# 3D Line¶

Line-like geometrical entities.

## Contains¶

LinearEntity3D Line3D Ray3D Segment3D

class diofant.geometry.line3d.Line3D[source]

An infinite 3D line in space.

A line is declared with two distinct points or a point and direction_ratio as defined using keyword $$direction_ratio$$.

Parameters: p1 (Point3D) pt (Point3D) direction_ratio (list)

Examples

>>> from diofant.abc import L
>>> L = Line3D(Point3D(2, 3, 4), Point3D(3, 5, 1))
>>> L
Line3D(Point3D(2, 3, 4), Point3D(3, 5, 1))
>>> L.points
(Point3D(2, 3, 4), Point3D(3, 5, 1))

contains(o)[source]

Return True if o is on this Line, or False otherwise.

Examples

>>> a = (0, 0, 0)
>>> b = (1, 1, 1)
>>> c = (2, 2, 2)
>>> l1 = Line3D(a, b)
>>> l2 = Line3D(b, a)
>>> l1 == l2
False
>>> l1 in l2
True

distance(o)[source]

Finds the shortest distance between a line and a point.

Raises: NotImplementedError is raised if o is not an instance of Point3D

Examples

>>> p1, p2 = Point3D(0, 0, 0), Point3D(1, 1, 1)
>>> s = Line3D(p1, p2)
>>> s.distance(Point3D(-1, 1, 1))
2*sqrt(6)/3
>>> s.distance((-1, 1, 1))
2*sqrt(6)/3

equals(other)[source]

Returns True if self and other are the same mathematical entities.

equation(x='x', y='y', z='z', k='k')[source]

The equation of the line in 3D

Parameters: x (str, optional) – The name to use for the x-axis, default value is ‘x’. y (str, optional) – The name to use for the y-axis, default value is ‘y’. z (str, optional) – The name to use for the x-axis, default value is ‘z’. equation (tuple)

Examples

>>> p1, p2 = Point3D(1, 0, 0), Point3D(5, 3, 0)
>>> l1 = Line3D(p1, p2)
>>> l1.equation()
(x/4 - 1/4, y/3, zoo*z, k)

plot_interval(parameter='t')[source]

The plot interval for the default geometric plot of line. Gives values that will produce a line that is +/- 5 units long (where a unit is the distance between the two points that define the line).

Parameters: parameter (str, optional) – Default value is ‘t’. plot_interval (list (plot interval)) – [parameter, lower_bound, upper_bound]

Examples

>>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 3, 1)
>>> l1 = Line3D(p1, p2)
>>> l1.plot_interval()
[t, -5, 5]

class diofant.geometry.line3d.LinearEntity3D[source]

An base class for all linear entities (line, ray and segment) in a 3-dimensional Euclidean space.

p1
p2
direction_ratio
direction_cosine
points

Notes

This is a base class and is not meant to be instantiated.

angle_between(other)[source]

The angle formed between the two linear entities.

Parameters: self (LinearEntity) other (LinearEntity) angle (angle in radians)

Notes

From the dot product of vectors v1 and v2 it is known that:

dot(v1, v2) = |v1|*|v2|*cos(A)

where A is the angle formed between the two vectors. We can get the directional vectors of the two lines and readily find the angle between the two using the above formula.

Examples

>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(-1, 2, 0)
>>> l1, l2 = Line3D(p1, p2), Line3D(p2, p3)
>>> l1.angle_between(l2)
acos(-sqrt(2)/3)

arbitrary_point(parameter='t')[source]

A parameterized point on the Line.

Parameters: parameter (str, optional) – The name of the parameter which will be used for the parametric point. The default value is ‘t’. When this parameter is 0, the first point used to define the line will be returned, and when it is 1 the second point will be returned. point (Point3D) ValueError – When parameter already appears in the Line’s definition.

Examples

>>> p1, p2 = Point3D(1, 0, 0), Point3D(5, 3, 1)
>>> l1 = Line3D(p1, p2)
>>> l1.arbitrary_point()
Point3D(4*t + 1, 3*t, t)

static are_concurrent(*lines)[source]

Is a sequence of linear entities concurrent?

Two or more linear entities are concurrent if they all intersect at a single point.

Parameters: lines (a sequence of linear entities.) True (if the set of linear entities are concurrent,) False (otherwise.)

Notes

Simply take the first two lines and find their intersection. If there is no intersection, then the first two lines were parallel and had no intersection so concurrency is impossible amongst the whole set. Otherwise, check to see if the intersection point of the first two lines is a member on the rest of the lines. If so, the lines are concurrent.

Examples

>>> p1, p2 = Point3D(0, 0, 0), Point3D(3, 5, 2)
>>> p3, p4 = Point3D(-2, -2, -2), Point3D(0, 2, 1)
>>> l1, l2, l3 = Line3D(p1, p2), Line3D(p1, p3), Line3D(p1, p4)
>>> Line3D.are_concurrent(l1, l2, l3)
True

>>> l4 = Line3D(p2, p3)
>>> Line3D.are_concurrent(l2, l3, l4)
False

contains(other)[source]

Subclasses should implement this method and should return True if other is on the boundaries of self; False if not on the boundaries of self; None if a determination cannot be made.

direction_cosine

The normalized direction ratio of a given line in 3D.

Examples

>>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 3, 1)
>>> l = Line3D(p1, p2)
>>> l.direction_cosine
[sqrt(35)/7, 3*sqrt(35)/35, sqrt(35)/35]
>>> sum(i**2 for i in _)
1

direction_ratio

The direction ratio of a given line in 3D.

Examples

>>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 3, 1)
>>> l = Line3D(p1, p2)
>>> l.direction_ratio
[5, 3, 1]

intersection(o)[source]

The intersection with another geometrical entity.

Parameters: o (Point or LinearEntity3D) intersection (list of geometrical entities)

Examples

>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(7, 7, 7)
>>> l1 = Line3D(p1, p2)
>>> l1.intersection(p3)
[Point3D(7, 7, 7)]

>>> l1 = Line3D(Point3D(4, 19, 12), Point3D(5, 25, 17))
>>> l2 = Line3D(Point3D(-3, -15, -19), direction_ratio=[2, 8, 8])
>>> l1.intersection(l2)
[Point3D(1, 1, -3)]

>>> p6, p7 = Point3D(0, 5, 2), Point3D(2, 6, 3)
>>> s1 = Segment3D(p6, p7)
>>> l1.intersection(s1)
[]

is_parallel(other)[source]

Are two linear entities parallel?

Parameters: self (LinearEntity) other (LinearEntity) True (if self and other are parallel,) False (otherwise.)

Examples

>>> p1, p2 = Point3D(0, 0, 0), Point3D(3, 4, 5)
>>> p3, p4 = Point3D(2, 1, 1), Point3D(8, 9, 11)
>>> l1, l2 = Line3D(p1, p2), Line3D(p3, p4)
>>> Line3D.is_parallel(l1, l2)
True

>>> p5 = Point3D(6, 6, 6)
>>> l3 = Line3D(p3, p5)
>>> Line3D.is_parallel(l1, l3)
False

is_perpendicular(other)[source]

Are two linear entities perpendicular?

Parameters: self (LinearEntity) other (LinearEntity) True (if self and other are perpendicular,) False (otherwise.)

Examples

>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(-1, 2, 0)
>>> l1, l2 = Line3D(p1, p2), Line3D(p2, p3)
>>> l1.is_perpendicular(l2)
False

>>> p4 = Point3D(5, 3, 7)
>>> l3 = Line3D(p1, p4)
>>> l1.is_perpendicular(l3)
False

is_similar(other)[source]

Return True if self and other are contained in the same line.

Examples

>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(2, 2, 2)
>>> l1 = Line3D(p1, p2)
>>> l2 = Line3D(p1, p3)
>>> l1.is_similar(l2)
True

length

The length of the line.

Examples

>>> p1, p2 = Point3D(0, 0, 0), Point3D(3, 5, 1)
>>> l1 = Line3D(p1, p2)
>>> l1.length
oo

p1

The first defining point of a linear entity.

Examples

>>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 3, 1)
>>> l = Line3D(p1, p2)
>>> l.p1
Point3D(0, 0, 0)

p2

The second defining point of a linear entity.

Examples

>>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 3, 1)
>>> l = Line3D(p1, p2)
>>> l.p2
Point3D(5, 3, 1)

parallel_line(p)[source]

Create a new Line parallel to this linear entity which passes through the point $$p$$.

Parameters: p (Point3D) line (Line3D)

Examples

>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(2, 3, 4), Point3D(-2, 2, 0)
>>> l1 = Line3D(p1, p2)
>>> l2 = l1.parallel_line(p3)
>>> p3 in l2
True
>>> l1.is_parallel(l2)
True

perpendicular_line(p)[source]

Create a new Line perpendicular to this linear entity which passes through the point $$p$$.

Parameters: p (Point3D) line (Line3D)

Examples

>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(2, 3, 4), Point3D(-2, 2, 0)
>>> l1 = Line3D(p1, p2)
>>> l2 = l1.perpendicular_line(p3)
>>> p3 in l2
True
>>> l1.is_perpendicular(l2)
True

perpendicular_segment(p)[source]

Create a perpendicular line segment from $$p$$ to this line.

The enpoints of the segment are p and the closest point in the line containing self. (If self is not a line, the point might not be in self.)

Parameters: p (Point3D) segment (Segment3D)

Notes

Returns $$p$$ itself if $$p$$ is on this linear entity.

Examples

>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(0, 2, 0)
>>> l1 = Line3D(p1, p2)
>>> s1 = l1.perpendicular_segment(p3)
>>> l1.is_perpendicular(s1)
True
>>> p3 in s1
True
>>> l1.perpendicular_segment(Point3D(4, 0, 0))
Segment3D(Point3D(4/3, 4/3, 4/3), Point3D(4, 0, 0))

points

The two points used to define this linear entity.

Returns: points (tuple of Points)

Examples

>>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 11, 1)
>>> l1 = Line3D(p1, p2)
>>> l1.points
(Point3D(0, 0, 0), Point3D(5, 11, 1))

projection(o)[source]

Project a point, line, ray, or segment onto this linear entity.

Parameters: other (Point or LinearEntity (Line, Ray, Segment)) projection (Point or LinearEntity (Line, Ray, Segment)) – The return type matches the type of the parameter other. diofant.geometry.exceptions.GeometryError – When method is unable to perform projection.

Notes

A projection involves taking the two points that define the linear entity and projecting those points onto a Line and then reforming the linear entity using these projections. A point P is projected onto a line L by finding the point on L that is closest to P. This point is the intersection of L and the line perpendicular to L that passes through P.

Examples

>>> p1, p2, p3 = Point3D(0, 0, 1), Point3D(1, 1, 2), Point3D(2, 0, 1)
>>> l1 = Line3D(p1, p2)
>>> l1.projection(p3)
Point3D(2/3, 2/3, 5/3)

>>> p4, p5 = Point3D(10, 0, 1), Point3D(12, 1, 3)
>>> s1 = Segment3D(p4, p5)
>>> l1.projection(s1)
[Segment3D(Point3D(10/3, 10/3, 13/3), Point3D(5, 5, 6))]

class diofant.geometry.line3d.Ray3D[source]

A Ray is a semi-line in the space with a source point and a direction.

Parameters: p1 (Point3D) – The source of the Ray p2 (Point or a direction vector) direction_ratio (Determines the direction in which the Ray propagates.)
source
xdirection
ydirection
zdirection

Examples

>>> from diofant.abc import r
>>> r = Ray3D(Point3D(2, 3, 4), Point3D(3, 5, 0))
>>> r
Ray3D(Point3D(2, 3, 4), Point3D(3, 5, 0))
>>> r.points
(Point3D(2, 3, 4), Point3D(3, 5, 0))
>>> r.source
Point3D(2, 3, 4)
>>> r.xdirection
oo
>>> r.ydirection
oo
>>> r.direction_ratio
[1, 2, -4]

contains(o)[source]

Is other GeometryEntity contained in this Ray?

distance(o)[source]

Finds the shortest distance between the ray and a point.

Raises: NotImplementedError is raised if o is not a Point

Examples

>>> p1, p2 = Point3D(0, 0, 0), Point3D(1, 1, 2)
>>> s = Ray3D(p1, p2)
>>> s.distance(Point3D(-1, -1, 2))
sqrt(6)
>>> s.distance((-1, -1, 2))
sqrt(6)

equals(other)[source]

Returns True if self and other are the same mathematical entities.

plot_interval(parameter='t')[source]

The plot interval for the default geometric plot of the Ray. Gives values that will produce a ray that is 10 units long (where a unit is the distance between the two points that define the ray).

Parameters: parameter (str, optional) – Default value is ‘t’. plot_interval (list) – [parameter, lower_bound, upper_bound]

Examples

>>> r = Ray3D(Point3D(0, 0, 0), Point3D(1, 1, 1))
>>> r.plot_interval()
[t, 0, 10]

source

The point from which the ray emanates.

Examples

>>> p1, p2 = Point3D(0, 0, 0), Point3D(4, 1, 5)
>>> r1 = Ray3D(p1, p2)
>>> r1.source
Point3D(0, 0, 0)

xdirection

The x direction of the ray.

Positive infinity if the ray points in the positive x direction, negative infinity if the ray points in the negative x direction, or 0 if the ray is vertical.

Examples

>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(0, -1, 0)
>>> r1, r2 = Ray3D(p1, p2), Ray3D(p1, p3)
>>> r1.xdirection
oo
>>> r2.xdirection
0

ydirection

The y direction of the ray.

Positive infinity if the ray points in the positive y direction, negative infinity if the ray points in the negative y direction, or 0 if the ray is horizontal.

Examples

>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(-1, -1, -1), Point3D(-1, 0, 0)
>>> r1, r2 = Ray3D(p1, p2), Ray3D(p1, p3)
>>> r1.ydirection
-oo
>>> r2.ydirection
0

zdirection

The z direction of the ray.

Positive infinity if the ray points in the positive z direction, negative infinity if the ray points in the negative z direction, or 0 if the ray is horizontal.

Examples

>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(-1, -1, -1), Point3D(-1, 0, 0)
>>> r1, r2 = Ray3D(p1, p2), Ray3D(p1, p3)
>>> r1.ydirection
-oo
>>> r2.ydirection
0
>>> r2.zdirection
0

class diofant.geometry.line3d.Segment3D[source]

A undirected line segment in a 3D space.

Parameters: p1 (Point3D) p2 (Point3D)
length
Type: Expr
midpoint
Type: Point3D

Examples

>>> from diofant.abc import s
>>> Segment3D((1, 0, 0), (1, 1, 1)) # tuples are interpreted as pts
Segment3D(Point3D(1, 0, 0), Point3D(1, 1, 1))
>>> s = Segment3D(Point3D(4, 3, 9), Point3D(1, 1, 7))
>>> s
Segment3D(Point3D(1, 1, 7), Point3D(4, 3, 9))
>>> s.points
(Point3D(1, 1, 7), Point3D(4, 3, 9))
>>> s.length
sqrt(17)
>>> s.midpoint
Point3D(5/2, 2, 8)

contains(other)[source]

Is the other GeometryEntity contained within this Segment?

Examples

>>> p1, p2 = Point3D(0, 1, 1), Point3D(3, 4, 5)
>>> s = Segment3D(p1, p2)
>>> s2 = Segment3D(p2, p1)
>>> s.contains(s2)
True

distance(o)[source]

Finds the shortest distance between a line segment and a point.

Raises: NotImplementedError is raised if o is not a Point3D

Examples

>>> p1, p2 = Point3D(0, 0, 3), Point3D(1, 1, 4)
>>> s = Segment3D(p1, p2)
>>> s.distance(Point3D(10, 15, 12))
sqrt(341)
>>> s.distance((10, 15, 12))
sqrt(341)

length

The length of the line segment.

Examples

>>> p1, p2 = Point3D(0, 0, 0), Point3D(4, 3, 3)
>>> s1 = Segment3D(p1, p2)
>>> s1.length
sqrt(34)

midpoint

The midpoint of the line segment.

Examples

>>> p1, p2 = Point3D(0, 0, 0), Point3D(4, 3, 3)
>>> s1 = Segment3D(p1, p2)
>>> s1.midpoint
Point3D(2, 3/2, 3/2)

plot_interval(parameter='t')[source]

The plot interval for the default geometric plot of the Segment gives values that will produce the full segment in a plot.

Parameters: parameter (str, optional) – Default value is ‘t’. plot_interval (list) – [parameter, lower_bound, upper_bound]

Examples

>>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 3, 0)
>>> s1 = Segment3D(p1, p2)
>>> s1.plot_interval()
[t, 0, 1]