A point in a ndimensional Euclidean space.
Parameters :  coords : sequence of ncoordinate values. In the special case where n=2 or 3, a Point2D or Point3D will be created as appropriate. 

Raises :  TypeError

See also
Examples
>>> from sympy.geometry import Point
>>> from sympy.abc import x
>>> Point(1, 2, 3)
Point3D(1, 2, 3)
>>> Point([1, 2])
Point2D(1, 2)
>>> Point(0, x)
Point2D(0, x)
Floats are automatically converted to Rational unless the evaluate flag is False:
>>> Point(0.5, 0.25)
Point2D(1/2, 1/4)
>>> Point(0.5, 0.25, evaluate=False)
Point2D(0.5, 0.25)
Attributes
length  
origin: A \(Point\) representing the origin of the  appropriatelydimensioned space. 
The dimension of the ambient space the point is in. I.e., if the point is in R^n, the ambient dimension will be n
The Euclidean distance from self to point p.
Parameters :  p : Point 

Returns :  distance : number or symbolic expression. 
See also
Examples
>>> from sympy.geometry import Point
>>> p1, p2 = Point(1, 1), Point(4, 5)
>>> p1.distance(p2)
5
>>> from sympy.abc import x, y
>>> p3 = Point(x, y)
>>> p3.distance(Point(0, 0))
sqrt(x**2 + y**2)
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 sympy import Point, Rational
>>> p1 = Point(Rational(1, 2), Rational(3, 2))
>>> p1
Point2D(1/2, 3/2)
>>> p1.evalf()
Point2D(0.5, 1.5)
The intersection between this point and another point.
Parameters :  other : Point 

Returns :  intersection : list of Points 
Notes
The return value will either be an empty list if there is no intersection, otherwise it will contain this point.
Examples
>>> from sympy import Point
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, 0)
>>> p1.intersection(p2)
[]
>>> p1.intersection(p3)
[Point2D(0, 0)]
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 :  is_collinear : boolean 
See also
Notes
Slope is preserved everywhere on a line, so the slope between any two points on the line should be the same. Take the first two points, p1 and p2, and create a translated point v1 with p1 as the origin. Now for every other point we create a translated point, vi with p1 also as the origin. Note that these translations preserve slope since everything is consistently translated to a new origin of p1. Since slope is preserved then we have the following equality:
 v1_slope = vi_slope
 v1.y/v1.x = vi.y/vi.x (due to translation)
 v1.y*vi.x = vi.y*v1.x
 v1.y*vi.x  vi.y*v1.x = 0 (*)
Hence, if we have a vi such that the equality in (*) is False then the points are not collinear. We do this test for every point in the list, and if all pass then they are collinear.
Examples
>>> from sympy import Point
>>> from sympy.abc import x
>>> p1, p2 = Point(0, 0), Point(1, 1)
>>> p3, p4, p5 = Point(2, 2), Point(x, x), Point(1, 2)
>>> Point.is_collinear(p1, p2, p3, p4)
True
>>> Point.is_collinear(p1, p2, p3, p5)
False
Returns whether \(p1\) and \(p2\) are scalar multiples of eachother.
Treating a Point as a Line, this returns 0 for the length of a Point.
Examples
>>> from sympy import Point
>>> p = Point(0, 1)
>>> p.length
0
The midpoint between self and point p.
Parameters :  p : Point 

Returns :  midpoint : Point 
See also
Examples
>>> from sympy.geometry import Point
>>> p1, p2 = Point(1, 1), Point(13, 5)
>>> p1.midpoint(p2)
Point2D(7, 3)
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 sympy import Point, Rational
>>> p1 = Point(Rational(1, 2), Rational(3, 2))
>>> p1
Point2D(1/2, 3/2)
>>> p1.evalf()
Point2D(0.5, 1.5)
The Taxicab Distance from self to point p.
Returns the sum of the horizontal and vertical distances to point p.
Parameters :  p : Point 

Returns :  taxicab_distance : The sum of the horizontal and vertical distances to point p. 
See also
sympy.geometry.Point.distance
Examples
>>> from sympy.geometry import Point
>>> p1, p2 = Point(1, 1), Point(4, 5)
>>> p1.taxicab_distance(p2)
7
A point in a 2dimensional Euclidean space.
Parameters :  coords : sequence of 2 coordinate values. 

Raises :  TypeError

See also
Examples
>>> from sympy.geometry import Point2D
>>> from sympy.abc import x
>>> Point2D(1, 2)
Point2D(1, 2)
>>> Point2D([1, 2])
Point2D(1, 2)
>>> Point2D(0, x)
Point2D(0, x)
Floats are automatically converted to Rational unless the evaluate flag is False:
>>> Point2D(0.5, 0.25)
Point2D(1/2, 1/4)
>>> Point2D(0.5, 0.25, evaluate=False)
Point2D(0.5, 0.25)
Attributes
x  
y  
length 
Return a tuple (xmin, ymin, xmax, ymax) representing the bounding rectangle for the geometric figure.
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

See also
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 sympy.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
Rotate angle radians counterclockwise about Point pt.
Examples
>>> from sympy import Point2D, pi
>>> t = Point2D(1, 0)
>>> t.rotate(pi/2)
Point2D(0, 1)
>>> t.rotate(pi/2, (2, 0))
Point2D(2, 1)
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).
Examples
>>> from sympy import Point2D
>>> t = Point2D(1, 1)
>>> t.scale(2)
Point2D(2, 1)
>>> t.scale(2, 2)
Point2D(2, 2)
Return the point after applying the transformation described by the 3x3 Matrix, matrix.
See also
geometry.entity.rotate, geometry.entity.scale, geometry.entity.translate
Shift the Point by adding x and y to the coordinates of the Point.
Examples
>>> from sympy 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)
A point in a 3dimensional Euclidean space.
Parameters :  coords : sequence of 3 coordinate values. 

Raises :  TypeError

Notes
Currently only 2dimensional and 3dimensional points are supported.
Examples
>>> from sympy import Point3D
>>> from sympy.abc import x
>>> Point3D(1, 2, 3)
Point3D(1, 2, 3)
>>> Point3D([1, 2, 3])
Point3D(1, 2, 3)
>>> Point3D(0, x, 3)
Point3D(0, x, 3)
Floats are automatically converted to Rational unless the evaluate flag is False:
>>> Point3D(0.5, 0.25, 2)
Point3D(1/2, 1/4, 2)
>>> Point3D(0.5, 0.25, 3, evaluate=False)
Point3D(0.5, 0.25, 3)
Attributes
x  
y  
z  
length 
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
Examples
>>> from sympy import Point3D, Matrix
>>> from sympy.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
This function tests whether passed points are coplanar or not. It uses the fact that the triple scalar product of three vectors vanishes if the vectors are coplanar. Which means that the volume of the solid described by them will have to be zero for coplanarity.
Parameters :  A set of points 3D points 

Returns :  boolean 
Examples
>>> from sympy import Point3D
>>> p1 = Point3D(1, 2, 2)
>>> p2 = Point3D(2, 7, 2)
>>> p3 = Point3D(0, 0, 2)
>>> p4 = Point3D(1, 1, 2)
>>> Point3D.are_coplanar(p1, p2, p3, p4)
True
>>> p5 = Point3D(0, 1, 3)
>>> Point3D.are_coplanar(p1, p2, p3, p5)
False
Gives the direction cosine between 2 points
Parameters :  p : Point3D 

Returns :  list 
Examples
>>> from sympy import Point3D
>>> p1 = Point3D(1, 2, 3)
>>> p1.direction_cosine(Point3D(2, 3, 5))
[sqrt(6)/6, sqrt(6)/6, sqrt(6)/3]
Gives the direction ratio between 2 points
Parameters :  p : Point3D 

Returns :  list 
Examples
>>> from sympy import Point3D
>>> p1 = Point3D(1, 2, 3)
>>> p1.direction_ratio(Point3D(2, 3, 5))
[1, 1, 2]
The intersection between this point and another point.
Parameters :  other : Point 

Returns :  intersection : list of Points 
Notes
The return value will either be an empty list if there is no intersection, otherwise it will contain this point.
Examples
>>> from sympy import Point3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(0, 0, 0)
>>> p1.intersection(p2)
[]
>>> p1.intersection(p3)
[Point3D(0, 0, 0)]
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
Examples
>>> from sympy import Point3D
>>> t = Point3D(1, 1, 1)
>>> t.scale(2)
Point3D(2, 1, 1)
>>> t.scale(2, 2)
Point3D(2, 2, 1)
Return the point after applying the transformation described by the 4x4 Matrix, matrix.
See also
geometry.entity.rotate, geometry.entity.scale, geometry.entity.translate
Shift the Point by adding x and y to the coordinates of the Point.
See also
rotate, scale
Examples
>>> from sympy import Point3D
>>> t = Point3D(0, 1, 1)
>>> t.translate(2)
Point3D(2, 1, 1)
>>> t.translate(2, 2)
Point3D(2, 3, 1)
>>> t + Point3D(2, 2, 2)
Point3D(2, 3, 3)
Returns the X coordinate of the Point.
Examples
>>> from sympy import Point3D
>>> p = Point3D(0, 1, 3)
>>> p.x
0