def rotate(self, angle, pt=None):
"""Rotate ``angle`` radians counterclockwise about Point ``pt``.
See Also
========
rotate, scale
Examples
========
>>> from diofant import Point2D, pi
>>> t = Point2D(1, 0)
>>> t.rotate(pi/2)
Point2D(0, 1)
>>> t.rotate(pi/2, (2, 0))
Point2D(2, -1)
"""
from diofant import cos, sin, Point
c = cos(angle)
s = sin(angle)
rv = self
if pt is not None:
pt = Point(pt)
rv -= pt
x, y = rv.args
rv = Point(c*x - s*y, s*x + c*y)
if pt is not None:
rv += pt
return rv
def scale(self, x=1, y=1, pt=None):
"""Scale the coordinates of the Point by multiplying by
``x`` and ``y`` after subtracting ``pt`` -- default is (0, 0) --
and then adding ``pt`` back again (i.e. ``pt`` is the point of
reference for the scaling).
See Also
========
rotate, translate
Examples
========
>>> from diofant import Point2D
>>> t = Point2D(1, 1)
>>> t.scale(2)
Point2D(2, 1)
>>> t.scale(2, 2)
Point2D(2, 2)
"""
if pt:
pt = Point(pt)
return self.translate(*(-pt).args).scale(x, y).translate(*pt.args)
return Point(self.x*x, self.y*y)
def are_collinear(*points):
"""Is a sequence of points collinear?
Test whether or not a set of points are collinear. Returns True if
the set of points are collinear, or False otherwise.
Parameters
==========
points : sequence of Point
Returns
=======
are_collinear : boolean
See Also
========
diofant.geometry.line3d.Line3D
Examples
========
>>> from diofant import Point3D, Matrix
>>> from diofant.abc import x
>>> p1, p2 = Point3D(0, 0, 0), Point3D(1, 1, 1)
>>> p3, p4, p5 = Point3D(2, 2, 2), Point3D(x, x, x), Point3D(1, 2, 6)
>>> Point3D.are_collinear(p1, p2, p3, p4)
True
>>> Point3D.are_collinear(p1, p2, p3, p5)
False
"""
return Point.is_collinear(*points)
def midpoint(self, p):
"""The midpoint between self and point p.
Parameters
==========
p : Point
Returns
=======
midpoint : Point
See Also
========
diofant.geometry.line.Segment.midpoint
Examples
========
>>> from diofant.geometry import Point
>>> p1, p2 = Point(1, 1), Point(13, 5)
>>> p1.midpoint(p2)
Point2D(7, 3)
"""
return Point([simplify((a + b)*S.Half) for a, b in zip(self.args, p.args)])
def distance(self, p):
"""The Euclidean distance from self to point p.
Parameters
==========
p : Point
Returns
=======
distance : number or symbolic expression.
See Also
========
diofant.geometry.line.Segment.length
Examples
========
>>> from diofant.geometry import Point
>>> p1, p2 = Point(1, 1), Point(4, 5)
>>> p1.distance(p2)
5
>>> from diofant.abc import x, y
>>> p3 = Point(x, y)
>>> p3.distance(Point(0, 0))
sqrt(x**2 + y**2)
"""
p = Point(p)
return sqrt(sum((a - b)**2 for a, b in zip(self.args, p.args)))
def __add__(self, other):
"""Add other to self by incrementing self's coordinates by those of other.
See Also
========
diofant.geometry.entity.GeometryEntity.translate
"""
if iterable(other) and len(other) == len(self):
return Point([simplify(a + b) for a, b in zip(self, other)])
else:
raise ValueError(
"Points must have the same number of dimensions")
def transform(self, matrix):
"""Return the point after applying the transformation described
by the 3x3 Matrix, ``matrix``.
See Also
========
diofant.geometry.entity.GeometryEntity.rotate
diofant.geometry.entity.GeometryEntity.scale
diofant.geometry.entity.GeometryEntity.translate
"""
try:
col, row = matrix.shape
valid_matrix = matrix.is_square and col == 3
except AttributeError:
# We hit this block if matrix argument is not actually a Matrix.
valid_matrix = False
if not valid_matrix:
raise ValueError("The argument to the transform function must be "
+ "a 3x3 matrix")
x, y = self.args
return Point(*(Matrix(1, 3, [x, y, 1])*matrix).tolist()[0][:2])
def translate(self, x=0, y=0):
"""Shift the Point by adding x and y to the coordinates of the Point.
See Also
========
rotate, scale
Examples
========
>>> from diofant import Point2D
>>> t = Point2D(0, 1)
>>> t.translate(2)
Point2D(2, 1)
>>> t.translate(2, 2)
Point2D(2, 3)
>>> t + Point2D(2, 2)
Point2D(2, 3)
"""
return Point(self.x + x, self.y + y)
def evalf(self, prec=None, **options):
"""Evaluate the coordinates of the point.
This method will, where possible, create and return a new Point
where the coordinates are evaluated as floating point numbers to
the precision indicated (default=15).
Returns
=======
point : Point
Examples
========
>>> from diofant import Point, Rational
>>> p1 = Point(Rational(1, 2), Rational(3, 2))
>>> p1
Point2D(1/2, 3/2)
>>> print(p1.evalf())
Point2D(0.5, 1.5)
"""
coords = [x.evalf(prec, **options) for x in self.args]
return Point(*coords, evaluate=False)
def is_concyclic(*points):
"""Is a sequence of points concyclic?
Test whether or not a sequence of points are concyclic (i.e., they lie
on a circle).
Parameters
==========
points : sequence of Points
Returns
=======
is_concyclic : boolean
True if points are concyclic, False otherwise.
See Also
========
diofant.geometry.ellipse.Circle
Notes
=====
No points are not considered to be concyclic. One or two points
are definitely concyclic and three points are conyclic iff they
are not collinear.
For more than three points, create a circle from the first three
points. If the circle cannot be created (i.e., they are collinear)
then all of the points cannot be concyclic. If the circle is created
successfully then simply check the remaining points for containment
in the circle.
Examples
========
>>> from diofant.geometry import Point
>>> p1, p2 = Point(-1, 0), Point(1, 0)
>>> p3, p4 = Point(0, 1), Point(-1, 2)
>>> Point.is_concyclic(p1, p2, p3)
True
>>> Point.is_concyclic(p1, p2, p3, p4)
False
"""
if len(points) == 0:
return False
if len(points) <= 2:
return True
points = [Point(p) for p in points]
if len(points) == 3:
return (not Point.is_collinear(*points))
try:
from .ellipse import Circle
c = Circle(points[0], points[1], points[2])
for point in points[3:]:
if point not in c:
return False
return True
except GeometryError:
# Circle could not be created, because of collinearity of the
# three points passed in, hence they are not concyclic.
return False
def __abs__(self):
"""Returns the distance between this point and the origin."""
origin = Point([0]*len(self))
return Point.distance(origin, self)
def __neg__(self):
"""Negate the point."""
return Point([-x for x in self.args])
def __div__(self, divisor):
"""Divide point's coordinates by a factor."""
divisor = sympify(divisor)
return Point([simplify(x/divisor) for x in self.args])
def __mul__(self, factor):
"""Multiply point's coordinates by a factor."""
factor = sympify(factor)
return Point([simplify(x*factor) for x in self.args])
def dot(self, p2):
"""Return dot product of self with another Point."""
p2 = Point(p2)
return Add(*[a*b for a, b in zip(self, p2)])
def origin(self):
"""A point of all zeros of the same ambient dimension
as the current point"""
return Point([0]*len(self))