# 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. TypeError When trying to add or subtract points with different dimensions. When $$intersection$$ is called with object other than a Point.

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)


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.
ambient_dimension[source]

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

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(p2)[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).

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)

intersection(o)[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 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]

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 is_collinear : boolean

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

is_scalar_multiple(p1, p2)[source]

Returns whether $$p1$$ and $$p2$$ are scalar multiples of eachother.

is_zero[source]

True if every coordinate is zero, otherwise False.

length[source]

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).

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)

origin[source]

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

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.

sympy.geometry.Point.distance

Examples

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

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.

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[source]

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

is_concyclic(*points)[source]

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 is_concyclic : boolean True if points are concyclic, False otherwise.

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, pt=None)[source]

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).

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.

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.

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[source]

Returns the X coordinate of the Point.

Examples

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

y[source]

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.

Notes

Currently only 2-dimensional and 3-dimensional 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
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

static are_coplanar(*points)[source]

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 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

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(o)[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).

translate

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.

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.

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[source]

Returns the X coordinate of the Point.

Examples

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

y[source]

Returns the Y coordinate of the Point.

Examples

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

z[source]

Returns the Z coordinate of the Point.

Examples

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


Utils

Lines