Points

class sympy.geometry.point.Point

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.

Raises:

TypeError

When trying to add or subtract points with different 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 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

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)

The Euclidean distance from self to point p.

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

See also

sympy.geometry.line.Segment.length

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)

Return dot product of self with another Point.

equals(other)

Returns whether the coordinates of self and other agree.

evalf(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)

intersection(o)

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_collinear(*args)

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

sympy.geometry.line.Line

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)

Returns whether \(p1\) and \(p2\) are scalar multiples of eachother.

is_zero

True if every coordinate is zero, otherwise False.

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)

The midpoint between self and point p.

Parameters:p : Point Returns:midpoint : Point

See also

sympy.geometry.line.Segment.midpoint

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

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

taxicab_distance(p)

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

class sympy.geometry.point.Point2D

A point in a 2-dimensional Euclidean space.

Parameters:

coords : sequence of 2 coordinate values.

Raises:

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.

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

sympy.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 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)

Rotate angle radians counterclockwise about Point pt.

See also

rotate, scale

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)

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

transform(matrix)

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)

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

See also

rotate, scale

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

A point in a 3-dimensional Euclidean space.

Parameters:

coords : sequence of 3 coordinate values.

Raises:

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)

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

sympy.geometry.line3d.Line3D

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)

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

direction_cosine(point)

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]

direction_ratio(point)

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]

intersection(o)

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(x=1, y=1, z=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

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)

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)

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