# Polygons¶

class sympy.geometry.polygon.Polygon(*args, n=0, **kwargs)[source]

A two-dimensional polygon.

A simple polygon in space. Can be constructed from a sequence of points or from a center, radius, number of sides and rotation angle.

Parameters:

vertices

A sequence of points.

n : int, optional

If $$> 0$$, an n-sided RegularPolygon is created. Default value is $$0$$.

Raises:

GeometryError

If all parameters are not Points.

Notes

Polygons are treated as closed paths rather than 2D areas so some calculations can be be negative or positive (e.g., area) based on the orientation of the points.

Any consecutive identical points are reduced to a single point and any points collinear and between two points will be removed unless they are needed to define an explicit intersection (see examples).

A Triangle, Segment or Point will be returned when there are 3 or fewer points provided.

Examples

>>> from sympy import Polygon, pi
>>> p1, p2, p3, p4, p5 = [(0, 0), (1, 0), (5, 1), (0, 1), (3, 0)]
>>> Polygon(p1, p2, p3, p4)
Polygon(Point2D(0, 0), Point2D(1, 0), Point2D(5, 1), Point2D(0, 1))
>>> Polygon(p1, p2)
Segment2D(Point2D(0, 0), Point2D(1, 0))
>>> Polygon(p1, p2, p5)
Segment2D(Point2D(0, 0), Point2D(3, 0))


The area of a polygon is calculated as positive when vertices are traversed in a ccw direction. When the sides of a polygon cross the area will have positive and negative contributions. The following defines a Z shape where the bottom right connects back to the top left.

>>> Polygon((0, 2), (2, 2), (0, 0), (2, 0)).area
0


When the keyword $$n$$ is used to define the number of sides of the Polygon then a RegularPolygon is created and the other arguments are interpreted as center, radius and rotation. The unrotated RegularPolygon will always have a vertex at Point(r, 0) where $$r$$ is the radius of the circle that circumscribes the RegularPolygon. Its method $$spin$$ can be used to increment that angle.

>>> p = Polygon((0,0), 1, n=3)
>>> p
RegularPolygon(Point2D(0, 0), 1, 3, 0)
>>> p.vertices[0]
Point2D(1, 0)
>>> p.args[0]
Point2D(0, 0)
>>> p.spin(pi/2)
>>> p.vertices[0]
Point2D(0, 1)


Attributes

 area angles perimeter vertices centroid sides
property angles

The internal angle at each vertex.

Returns:

angles : dict

A dictionary where each key is a vertex and each value is the internal angle at that vertex. The vertices are represented as Points.

Examples

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.angles[p1]
pi/2
>>> poly.angles[p2]
acos(-4*sqrt(17)/17)

arbitrary_point(parameter='t')[source]

A parameterized point on the polygon.

The parameter, varying from 0 to 1, assigns points to the position on the perimeter that is that fraction of the total perimeter. So the point evaluated at t=1/2 would return the point from the first vertex that is 1/2 way around the polygon.

Parameters:

parameter : str, optional

Default value is ‘t’.

Returns:

arbitrary_point : Point

Raises:

ValueError

When $$parameter$$ already appears in the Polygon’s definition.

Examples

>>> from sympy import Polygon, Symbol
>>> t = Symbol('t', real=True)
>>> tri = Polygon((0, 0), (1, 0), (1, 1))
>>> p = tri.arbitrary_point('t')
>>> perimeter = tri.perimeter
>>> s1, s2 = [s.length for s in tri.sides[:2]]
>>> p.subs(t, (s1 + s2/2)/perimeter)
Point2D(1, 1/2)

property area

The area of the polygon.

Notes

The area calculation can be positive or negative based on the orientation of the points. If any side of the polygon crosses any other side, there will be areas having opposite signs.

Examples

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.area
3


In the Z shaped polygon (with the lower right connecting back to the upper left) the areas cancel out:

>>> Z = Polygon((0, 1), (1, 1), (0, 0), (1, 0))
>>> Z.area
0


In the M shaped polygon, areas do not cancel because no side crosses any other (though there is a point of contact).

>>> M = Polygon((0, 0), (0, 1), (2, 0), (3, 1), (3, 0))
>>> M.area
-3/2

bisectors(prec=None)[source]

Returns angle bisectors of a polygon. If prec is given then approximate the point defining the ray to that precision.

The distance between the points defining the bisector ray is 1.

Examples

>>> from sympy import Polygon, Point
>>> p = Polygon(Point(0, 0), Point(2, 0), Point(1, 1), Point(0, 3))
>>> p.bisectors(2)
{Point2D(0, 0): Ray2D(Point2D(0, 0), Point2D(0.71, 0.71)),
Point2D(0, 3): Ray2D(Point2D(0, 3), Point2D(0.23, 2.0)),
Point2D(1, 1): Ray2D(Point2D(1, 1), Point2D(0.19, 0.42)),
Point2D(2, 0): Ray2D(Point2D(2, 0), Point2D(1.1, 0.38))}

property bounds

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

property centroid

The centroid of the polygon.

Returns:

centroid : Point

Examples

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.centroid
Point2D(31/18, 11/18)

cut_section(line)[source]

Returns a tuple of two polygon segments that lie above and below the intersecting line respectively.

Parameters:

line: Line object of geometry module

line which cuts the Polygon. The part of the Polygon that lies above and below this line is returned.

Returns:

upper_polygon, lower_polygon: Polygon objects or None

upper_polygon is the polygon that lies above the given line. lower_polygon is the polygon that lies below the given line. upper_polygon and lower polygon are None when no polygon exists above the line or below the line.

Raises:

ValueError: When the line does not intersect the polygon

Examples

>>> from sympy import Polygon, Line
>>> a, b = 20, 10
>>> p1, p2, p3, p4 = [(0, b), (0, 0), (a, 0), (a, b)]
>>> rectangle = Polygon(p1, p2, p3, p4)
>>> t = rectangle.cut_section(Line((0, 5), slope=0))
>>> t
(Polygon(Point2D(0, 10), Point2D(0, 5), Point2D(20, 5), Point2D(20, 10)),
Polygon(Point2D(0, 5), Point2D(0, 0), Point2D(20, 0), Point2D(20, 5)))
>>> upper_segment, lower_segment = t
>>> upper_segment.area
100
>>> upper_segment.centroid
Point2D(10, 15/2)
>>> lower_segment.centroid
Point2D(10, 5/2)


References

distance(o)[source]

Returns the shortest distance between self and o.

If o is a point, then self does not need to be convex. If o is another polygon self and o must be convex.

Examples

>>> from sympy import Point, Polygon, RegularPolygon
>>> p1, p2 = map(Point, [(0, 0), (7, 5)])
>>> poly = Polygon(*RegularPolygon(p1, 1, 3).vertices)
>>> poly.distance(p2)
sqrt(61)

encloses_point(p)[source]

Return True if p is enclosed by (is inside of) self.

Parameters:

p : Point

Returns:

encloses_point : True, False or None

Notes

Being on the border of self is considered False.

Examples

>>> from sympy import Polygon, Point
>>> p = Polygon((0, 0), (4, 0), (4, 4))
>>> p.encloses_point(Point(2, 1))
True
>>> p.encloses_point(Point(2, 2))
False
>>> p.encloses_point(Point(5, 5))
False


References

first_moment_of_area(point=None)[source]

Returns the first moment of area of a two-dimensional polygon with respect to a certain point of interest.

First moment of area is a measure of the distribution of the area of a polygon in relation to an axis. The first moment of area of the entire polygon about its own centroid is always zero. Therefore, here it is calculated for an area, above or below a certain point of interest, that makes up a smaller portion of the polygon. This area is bounded by the point of interest and the extreme end (top or bottom) of the polygon. The first moment for this area is is then determined about the centroidal axis of the initial polygon.

Parameters:

point: Point, two-tuple of sympifyable objects, or None (default=None)

point is the point above or below which the area of interest lies If point=None then the centroid acts as the point of interest.

Returns:

Q_x, Q_y: number or SymPy expressions

Q_x is the first moment of area about the x-axis Q_y is the first moment of area about the y-axis A negative sign indicates that the section modulus is determined for a section below (or left of) the centroidal axis

Examples

>>> from sympy import Point, Polygon
>>> a, b = 50, 10
>>> p1, p2, p3, p4 = [(0, b), (0, 0), (a, 0), (a, b)]
>>> p = Polygon(p1, p2, p3, p4)
>>> p.first_moment_of_area()
(625, 3125)
>>> p.first_moment_of_area(point=Point(30, 7))
(525, 3000)


References

intersection(o)[source]

The intersection of polygon and geometry entity.

The intersection may be empty and can contain individual Points and complete Line Segments.

Parameters:

other: GeometryEntity

Returns:

intersection : list

The list of Segments and Points

Examples

>>> from sympy import Point, Polygon, Line
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly1 = Polygon(p1, p2, p3, p4)
>>> p5, p6, p7 = map(Point, [(3, 2), (1, -1), (0, 2)])
>>> poly2 = Polygon(p5, p6, p7)
>>> poly1.intersection(poly2)
[Point2D(1/3, 1), Point2D(2/3, 0), Point2D(9/5, 1/5), Point2D(7/3, 1)]
>>> poly1.intersection(Line(p1, p2))
[Segment2D(Point2D(0, 0), Point2D(1, 0))]
>>> poly1.intersection(p1)
[Point2D(0, 0)]

is_convex()[source]

Is the polygon convex?

A polygon is convex if all its interior angles are less than 180 degrees and there are no intersections between sides.

Returns:

is_convex : boolean

True if this polygon is convex, False otherwise.

Examples

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.is_convex()
True

property perimeter

The perimeter of the polygon.

Returns:

perimeter : number or Basic instance

Examples

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.perimeter
sqrt(17) + 7

plot_interval(parameter='t')[source]

The plot interval for the default geometric plot of the polygon.

Parameters:

parameter : str, optional

Default value is ‘t’.

Returns:

plot_interval : list (plot interval)

[parameter, lower_bound, upper_bound]

Examples

>>> from sympy import Polygon
>>> p = Polygon((0, 0), (1, 0), (1, 1))
>>> p.plot_interval()
[t, 0, 1]

polar_second_moment_of_area()[source]

Returns the polar modulus of a two-dimensional polygon

It is a constituent of the second moment of area, linked through the perpendicular axis theorem. While the planar second moment of area describes an object’s resistance to deflection (bending) when subjected to a force applied to a plane parallel to the central axis, the polar second moment of area describes an object’s resistance to deflection when subjected to a moment applied in a plane perpendicular to the object’s central axis (i.e. parallel to the cross-section)

Examples

>>> from sympy import Polygon, symbols
>>> a, b = symbols('a, b')
>>> rectangle = Polygon((0, 0), (a, 0), (a, b), (0, b))
>>> rectangle.polar_second_moment_of_area()
a**3*b/12 + a*b**3/12


References

second_moment_of_area(point=None)[source]

Returns the second moment and product moment of area of a two dimensional polygon.

Parameters:

point : Point, two-tuple of sympifyable objects, or None(default=None)

point is the point about which second moment of area is to be found. If “point=None” it will be calculated about the axis passing through the centroid of the polygon.

Returns:

I_xx, I_yy, I_xy : number or SymPy expression

I_xx, I_yy are second moment of area of a two dimensional polygon. I_xy is product moment of area of a two dimensional polygon.

Examples

>>> from sympy import Polygon, symbols
>>> a, b = symbols('a, b')
>>> p1, p2, p3, p4, p5 = [(0, 0), (a, 0), (a, b), (0, b), (a/3, b/3)]
>>> rectangle = Polygon(p1, p2, p3, p4)
>>> rectangle.second_moment_of_area()
(a*b**3/12, a**3*b/12, 0)
>>> rectangle.second_moment_of_area(p5)
(a*b**3/9, a**3*b/9, a**2*b**2/36)


References

section_modulus(point=None)[source]

Returns a tuple with the section modulus of a two-dimensional polygon.

Section modulus is a geometric property of a polygon defined as the ratio of second moment of area to the distance of the extreme end of the polygon from the centroidal axis.

Parameters:

point : Point, two-tuple of sympifyable objects, or None(default=None)

point is the point at which section modulus is to be found. If “point=None” it will be calculated for the point farthest from the centroidal axis of the polygon.

Returns:

S_x, S_y: numbers or SymPy expressions

S_x is the section modulus with respect to the x-axis S_y is the section modulus with respect to the y-axis A negative sign indicates that the section modulus is determined for a point below the centroidal axis

Examples

>>> from sympy import symbols, Polygon, Point
>>> a, b = symbols('a, b', positive=True)
>>> rectangle = Polygon((0, 0), (a, 0), (a, b), (0, b))
>>> rectangle.section_modulus()
(a*b**2/6, a**2*b/6)
>>> rectangle.section_modulus(Point(a/4, b/4))
(-a*b**2/3, -a**2*b/3)


References

property sides

The directed line segments that form the sides of the polygon.

Returns:

sides : list of sides

Each side is a directed Segment.

Examples

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.sides
[Segment2D(Point2D(0, 0), Point2D(1, 0)),
Segment2D(Point2D(1, 0), Point2D(5, 1)),
Segment2D(Point2D(5, 1), Point2D(0, 1)), Segment2D(Point2D(0, 1), Point2D(0, 0))]

property vertices

The vertices of the polygon.

Returns:

vertices : list of Points

Notes

When iterating over the vertices, it is more efficient to index self rather than to request the vertices and index them. Only use the vertices when you want to process all of them at once. This is even more important with RegularPolygons that calculate each vertex.

Examples

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.vertices
[Point2D(0, 0), Point2D(1, 0), Point2D(5, 1), Point2D(0, 1)]
>>> poly.vertices[0]
Point2D(0, 0)

class sympy.geometry.polygon.RegularPolygon(c, r, n, rot=0, **kwargs)[source]

A regular polygon.

Such a polygon has all internal angles equal and all sides the same length.

Parameters:

center : Point

radius : number or Basic instance

The distance from the center to a vertex

n : int

The number of sides

Raises:

GeometryError

If the $$center$$ is not a Point, or the $$radius$$ is not a number or Basic instance, or the number of sides, $$n$$, is less than three.

Notes

A RegularPolygon can be instantiated with Polygon with the kwarg n.

Regular polygons are instantiated with a center, radius, number of sides and a rotation angle. Whereas the arguments of a Polygon are vertices, the vertices of the RegularPolygon must be obtained with the vertices method.

Examples

>>> from sympy import RegularPolygon, Point
>>> r = RegularPolygon(Point(0, 0), 5, 3)
>>> r
RegularPolygon(Point2D(0, 0), 5, 3, 0)
>>> r.vertices[0]
Point2D(5, 0)


Attributes

 vertices center radius rotation apothem interior_angle exterior_angle circumcircle incircle angles
property angles

Returns a dictionary with keys, the vertices of the Polygon, and values, the interior angle at each vertex.

Examples

>>> from sympy import RegularPolygon, Point
>>> r = RegularPolygon(Point(0, 0), 5, 3)
>>> r.angles
{Point2D(-5/2, -5*sqrt(3)/2): pi/3,
Point2D(-5/2, 5*sqrt(3)/2): pi/3,
Point2D(5, 0): pi/3}

property apothem

Returns:

apothem : number or instance of Basic

Examples

>>> from sympy import Symbol
>>> from sympy import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), radius, 4)
>>> rp.apothem
sqrt(2)*r/2

property area

Returns the area.

Examples

>>> from sympy import RegularPolygon
>>> square = RegularPolygon((0, 0), 1, 4)
>>> square.area
2
>>> _ == square.length**2
True

property args

Returns the center point, the radius, the number of sides, and the orientation angle.

Examples

>>> from sympy import RegularPolygon, Point
>>> r = RegularPolygon(Point(0, 0), 5, 3)
>>> r.args
(Point2D(0, 0), 5, 3, 0)

property center

The center of the RegularPolygon

This is also the center of the circumscribing circle.

Returns:

center : Point

Examples

>>> from sympy import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 5, 4)
>>> rp.center
Point2D(0, 0)

property centroid

The center of the RegularPolygon

This is also the center of the circumscribing circle.

Returns:

center : Point

Examples

>>> from sympy import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 5, 4)
>>> rp.center
Point2D(0, 0)

property circumcenter

Alias for center.

Examples

>>> from sympy import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 5, 4)
>>> rp.circumcenter
Point2D(0, 0)

property circumcircle

The circumcircle of the RegularPolygon.

Returns:

circumcircle : Circle

Examples

>>> from sympy import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 4, 8)
>>> rp.circumcircle
Circle(Point2D(0, 0), 4)


Examples

>>> from sympy import Symbol
>>> from sympy import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), radius, 4)
r

encloses_point(p)[source]

Return True if p is enclosed by (is inside of) self.

Parameters:

p : Point

Returns:

encloses_point : True, False or None

Notes

Being on the border of self is considered False.

The general Polygon.encloses_point method is called only if a point is not within or beyond the incircle or circumcircle, respectively.

Examples

>>> from sympy import RegularPolygon, S, Point, Symbol
>>> p = RegularPolygon((0, 0), 3, 4)
>>> p.encloses_point(Point(0, 0))
True
>>> p.encloses_point(Point((r + R)/2, 0))
True
>>> p.encloses_point(Point(R/2, R/2 + (R - r)/10))
False
>>> t = Symbol('t', real=True)
>>> p.encloses_point(p.arbitrary_point().subs(t, S.Half))
False
>>> p.encloses_point(Point(5, 5))
False

property exterior_angle

Measure of the exterior angles.

Returns:

exterior_angle : number

Examples

>>> from sympy import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 4, 8)
>>> rp.exterior_angle
pi/4

property incircle

The incircle of the RegularPolygon.

Returns:

incircle : Circle

Examples

>>> from sympy import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 4, 7)
>>> rp.incircle
Circle(Point2D(0, 0), 4*cos(pi/7))


Alias for apothem.

Examples

>>> from sympy import Symbol
>>> from sympy import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), radius, 4)
sqrt(2)*r/2

property interior_angle

Measure of the interior angles.

Returns:

interior_angle : number

Examples

>>> from sympy import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 4, 8)
>>> rp.interior_angle
3*pi/4

property length

Returns the length of the sides.

The half-length of the side and the apothem form two legs of a right triangle whose hypotenuse is the radius of the regular polygon.

Examples

>>> from sympy import RegularPolygon
>>> from sympy import sqrt
>>> s = square_in_unit_circle = RegularPolygon((0, 0), 1, 4)
>>> s.length
sqrt(2)
>>> sqrt((_/2)**2 + s.apothem**2) == s.radius
True


This is also the radius of the circumscribing circle.

Returns:

radius : number or instance of Basic

Examples

>>> from sympy import Symbol
>>> from sympy import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), radius, 4)
r

reflect(line)[source]

Override GeometryEntity.reflect since this is not made of only points.

Examples

>>> from sympy import RegularPolygon, Line

>>> RegularPolygon((0, 0), 1, 4).reflect(Line((0, 1), slope=-2))
RegularPolygon(Point2D(4/5, 2/5), -1, 4, atan(4/3))

rotate(angle, pt=None)[source]

Override GeometryEntity.rotate to first rotate the RegularPolygon about its center.

>>> from sympy import Point, RegularPolygon, pi
>>> t = RegularPolygon(Point(1, 0), 1, 3)
>>> t.vertices[0] # vertex on x-axis
Point2D(2, 0)
>>> t.rotate(pi/2).vertices[0] # vertex on y axis now
Point2D(0, 2)


rotation

spin

Rotates a RegularPolygon in place

property rotation

CCW angle by which the RegularPolygon is rotated

Returns:

rotation : number or instance of Basic

Examples

>>> from sympy import pi
>>> from sympy.abc import a
>>> from sympy import RegularPolygon, Point
>>> RegularPolygon(Point(0, 0), 3, 4, pi/4).rotation
pi/4


Numerical rotation angles are made canonical:

>>> RegularPolygon(Point(0, 0), 3, 4, a).rotation
a
>>> RegularPolygon(Point(0, 0), 3, 4, pi).rotation
0

scale(x=1, y=1, pt=None)[source]

Override GeometryEntity.scale since it is the radius that must be scaled (if x == y) or else a new Polygon must be returned.

>>> from sympy import RegularPolygon


Symmetric scaling returns a RegularPolygon:

>>> RegularPolygon((0, 0), 1, 4).scale(2, 2)
RegularPolygon(Point2D(0, 0), 2, 4, 0)


Asymmetric scaling returns a kite as a Polygon:

>>> RegularPolygon((0, 0), 1, 4).scale(2, 1)
Polygon(Point2D(2, 0), Point2D(0, 1), Point2D(-2, 0), Point2D(0, -1))

spin(angle)[source]

Increment in place the virtual Polygon’s rotation by ccw angle.

>>> from sympy import Polygon, Point, pi
>>> r = Polygon(Point(0,0), 1, n=3)
>>> r.vertices[0]
Point2D(1, 0)
>>> r.spin(pi/6)
>>> r.vertices[0]
Point2D(sqrt(3)/2, 1/2)


rotation

rotate

Creates a copy of the RegularPolygon rotated about a Point

property vertices

The vertices of the RegularPolygon.

Returns:

vertices : list

Each vertex is a Point.

Examples

>>> from sympy import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 5, 4)
>>> rp.vertices
[Point2D(5, 0), Point2D(0, 5), Point2D(-5, 0), Point2D(0, -5)]

class sympy.geometry.polygon.Triangle(*args, **kwargs)[source]

A polygon with three vertices and three sides.

Parameters:

points : sequence of Points

keyword: asa, sas, or sss to specify sides/angles of the triangle

Raises:

GeometryError

If the number of vertices is not equal to three, or one of the vertices is not a Point, or a valid keyword is not given.

Examples

>>> from sympy import Triangle, Point
>>> Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
Triangle(Point2D(0, 0), Point2D(4, 0), Point2D(4, 3))


Keywords sss, sas, or asa can be used to give the desired side lengths (in order) and interior angles (in degrees) that define the triangle:

>>> Triangle(sss=(3, 4, 5))
Triangle(Point2D(0, 0), Point2D(3, 0), Point2D(3, 4))
>>> Triangle(asa=(30, 1, 30))
Triangle(Point2D(0, 0), Point2D(1, 0), Point2D(1/2, sqrt(3)/6))
>>> Triangle(sas=(1, 45, 2))
Triangle(Point2D(0, 0), Point2D(2, 0), Point2D(sqrt(2)/2, sqrt(2)/2))


Attributes

property altitudes

The altitudes of the triangle.

An altitude of a triangle is a segment through a vertex, perpendicular to the opposite side, with length being the height of the vertex measured from the line containing the side.

Returns:

altitudes : dict

The dictionary consists of keys which are vertices and values which are Segments.

Examples

>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.altitudes[p1]
Segment2D(Point2D(0, 0), Point2D(1/2, 1/2))

bisectors()[source]

The angle bisectors of the triangle.

An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half.

Returns:

bisectors : dict

Each key is a vertex (Point) and each value is the corresponding bisector (Segment).

Examples

>>> from sympy import Point, Triangle, Segment
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> from sympy import sqrt
>>> t.bisectors()[p2] == Segment(Point(1, 0), Point(0, sqrt(2) - 1))
True

property circumcenter

The circumcenter of the triangle

The circumcenter is the center of the circumcircle.

Returns:

circumcenter : Point

Examples

>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.circumcenter
Point2D(1/2, 1/2)

property circumcircle

The circle which passes through the three vertices of the triangle.

Returns:

circumcircle : Circle

Examples

>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.circumcircle
Circle(Point2D(1/2, 1/2), sqrt(2)/2)


The radius of the circumcircle of the triangle.

Returns:

circumradius : number of Basic instance

Examples

>>> from sympy import Symbol
>>> from sympy import Point, Triangle
>>> a = Symbol('a')
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, a)
>>> t = Triangle(p1, p2, p3)
sqrt(a**2/4 + 1/4)

property eulerline

The Euler line of the triangle.

The line which passes through circumcenter, centroid and orthocenter.

Returns:

eulerline : Line (or Point for equilateral triangles in which case all

centers coincide)

Examples

>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.eulerline
Line2D(Point2D(0, 0), Point2D(1/2, 1/2))

property excenters

Excenters of the triangle.

An excenter is the center of a circle that is tangent to a side of the triangle and the extensions of the other two sides.

Returns:

excenters : dict

Examples

The excenters are keyed to the side of the triangle to which their corresponding excircle is tangent: The center is keyed, e.g. the excenter of a circle touching side 0 is:

>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(6, 0), Point(0, 2)
>>> t = Triangle(p1, p2, p3)
>>> t.excenters[t.sides[0]]
Point2D(12*sqrt(10), 2/3 + sqrt(10)/3)


References

The radius of excircles of a triangle.

An excircle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two.

Returns:

Examples

The exradius touches the side of the triangle to which it is keyed, e.g. the exradius touching side 2 is:

>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(6, 0), Point(0, 2)
>>> t = Triangle(p1, p2, p3)
-2 + sqrt(10)


References

property incenter

The center of the incircle.

The incircle is the circle which lies inside the triangle and touches all three sides.

Returns:

incenter : Point

Examples

>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.incenter
Point2D(1 - sqrt(2)/2, 1 - sqrt(2)/2)

property incircle

The incircle of the triangle.

The incircle is the circle which lies inside the triangle and touches all three sides.

Returns:

incircle : Circle

Examples

>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(2, 0), Point(0, 2)
>>> t = Triangle(p1, p2, p3)
>>> t.incircle
Circle(Point2D(2 - sqrt(2), 2 - sqrt(2)), 2 - sqrt(2))


Returns:

inradius : number of Basic instance

Examples

>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(4, 0), Point(0, 3)
>>> t = Triangle(p1, p2, p3)
1

is_equilateral()[source]

Are all the sides the same length?

Returns:

is_equilateral : boolean

Examples

>>> from sympy import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
>>> t1.is_equilateral()
False

>>> from sympy import sqrt
>>> t2 = Triangle(Point(0, 0), Point(10, 0), Point(5, 5*sqrt(3)))
>>> t2.is_equilateral()
True

is_isosceles()[source]

Are two or more of the sides the same length?

Returns:

is_isosceles : boolean

Examples

>>> from sympy import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(2, 4))
>>> t1.is_isosceles()
True

is_right()[source]

Is the triangle right-angled.

Returns:

is_right : boolean

Examples

>>> from sympy import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
>>> t1.is_right()
True

is_scalene()[source]

Are all the sides of the triangle of different lengths?

Returns:

is_scalene : boolean

Examples

>>> from sympy import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(1, 4))
>>> t1.is_scalene()
True

is_similar(t2)[source]

Is another triangle similar to this one.

Two triangles are similar if one can be uniformly scaled to the other.

Parameters:

other: Triangle

Returns:

is_similar : boolean

Examples

>>> from sympy import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
>>> t2 = Triangle(Point(0, 0), Point(-4, 0), Point(-4, -3))
>>> t1.is_similar(t2)
True

>>> t2 = Triangle(Point(0, 0), Point(-4, 0), Point(-4, -4))
>>> t1.is_similar(t2)
False

property medial

The medial triangle of the triangle.

The triangle which is formed from the midpoints of the three sides.

Returns:

medial : Triangle

Examples

>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.medial
Triangle(Point2D(1/2, 0), Point2D(1/2, 1/2), Point2D(0, 1/2))

property medians

The medians of the triangle.

A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas.

Returns:

medians : dict

Each key is a vertex (Point) and each value is the median (Segment) at that point.

Examples

>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.medians[p1]
Segment2D(Point2D(0, 0), Point2D(1/2, 1/2))

property nine_point_circle

The nine-point circle of the triangle.

Nine-point circle is the circumcircle of the medial triangle, which passes through the feet of altitudes and the middle points of segments connecting the vertices and the orthocenter.

Returns:

nine_point_circle : Circle

Examples

>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.nine_point_circle
Circle(Point2D(1/4, 1/4), sqrt(2)/4)

property orthocenter

The orthocenter of the triangle.

The orthocenter is the intersection of the altitudes of a triangle. It may lie inside, outside or on the triangle.

Returns:

orthocenter : Point

Examples

>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.orthocenter
Point2D(0, 0)

property vertices

The triangle’s vertices

Returns:

vertices : tuple

Each element in the tuple is a Point

Examples

>>> from sympy import Triangle, Point
>>> t = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
>>> t.vertices
(Point2D(0, 0), Point2D(4, 0), Point2D(4, 3))