, 3, 1)
    assert p.translate(1, 2) == Point3D(2, 3, 1)
    assert p.translate(1) == Point3D(2, 1, 1)
    assert p.translate(z=1) == Point3D(1, 1, 2)
    assert p.translate(*p.args) == Point3D(2, 2, 2)

def test_composite_sums():
    f = Rational(1, 2)*(7 - 6*n + Rational(1, 7)*n**3)
    s = summation(f, (n, a, b))
    assert not isinstance(s, Sum)
    A = 0
    for i in range(-3, 5):
        A += f.subs(n, i)
    B = s.subs(a, -3).subs(b, 4)
    assert A == B

def test_behavior2():
    x = Symbol('x')
    p = Wild('p')

    e = Rational(6)
    assert e.match(2*p) == {p: 3}

    e = 3*x + 3 + 6/x
    a = Wild('a', exclude = [x])
    assert e.expand().match(a*x**2 + a*x + 2*a) == None
    assert e.expand().match(p*x**2 + p*x + 2*p) == {p: 3/x}

def test_symbol():
    x = Symbol('x')
    a,b,c,p,q = map(Wild, 'abcpq')

    e = x
    assert e.match(x) == {}
    assert e.match(a) == {a: x}

    e = Rational(5)
    assert e.match(c) == {c: 5}
    assert e.match(e) == {}
    assert e.match(e+1) == None

 def __new__(cls, p, q=None, prec=15):
     rat = Rational.__new__(cls, p, q)
     if isinstance(rat, (Integer, Infinity)):
         return rat
     obj = Expr.__new__(cls)
     obj.p = rat.p
     obj.q = rat.q
     obj.prec = prec
     return obj

def test_pow_im():
    for m in (-2, -1, 2):
        for d in (3, 4, 5):
            b = m * I
            for i in range(1, 4 * d + 1):
                e = Rational(i, d)
                assert (b ** e - b.n() ** e.n()).n(2, chop=1e-10) == 0

    e = Rational(7, 3)
    assert (2 * x * I) ** e == 4 * 2 ** Rational(1, 3) * (I * x) ** e  # same as Wolfram Alpha
    im = symbols("im", imaginary=True)
    assert (2 * im * I) ** e == 4 * 2 ** Rational(1, 3) * (I * im) ** e

    args = [I, I, I, I, 2]
    e = Rational(1, 3)
    ans = 2 ** e
    assert Mul(*args, **dict(evaluate=False)) ** e == ans
    assert Mul(*args) ** e == ans
    args = [I, I, I, 2]
    e = Rational(1, 3)
    ans = -(-1) ** Rational(5, 6) * 2 ** e
    assert Mul(*args, **dict(evaluate=False)) ** e == ans
    assert Mul(*args) ** e == ans
    args = [I, I, 2]
    e = Rational(1, 3)
    ans = (-2) ** e
    assert Mul(*args, **dict(evaluate=False)) ** e == ans
    assert Mul(*args) ** e == ans

def test_pow_im():
    for m in (-2, -1, 2):
        for d in (3, 4, 5):
            b = m*I
            for i in range(1, 4*d + 1):
                e = Rational(i, d)
                assert (b**e - b.n()**e.n()).n(2, chop=1e-10) == 0

    e = Rational(7, 3)
    assert (2*x*I)**e == 4*2**Rational(1, 3)*(I*x)**e  # same as Wolfram Alpha
    im = symbols('im', imaginary=True)
    assert (2*im*I)**e == 4*2**Rational(1, 3)*(I*im)**e

    args = [I, I, I, I, 2]
    e = Rational(1, 3)
    ans = 2**e
    assert Mul(*args, evaluate=False)**e == ans
    assert Mul(*args)**e == ans
    args = [I, I, I, 2]
    e = Rational(1, 3)
    ans = 2**e*(-I)**e
    assert Mul(*args, evaluate=False)**e == ans
    assert Mul(*args)**e == ans
    args.append(-3)
    ans = (6*I)**e
    assert Mul(*args, evaluate=False)**e == ans
    assert Mul(*args)**e == ans
    args.append(-1)
    ans = (-6*I)**e
    assert Mul(*args, evaluate=False)**e == ans
    assert Mul(*args)**e == ans

    args = [I, I, 2]
    e = Rational(1, 3)
    ans = (-2)**e
    assert Mul(*args, evaluate=False)**e == ans
    assert Mul(*args)**e == ans
    args.append(-3)
    ans = (6)**e
    assert Mul(*args, evaluate=False)**e == ans
    assert Mul(*args)**e == ans
    args.append(-1)
    ans = (-6)**e
    assert Mul(*args, evaluate=False)**e == ans
    assert Mul(*args)**e == ans
    assert Mul(Pow(-1, Rational(3, 2), evaluate=False), I, I) == I
    assert Mul(I*Pow(I, S.Half, evaluate=False)) == (-1)**Rational(3, 4)

def test_point():
    p1 = Point(x1, x2)
    p2 = Point(y1, y2)
    p3 = Point(0, 0)
    p4 = Point(1, 1)

    assert p1 in p1
    assert p1 not in p2
    assert p2.y == y2
    assert (p3 + p4) == p4
    assert (p2 - p1) == Point(y1 - x1, y2 - x2)
    assert p4 * 5 == Point(5, 5)
    assert -p2 == Point(-y1, -y2)

    assert Point.midpoint(p3, p4) == Point(half, half)
    assert Point.midpoint(p1, p4) == Point(half + half * x1, half + half * x2)
    assert Point.midpoint(p2, p2) == p2
    assert p2.midpoint(p2) == p2

    assert Point.distance(p3, p4) == sqrt(2)
    assert Point.distance(p1, p1) == 0
    assert Point.distance(p3, p2) == sqrt(p2.x ** 2 + p2.y ** 2)

    p1_1 = Point(x1, x1)
    p1_2 = Point(y2, y2)
    p1_3 = Point(x1 + 1, x1)
    assert Point.is_collinear(p3)
    assert Point.is_collinear(p3, p4)
    assert Point.is_collinear(p3, p4, p1_1, p1_2)
    assert Point.is_collinear(p3, p4, p1_1, p1_3) == False

    assert p3.intersection(Point(0, 0)) == [p3]
    assert p3.intersection(p4) == []

    x_pos = Symbol("x", real=True, positive=True)
    p2_1 = Point(x_pos, 0)
    p2_2 = Point(0, x_pos)
    p2_3 = Point(-x_pos, 0)
    p2_4 = Point(0, -x_pos)
    p2_5 = Point(x_pos, 5)
    assert Point.is_concyclic(p2_1)
    assert Point.is_concyclic(p2_1, p2_2)
    assert Point.is_concyclic(p2_1, p2_2, p2_3, p2_4)
    assert Point.is_concyclic(p2_1, p2_2, p2_3, p2_5) == False
    assert Point.is_concyclic(p4, p4 * 2, p4 * 3) == False

    assert p4.scale(2, 3) == Point(2, 3)
    assert p3.scale(2, 3) == p3

    assert p4.rotate(pi, Point(0.5, 0.5)) == p3
    assert p1.__radd__(p2) == p1.midpoint(p2).scale(2, 2)
    assert (-p3).__rsub__(p4) == p3.midpoint(p4).scale(2, 2)

    assert p4 * 5 == Point(5, 5)
    assert p4 / 5 == Point(0.2, 0.2)

    raises(ValueError, lambda: Point(0, 0) + 10)

    # Point differences should be simplified
    assert Point(x * (x - 1), y) - Point(x ** 2 - x, y + 1) == Point(0, -1)

    a, b = Rational(1, 2), Rational(1, 3)
    assert Point(a, b).evalf(2) == Point(a.n(2), b.n(2))
    raises(ValueError, lambda: Point(1, 2) + 1)

    # test transformations
    p = Point(1, 0)
    assert p.rotate(pi / 2) == Point(0, 1)
    assert p.rotate(pi / 2, p) == p
    p = Point(1, 1)
    assert p.scale(2, 3) == Point(2, 3)
    assert p.translate(1, 2) == Point(2, 3)
    assert p.translate(1) == Point(2, 1)
    assert p.translate(y=1) == Point(1, 2)
    assert p.translate(*p.args) == Point(2, 2)

def test_match_exclude():
    x = Symbol('x')
    y = Symbol('y')
    p = Wild("p")
    q = Wild("q")
    r = Wild("r")

    e = Rational(6)
    assert e.match(2*p) == {p: 3}

    e = 3/(4*x + 5)
    assert e.match(3/(p*x + q)) == {p: 4, q: 5}

    e = 3/(4*x + 5)
    assert e.match(p/(q*x + r)) == {p: 3, q: 4, r: 5}

    e = 2/(x + 1)
    assert e.match(p/(q*x + r)) == {p: 2, q: 1, r: 1}

    e = 1/(x + 1)
    assert e.match(p/(q*x + r)) == {p: 1, q: 1, r: 1}

    e = 4*x + 5
    assert e.match(p*x + q) == {p: 4, q: 5}

    e = 4*x + 5*y + 6
    assert e.match(p*x + q*y + r) == {p: 4, q: 5, r: 6}

    a = Wild('a', exclude=[x])

    e = 3*x
    assert e.match(p*x) == {p: 3}
    assert e.match(a*x) == {a: 3}

    e = 3*x**2
    assert e.match(p*x) == {p: 3*x}
    assert e.match(a*x) is None

    e = 3*x + 3 + 6/x
    assert e.match(p*x**2 + p*x + 2*p) == {p: 3/x}
    assert e.match(a*x**2 + a*x + 2*a) is None

def test_point():
    x = Symbol('x', real=True)
    y = Symbol('y', real=True)
    x1 = Symbol('x1', real=True)
    x2 = Symbol('x2', real=True)
    y1 = Symbol('y1', real=True)
    y2 = Symbol('y2', real=True)
    half = Rational(1, 2)
    p1 = Point(x1, x2)
    p2 = Point(y1, y2)
    p3 = Point(0, 0)
    p4 = Point(1, 1)
    p5 = Point(0, 1)

    assert p1 in p1
    assert p1 not in p2
    assert p2.y == y2
    assert (p3 + p4) == p4
    assert (p2 - p1) == Point(y1 - x1, y2 - x2)
    assert p4*5 == Point(5, 5)
    assert -p2 == Point(-y1, -y2)
    raises(ValueError, lambda: Point(3, I))
    raises(ValueError, lambda: Point(2*I, I))
    raises(ValueError, lambda: Point(3 + I, I))

    assert Point(34.05, sqrt(3)) == Point(Rational(681, 20), sqrt(3))
    assert Point.midpoint(p3, p4) == Point(half, half)
    assert Point.midpoint(p1, p4) == Point(half + half*x1, half + half*x2)
    assert Point.midpoint(p2, p2) == p2
    assert p2.midpoint(p2) == p2

    assert Point.distance(p3, p4) == sqrt(2)
    assert Point.distance(p1, p1) == 0
    assert Point.distance(p3, p2) == sqrt(p2.x**2 + p2.y**2)

    assert Point.taxicab_distance(p4, p3) == 2

    p1_1 = Point(x1, x1)
    p1_2 = Point(y2, y2)
    p1_3 = Point(x1 + 1, x1)
    assert Point.is_collinear(p3)
    assert Point.is_collinear(p3, p4)
    assert Point.is_collinear(p3, p4, p1_1, p1_2)
    assert Point.is_collinear(p3, p4, p1_1, p1_3) is False
    assert Point.is_collinear(p3, p3, p4, p5) is False
    line = Line(Point(1,0), slope = 1)
    raises(TypeError, lambda: Point.is_collinear(line))
    raises(TypeError, lambda: p1_1.is_collinear(line))

    assert p3.intersection(Point(0, 0)) == [p3]
    assert p3.intersection(p4) == []

    assert p1.dot(p4) == x1 + x2
    assert p3.dot(p4) == 0
    assert p4.dot(p5) == 1

    x_pos = Symbol('x', real=True, positive=True)
    p2_1 = Point(x_pos, 0)
    p2_2 = Point(0, x_pos)
    p2_3 = Point(-x_pos, 0)
    p2_4 = Point(0, -x_pos)
    p2_5 = Point(x_pos, 5)
    assert Point.is_concyclic(p2_1)
    assert Point.is_concyclic(p2_1, p2_2)
    assert Point.is_concyclic(p2_1, p2_2, p2_3, p2_4)
    assert Point.is_concyclic(p2_1, p2_2, p2_3, p2_5) is False
    assert Point.is_concyclic(p4, p4 * 2, p4 * 3) is False

    assert p4.scale(2, 3) == Point(2, 3)
    assert p3.scale(2, 3) == p3

    assert p4.rotate(pi, Point(0.5, 0.5)) == p3
    assert p1.__radd__(p2) == p1.midpoint(p2).scale(2, 2)
    assert (-p3).__rsub__(p4) == p3.midpoint(p4).scale(2, 2)

    assert p4 * 5 == Point(5, 5)
    assert p4 / 5 == Point(0.2, 0.2)

    raises(ValueError, lambda: Point(0, 0) + 10)

    # Point differences should be simplified
    assert Point(x*(x - 1), y) - Point(x**2 - x, y + 1) == Point(0, -1)

    a, b = Rational(1, 2), Rational(1, 3)
    assert Point(a, b).evalf(2) == \
        Point(a.n(2), b.n(2))
    raises(ValueError, lambda: Point(1, 2) + 1)

    # test transformations
    p = Point(1, 0)
    assert p.rotate(pi/2) == Point(0, 1)
    assert p.rotate(pi/2, p) == p
    p = Point(1, 1)
    assert p.scale(2, 3) == Point(2, 3)
    assert p.translate(1, 2) == Point(2, 3)
    assert p.translate(1) == Point(2, 1)
    assert p.translate(y=1) == Point(1, 2)
    assert p.translate(*p.args) == Point(2, 2)

    # Check invalid input for transform
#.........这里部分代码省略.........

def test_point3D():
    x = Symbol('x', real=True)
    y = Symbol('y', real=True)
    x1 = Symbol('x1', real=True)
    x2 = Symbol('x2', real=True)
    x3 = Symbol('x3', real=True)
    y1 = Symbol('y1', real=True)
    y2 = Symbol('y2', real=True)
    y3 = Symbol('y3', real=True)
    half = Rational(1, 2)
    p1 = Point3D(x1, x2, x3)
    p2 = Point3D(y1, y2, y3)
    p3 = Point3D(0, 0, 0)
    p4 = Point3D(1, 1, 1)
    p5 = Point3D(0, 1, 2)

    assert p1 in p1
    assert p1 not in p2
    assert p2.y == y2
    assert (p3 + p4) == p4
    assert (p2 - p1) == Point3D(y1 - x1, y2 - x2, y3 - x3)
    assert p4*5 == Point3D(5, 5, 5)
    assert -p2 == Point3D(-y1, -y2, -y3)

    assert Point(34.05, sqrt(3)) == Point(Rational(681, 20), sqrt(3))
    assert Point3D.midpoint(p3, p4) == Point3D(half, half, half)
    assert Point3D.midpoint(p1, p4) == Point3D(half + half*x1, half + half*x2,
                                         half + half*x3)
    assert Point3D.midpoint(p2, p2) == p2
    assert p2.midpoint(p2) == p2

    assert Point3D.distance(p3, p4) == sqrt(3)
    assert Point3D.distance(p1, p1) == 0
    assert Point3D.distance(p3, p2) == sqrt(p2.x**2 + p2.y**2 + p2.z**2)

    p1_1 = Point3D(x1, x1, x1)
    p1_2 = Point3D(y2, y2, y2)
    p1_3 = Point3D(x1 + 1, x1, x1)
    # according to the description in the docs, points are collinear
    # if they like on a single line.  Thus a single point should always
    # be collinear
    assert Point3D.are_collinear(p3)
    assert Point3D.are_collinear(p3, p4)
    assert Point3D.are_collinear(p3, p4, p1_1, p1_2)
    assert Point3D.are_collinear(p3, p4, p1_1, p1_3) is False
    assert Point3D.are_collinear(p3, p3, p4, p5) is False

    assert p3.intersection(Point3D(0, 0, 0)) == [p3]
    assert p3.intersection(p4) == []


    assert p4 * 5 == Point3D(5, 5, 5)
    assert p4 / 5 == Point3D(0.2, 0.2, 0.2)

    raises(ValueError, lambda: Point3D(0, 0, 0) + 10)

    # Point differences should be simplified
    assert Point3D(x*(x - 1), y, 2) - Point3D(x**2 - x, y + 1, 1) == \
        Point3D(0, -1, 1)

    a, b = Rational(1, 2), Rational(1, 3)
    assert Point(a, b).evalf(2) == \
        Point(a.n(2), b.n(2))
    raises(ValueError, lambda: Point(1, 2) + 1)

    # test transformations
    p = Point3D(1, 1, 1)
    assert p.scale(2, 3) == Point3D(2, 3, 1)
    assert p.translate(1, 2) == Point3D(2, 3, 1)
    assert p.translate(1) == Point3D(2, 1, 1)
    assert p.translate(z=1) == Point3D(1, 1, 2)
    assert p.translate(*p.args) == Point3D(2, 2, 2)

    # Test __new__
    assert Point3D(Point3D(1, 2, 3), 4, 5, evaluate=False) ==  Point3D(1, 2, 3)


    # Test length property returns correctly
    assert p.length == 0
    assert p1_1.length == 0
    assert p1_2.length == 0

    # Test are_colinear type error
    raises(TypeError, lambda: Point3D.are_collinear(p, x))

    # Test are_coplanar
    planar2 = Point3D(1, -1, 1)
    planar3 = Point3D(-1, 1, 1)
    assert Point3D.are_coplanar(p, planar2, planar3) == True
    assert Point3D.are_coplanar(p, planar2, planar3, p3) == False
    raises(ValueError, lambda: Point3D.are_coplanar(p, planar2))
    planar2 = Point3D(1, 1, 2)
    planar3 = Point3D(1, 1, 3)
    raises(ValueError, lambda: Point3D.are_coplanar(p, planar2, planar3))

    # Test Intersection
    assert planar2.intersection(Line3D(p, planar3)) == [Point3D(1, 1, 2)]

    # Test Scale
    assert planar2.scale(1, 1, 1) == planar2
#.........这里部分代码省略.........

def test_point3D():
    x = Symbol('x', real=True)
    y = Symbol('y', real=True)
    x1 = Symbol('x1', real=True)
    x2 = Symbol('x2', real=True)
    x3 = Symbol('x3', real=True)
    y1 = Symbol('y1', real=True)
    y2 = Symbol('y2', real=True)
    y3 = Symbol('y3', real=True)
    half = Rational(1, 2)
    p1 = Point3D(x1, x2, x3)
    p2 = Point3D(y1, y2, y3)
    p3 = Point3D(0, 0, 0)
    p4 = Point3D(1, 1, 1)
    p5 = Point3D(0, 1, 2)

    assert p1 in p1
    assert p1 not in p2
    assert p2.y == y2
    assert (p3 + p4) == p4
    assert (p2 - p1) == Point3D(y1 - x1, y2 - x2, y3 - x3)
    assert p4*5 == Point3D(5, 5, 5)
    assert -p2 == Point3D(-y1, -y2, -y3)

    assert Point(34.05, sqrt(3)) == Point(Rational(681, 20), sqrt(3))
    assert Point3D.midpoint(p3, p4) == Point3D(half, half, half)
    assert Point3D.midpoint(p1, p4) == Point3D(half + half*x1, half + half*x2,
                                         half + half*x3)
    assert Point3D.midpoint(p2, p2) == p2
    assert p2.midpoint(p2) == p2

    assert Point3D.distance(p3, p4) == sqrt(3)
    assert Point3D.distance(p1, p1) == 0
    assert Point3D.distance(p3, p2) == sqrt(p2.x**2 + p2.y**2 + p2.z**2)

    p1_1 = Point3D(x1, x1, x1)
    p1_2 = Point3D(y2, y2, y2)
    p1_3 = Point3D(x1 + 1, x1, x1)
    Point3D.are_collinear(p3)
    assert Point3D.are_collinear(p3, p4)
    assert Point3D.are_collinear(p3, p4, p1_1, p1_2)
    assert Point3D.are_collinear(p3, p4, p1_1, p1_3) is False
    assert Point3D.are_collinear(p3, p3, p4, p5) is False

    assert p3.intersection(Point3D(0, 0, 0)) == [p3]
    assert p3.intersection(p4) == []


    assert p4 * 5 == Point3D(5, 5, 5)
    assert p4 / 5 == Point3D(0.2, 0.2, 0.2)

    raises(ValueError, lambda: Point3D(0, 0, 0) + 10)

    # Point differences should be simplified
    assert Point3D(x*(x - 1), y, 2) - Point3D(x**2 - x, y + 1, 1) == \
        Point3D(0, -1, 1)

    a, b = Rational(1, 2), Rational(1, 3)
    assert Point(a, b).evalf(2) == \
        Point(a.n(2), b.n(2))
    raises(ValueError, lambda: Point(1, 2) + 1)

    # test transformations
    p = Point3D(1, 1, 1)
    assert p.scale(2, 3) == Point3D(2, 3, 1)
    assert p.translate(1, 2) == Point3D(2, 3, 1)
    assert p.translate(1) == Point3D(2, 1, 1)
    assert p.translate(z=1) == Point3D(1, 1, 2)
    assert p.translate(*p.args) == Point3D(2, 2, 2)

    # Test __new__
    assert Point3D(0.1, 0.2, evaluate=False, on_morph='ignore').args[0].is_Float


    # Test length property returns correctly
    assert p.length == 0
    assert p1_1.length == 0
    assert p1_2.length == 0

    # Test are_colinear type error
    raises(TypeError, lambda: Point3D.are_collinear(p, x))

    # Test are_coplanar
    assert Point.are_coplanar()
    assert Point.are_coplanar((1, 2, 0), (1, 2, 0), (1, 3, 0))
    assert Point.are_coplanar((1, 2, 0), (1, 2, 3))
    with warnings.catch_warnings(record=True) as w:
        raises(ValueError, lambda: Point2D.are_coplanar((1, 2), (1, 2, 3)))
    assert Point3D.are_coplanar((1, 2, 0), (1, 2, 3))
    assert Point.are_coplanar((0, 0, 0), (1, 1, 0), (1, 1, 1), (1, 2, 1)) is False
    planar2 = Point3D(1, -1, 1)
    planar3 = Point3D(-1, 1, 1)
    assert Point3D.are_coplanar(p, planar2, planar3) == True
    assert Point3D.are_coplanar(p, planar2, planar3, p3) == False
    assert Point.are_coplanar(p, planar2)
    planar2 = Point3D(1, 1, 2)
    planar3 = Point3D(1, 1, 3)
    assert Point3D.are_coplanar(p, planar2, planar3)  # line, not plane
    plane = Plane((1, 2, 1), (2, 1, 0), (3, 1, 2))
    assert Point.are_coplanar(*[plane.projection(((-1)**i, i)) for i in range(4)])
#.........这里部分代码省略.........

LaguerreF = binomial(n + alpha, n - k) * (-x) ** k / factorial(k)
LaguerreP = Sum(LaguerreF, (k, 0, n))

for alphavar in range(4):
    for nvar in range(4):
        print alphavar, nvar, LaguerreP.subs(n, nvar).subs(alpha, alphavar).doit(), LaguerreP.subs(n, nvar).subs(
            alpha, alphavar
        ).doit() == laguerre_poly(
            nvar, x, alpha=alphavar
        )  # True

(LaguerreF.subs(n, n + 1) / LaguerreF).simplify()  # (alpha + n + 1)/(-k + n + 1)
(LaguerreF.subs(k, k + 1) / LaguerreF).simplify()  # x*(k - n)/((k + 1)*(alpha + k + 1))

LegendreF = Rat(1) / (Rat(2) ** n) * (binomial(n, k)) ** 2 * (x - Rat(1)) ** k * (x + Rat(1)) ** (n - k)
LegendreP = Sum(LegendreF, (k, 0, n))

for nvar in range(4):
    legendre_poly(nvar, x) == LegendreP.subs(n, nvar).doit().expand()  # True

(LegendreF.subs(n, n + 1) / LegendreF).simplify()  # (n + 1)**2*(x + 1)/(2*(k - n - 1)**2)
(LegendreF.subs(k, k + 1) / LegendreF).simplify()  # (k - n)**2*(x - 1)/((k + 1)**2*(x + 1))

JacobiF = (
    Rat(1)
    / (Rat(2) ** n)
    * binomial(n + alpha, n - k)
    * binomial(n + beta, k)
    * (x - Rat(1)) ** k
    * (x + Rat(1)) ** (n - k)