/

# Points¶

class sympy.geometry.point.Point[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 Point
>>> from sympy.abc import x
>>> Point(1, 2)
Point(1, 2)
>>> Point([1, 2])
Point(1, 2)
>>> Point(0, x)
Point(0, x)


Floats are automatically converted to Rational unless the evaluate flag is False:

>>> Point(0.5, 0.25)
Point(1/2, 1/4)
>>> Point(0.5, 0.25, evaluate=False)
Point(0.5, 0.25)


Attributes

 x y length
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.

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
Point(1/2, 3/2)
>>> p1.evalf()
Point(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)
[Point(0, 0)]

is_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 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_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

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)
Point(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
Point(1/2, 3/2)
>>> p1.evalf()
Point(0.5, 1.5)

rotate(angle, pt=None)[source]

Examples

>>> from sympy import Point, pi
>>> t = Point(1, 0)
>>> t.rotate(pi/2)
Point(0, 1)
>>> t.rotate(pi/2, (2, 0))
Point(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 Point
>>> t = Point(1, 1)
>>> t.scale(2)
Point(2, 1)
>>> t.scale(2, 2)
Point(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 Point
>>> t = Point(0, 1)
>>> t.translate(2)
Point(2, 1)
>>> t.translate(2, 2)
Point(2, 3)
>>> t + Point(2, 2)
Point(2, 3)

x[source]

Returns the X coordinate of the Point.

Examples

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

y[source]

Returns the Y coordinate of the Point.

Examples

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


Utils

3D Point