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.
- Raises
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.
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)
See also
sympy.geometry.line.Segment
Connects two Points
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
- Returns
boolean
- Raises
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
- Returns
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.
- Raises
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
See also
-
distance
(other)[source]¶ The Euclidean distance between self and another GeometricEntity.
- Returns
distance : number or symbolic expression.
- Raises
TypeError : if other is not recognized as a GeometricEntity or is a
GeometricEntity for which distance is not defined.
Examples
>>> from sympy.geometry import Point, Line >>> p1, p2 = Point(1, 1), Point(4, 5) >>> l = Line((3, 1), (2, 2)) >>> p1.distance(p2) 5 >>> p1.distance(l) sqrt(2)
The computed distance may be symbolic, too:
>>> from sympy.abc import x, y >>> p3 = Point(x, y) >>> p3.distance((0, 0)) sqrt(x**2 + y**2)
-
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
- 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
(other)[source]¶ The intersection between this point and another GeometryEntity.
- 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)[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
- Returns
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
See also
-
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
- Returns
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
- Returns
midpoint : Point
Examples
>>> from sympy.geometry import Point >>> p1, p2 = Point(1, 1), Point(13, 5) >>> p1.midpoint(p2) Point2D(7, 3)
See also
-
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
- 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
-
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
- Returns
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
- Returns
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
See also
-
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.
- 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.
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)
See also
sympy.geometry.line.Segment
Connects two Points
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 Pointpt
.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
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 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.
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.
- Raises
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
- Returns
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
See also
-
direction_cosine
(point)[source]¶ 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)[source]¶ 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
(other)[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 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
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 Point3D >>> t = Point3D(1, 1, 1) >>> t.scale(2) Point3D(2, 1, 1) >>> t.scale(2, 2) Point3D(2, 2, 1)
See also
-
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.
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)
See also
rotate
,scale
-
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