Points¶
-
class
sympy.geometry.point.
Point
[source]¶ 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 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 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)
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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 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) [Point(0, 0)]
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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 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
-
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
Returns: is_concyclic : boolean
True if points are concyclic, False otherwise.
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
-
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
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midpoint
(p)[source]¶ 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) 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]¶ Rotate
angle
radians counterclockwise about Pointpt
.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)
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scale
(x=1, y=1, pt=None)[source]¶ Scale the coordinates of the Point by multiplying by
x
andy
after subtractingpt
– default is (0, 0) – and then addingpt
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
.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.
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
¶ Returns the X coordinate of the Point.
Examples
>>> from sympy import Point >>> p = Point(0, 1) >>> p.x 0
-
y
¶ Returns the Y coordinate of the Point.
Examples
>>> from sympy import Point >>> p = Point(0, 1) >>> p.y 1