# Points¶

class sympy.geometry.point.Point[source]

A point in a n-dimensional Euclidean space.

Parameters: coords : sequence of n-coordinate values. In the special case where n=2 or 3, a Point2D or Point3D will be created as appropriate. evaluate : if $$True$$ (default), all floats are turn into exact types. dim : number of coordinates the point should have. If coordinates are unspecified, they are padded with zeros. on_morph : indicates what should happen when the number of coordinates of a point need to be changed by adding or removing zeros. Possible values are $$'warn'$$, $$'error'$$, or $$ignore$$ (default). No warning or error is given when $$*args$$ is empty and $$dim$$ is given. An error is always raised when trying to remove nonzero coordinates. TypeError : When instantiating with anything but a Point or sequence ValueError : when instantiating with a sequence with length < 2 or when trying to reduce dimensions if keyword $$on_morph='error'$$ is set.

See also

sympy.geometry.line.Segment
Connects two Points

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)
>>> Point(dim=4)
Point(0, 0, 0, 0)


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 appropriately-dimensioned space.
static affine_rank(*args)[source]

The affine rank of a set of points is the dimension of the smallest affine space containing all the points. For example, if the points lie on a line (and are not all the same) their affine rank is 1. If the points lie on a plane but not a line, their affine rank is 2. By convention, the empty set has affine rank -1.

ambient_dimension

Number of components this point has.

classmethod are_coplanar(*points)[source]

Return True if there exists a plane in which all the points lie. A trivial True value is returned if $$len(points) < 3$$ or all Points are 2-dimensional.

Parameters: A set of points boolean ValueError : if less than 3 unique points are given

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

canberra_distance(p)[source]

The Canberra Distance from self to point p.

Returns the weighted sum of horizontal and vertical distances to point p.

Parameters: p : Point canberra_distance : The weighted sum of horizontal and vertical distances to point p. The weight used is the sum of absolute values of the coordinates. ValueError when both vectors are zero.

Examples

>>> from sympy.geometry import Point
>>> p1, p2 = Point(1, 1), Point(3, 3)
>>> p1.canberra_distance(p2)
1
>>> p1, p2 = Point(0, 0), Point(3, 3)
>>> p1.canberra_distance(p2)
2

distance(p)[source]

The Euclidean distance from self to point p.

Parameters: p : Point distance : number or symbolic expression.

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)

dot(p)[source]

Return dot product of self with another Point.

equals(other)[source]

Returns whether the coordinates of self and other agree.

evalf(prec=None, **options)[source]

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).

Parameters: prec : int 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)

intersection(other)[source]

The intersection between this point and another GeometryEntity.

Parameters: other : Point 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_collinear(*args)[source]

Returns $$True$$ if there exists a line that contains $$self$$ and $$points$$. Returns $$False$$ otherwise. A trivially True value is returned if no points are given.

Parameters: args : sequence of Points is_collinear : boolean

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

is_concyclic(*args)[source]

Do $$self$$ and the given sequence of points lie in a circle?

Returns True if the set of points are concyclic and False otherwise. A trivial value of True is returned if there are fewer than 2 other points.

Parameters: args : sequence of Points is_concyclic : boolean

Examples

>>> from sympy import Point


Define 4 points that are on the unit circle:

>>> p1, p2, p3, p4 = Point(1, 0), (0, 1), (-1, 0), (0, -1)

>>> p1.is_concyclic() == p1.is_concyclic(p2, p3, p4) == True
True


Define a point not on that circle:

>>> p = Point(1, 1)

>>> p.is_concyclic(p1, p2, p3)
False

is_nonzero

True if any coordinate is nonzero, False if every coordinate is zero, and None if it cannot be determined.

is_scalar_multiple(p)[source]

Returns whether each coordinate of $$self$$ is a scalar multiple of the corresponding coordinate in point p.

is_zero

True if every coordinate is zero, False if any coordinate is not zero, and None if it cannot be determined.

length

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

midpoint(p)[source]

The midpoint between self and point p.

Parameters: p : Point midpoint : Point

Examples

>>> from sympy.geometry import Point
>>> p1, p2 = Point(1, 1), Point(13, 5)
>>> p1.midpoint(p2)
Point2D(7, 3)

n(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).

Parameters: prec : int 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)

origin

A point of all zeros of the same ambient dimension as the current point

orthogonal_direction

Returns a non-zero point that is orthogonal to the line containing $$self$$ and the origin.

Examples

>>> from sympy.geometry import Line, Point
>>> a = Point(1, 2, 3)
>>> a.orthogonal_direction
Point3D(-2, 1, 0)
>>> b = _
>>> Line(b, b.origin).is_perpendicular(Line(a, a.origin))
True

static project(a, b)[source]

Project the point $$a$$ onto the line between the origin and point $$b$$ along the normal direction.

Parameters: a : Point b : Point p : Point

Examples

>>> from sympy.geometry import Line, Point
>>> a = Point(1, 2)
>>> b = Point(2, 5)
>>> z = a.origin
>>> p = Point.project(a, b)
>>> Line(p, a).is_perpendicular(Line(p, b))
True
>>> Point.is_collinear(z, p, b)
True

taxicab_distance(p)[source]

The Taxicab Distance from self to point p.

Returns the sum of the horizontal and vertical distances to point p.

Parameters: p : Point taxicab_distance : The sum of the horizontal and vertical distances to point p.

Examples

>>> from sympy.geometry import Point
>>> p1, p2 = Point(1, 1), Point(4, 5)
>>> p1.taxicab_distance(p2)
7

unit

Return the Point that is in the same direction as $$self$$ and a distance of 1 from the origin

class sympy.geometry.point.Point2D[source]

A point in a 2-dimensional Euclidean space.

Parameters: coords : sequence of 2 coordinate values. TypeError When trying to add or subtract points with different dimensions. When trying to create a point with more than two dimensions. When $$intersection$$ is called with object other than a Point.

See also

sympy.geometry.line.Segment
Connects two Points

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
bounds

Return a tuple (xmin, ymin, xmax, ymax) representing the bounding rectangle for the geometric figure.

rotate(angle, pt=None)[source]

Rotate angle radians counterclockwise about Point pt.

See also

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(x=1, y=1, pt=None)[source]

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 Point2D
>>> t = Point2D(1, 1)
>>> t.scale(2)
Point2D(2, 1)
>>> t.scale(2, 2)
Point2D(2, 2)

transform(matrix)[source]

Return the point after applying the transformation described by the 3x3 Matrix, matrix.

See also

geometry.entity.rotate, geometry.entity.scale, geometry.entity.translate

translate(x=0, y=0)[source]

Shift the Point by adding x and y to the coordinates of the Point.

See also

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)

x

Returns the X coordinate of the Point.

Examples

>>> from sympy import Point2D
>>> p = Point2D(0, 1)
>>> p.x
0

y

Returns the Y coordinate of the Point.

Examples

>>> from sympy import Point2D
>>> p = Point2D(0, 1)
>>> p.y
1

class sympy.geometry.point.Point3D[source]

A point in a 3-dimensional Euclidean space.

Parameters: coords : sequence of 3 coordinate values. TypeError When trying to add or subtract points with different dimensions. When $$intersection$$ is called with object other than a Point.

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
static are_collinear(*points)[source]

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 are_collinear : boolean

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

direction_cosine(point)[source]

Gives the direction cosine between 2 points

Parameters: p : Point3D 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]

direction_ratio(point)[source]

Gives the direction ratio between 2 points

Parameters: p : Point3D list

Examples

>>> from sympy import Point3D
>>> p1 = Point3D(1, 2, 3)
>>> p1.direction_ratio(Point3D(2, 3, 5))
[1, 1, 2]

intersection(other)[source]

The intersection between this point and another point.

Parameters: other : Point 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(x=1, y=1, z=1, pt=None)[source]

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 Point3D
>>> t = Point3D(1, 1, 1)
>>> t.scale(2)
Point3D(2, 1, 1)
>>> t.scale(2, 2)
Point3D(2, 2, 1)

transform(matrix)[source]

Return the point after applying the transformation described by the 4x4 Matrix, matrix.

See also

geometry.entity.rotate, geometry.entity.scale, geometry.entity.translate

translate(x=0, y=0, z=0)[source]

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)

x

Returns the X coordinate of the Point.

Examples

>>> from sympy import Point3D
>>> p = Point3D(0, 1, 3)
>>> p.x
0

y

Returns the Y coordinate of the Point.

Examples

>>> from sympy import Point3D
>>> p = Point3D(0, 1, 2)
>>> p.y
1

z

Returns the Z coordinate of the Point.

Examples

>>> from sympy import Point3D
>>> p = Point3D(0, 1, 1)
>>> p.z
1