Points¶

class
diofant.geometry.point.
Point
[source]¶ A point in a ndimensional Euclidean space.
 Parameters
coords (sequence of ncoordinate values. In the special)
case where n=2 or 3, a Point2D or Point3D will be created
as appropriate.

length
¶

origin
¶ appropriatelydimensioned space.
 Type
A \(Point\) representing the origin of the
 Raises
TypeError – When trying to add or subtract points with different dimensions. When \(intersection\) is called with object other than a Point.
See also
diofant.geometry.line.Segment
Connects two Points
Examples
>>> 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) >>> print(Point(0.5, 0.25, evaluate=False)) Point2D(0.5, 0.25)

property
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)[source]¶ The Euclidean distance from self to point p.
 Parameters
p (Point)
 Returns
distance (number or symbolic expression.)
Examples
>>> p1, p2 = Point(1, 1), Point(4, 5) >>> p1.distance(p2) 5
>>> p3 = Point(x, y) >>> p3.distance(Point(0, 0)) sqrt(x**2 + y**2)

evalf
(dps=15, **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 decimal precision dps.
 Returns
point (Point)
Examples
>>> p1 = Point(Rational(1, 2), Rational(3, 2)) >>> p1 Point2D(1/2, 3/2) >>> print(p1.evalf()) Point2D(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
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, 0) >>> p1.intersection(p2) [] >>> p1.intersection(p3) [Point2D(0, 0)]

is_collinear
()[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)
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.
See also
Examples
>>> 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
(other)[source]¶ Returns whether \(self\) and \(other\) are scalar multiples of each other.

property
is_zero
¶ True if every coordinate is zero, otherwise False.

property
length
Treating a Point as a Line, this returns 0 for the length of a Point.
Examples
>>> p = Point(0, 1) >>> p.length 0

midpoint
(p)[source]¶ The midpoint between self and point p.
 Parameters
p (Point)
 Returns
midpoint (Point)
Examples
>>> p1, p2 = Point(1, 1), Point(13, 5) >>> p1.midpoint(p2) Point2D(7, 3)

n
(dps=15, **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 decimal precision dps.
 Returns
point (Point)
Examples
>>> p1 = Point(Rational(1, 2), Rational(3, 2)) >>> p1 Point2D(1/2, 3/2) >>> print(p1.evalf()) Point2D(0.5, 1.5)

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

class
diofant.geometry.point.
Point2D
[source]¶ A point in a 2dimensional Euclidean space.
 Parameters
coords (sequence of 2 coordinate values.)

x
¶

y
¶

diofant.geometry.Point.
length
¶
 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
diofant.geometry.line.Segment
Connects two Points
Examples
>>> 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) >>> print(Point2D(0.5, 0.25, evaluate=False)) Point2D(0.5, 0.25)

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

is_concyclic
()[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
>>> 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]¶ Rotate
angle
radians counterclockwise about Pointpt
.Examples
>>> 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).See also
Examples
>>> 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
.

translate
(x=0, y=0)[source]¶ Shift the Point by adding x and y to the coordinates of the Point.
Examples
>>> t = Point2D(0, 1) >>> t.translate(2) Point2D(2, 1) >>> t.translate(2, 2) Point2D(2, 3) >>> t + Point2D(2, 2) Point2D(2, 3)

property
x
Returns the X coordinate of the Point.
Examples
>>> p = Point2D(0, 1) >>> p.x 0

property
y
Returns the Y coordinate of the Point.
Examples
>>> p = Point2D(0, 1) >>> p.y 1

class
diofant.geometry.point.
Point3D
[source]¶ A point in a 3dimensional Euclidean space.
 Parameters
coords (sequence of 3 coordinate values.)

x
¶

y
¶

z
¶

diofant.geometry.Point.
length
¶
 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 2dimensional and 3dimensional points are supported.
Examples
>>> 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) >>> print(Point3D(0.5, 0.25, 3, evaluate=False)) Point3D(0.5, 0.25, 3)

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)
See also
Examples
>>> 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
 Returns
boolean
Examples
>>> 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)
 Returns
list
Examples
>>> 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
>>> 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)
 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
>>> 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).See also
Examples
>>> 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
.

translate
(x=0, y=0, z=0)[source]¶ Shift the Point by adding x and y to the coordinates of the Point.
See also
diofant.geometry.entity.GeometryEntity.rotate()
,diofant.geometry.entity.GeometryEntity.scale()
Examples
>>> 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)

property
x
Returns the X coordinate of the Point.
Examples
>>> p = Point3D(0, 1, 3) >>> p.x 0

property
y
Returns the Y coordinate of the Point.
Examples
>>> p = Point3D(0, 1, 2) >>> p.y 1

property
z
Returns the Z coordinate of the Point.
Examples
>>> p = Point3D(0, 1, 1) >>> p.z 1