Polygons¶

class sympy.geometry.polygon.Polygon[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 : sequence of Points
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 Point, 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 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

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, S, 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)

See also

sympy.geometry.point.Point

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

See also

sympy.geometry.ellipse.Ellipse.area

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

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)

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
>>> from sympy.abc import t
>>> 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

[1] http://paulbourke.net/geometry/polygonmesh/#insidepoly

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

See also

sympy.geometry.util.convex_hull

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

See also

sympy.geometry.line.Segment.length

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]

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 sympifiable 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 Point, 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

https://en.wikipedia.org/wiki/Second_moment_of_area

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))]

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)

See also

sympy.geometry.point.Point

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

See also

sympy.geometry.point.Point, Polygon

Attributes

vertices
center
radius
rotation
apothem
interior_angle
exterior_angle
circumcircle
incircle
angles

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}

apothem¶

The inradius of the RegularPolygon.

The apothem/inradius is the radius of the inscribed circle.

Returns: apothem : number or instance of Basic

Examples

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

area¶

Returns the area.

Examples

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

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)

center¶

The center of the RegularPolygon

This is also the center of the circumscribing circle.

Returns: center : Point

Examples

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

centroid¶

The center of the RegularPolygon

This is also the center of the circumscribing circle.

Returns: center : Point

Examples

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

circumcenter¶

Alias for center.

Examples

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

circumcircle¶

The circumcircle of the RegularPolygon.

Returns: circumcircle : Circle

Examples

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

circumradius¶

Alias for radius.

Examples

>>> from sympy import Symbol
>>> from sympy.geometry import RegularPolygon, Point
>>> radius = Symbol('r')
>>> rp = RegularPolygon(Point(0, 0), radius, 4)
>>> rp.circumradius
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
>>> r, R = p.inradius, p.circumradius
>>> 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

exterior_angle¶

Measure of the exterior angles.

Returns: exterior_angle : number

Examples

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

incircle¶

The incircle of the RegularPolygon.

Returns: incircle : Circle

Examples

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

inradius¶

Alias for apothem.

Examples

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

interior_angle¶

Measure of the interior angles.

Returns: interior_angle : number

Examples

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

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.geometry 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

radius¶

Radius of the RegularPolygon

This is also the radius of the circumscribing circle.

Returns: radius : number or instance of Basic

Examples

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

See also

rotation

spin: Rotates a RegularPolygon in place

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.geometry 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.

See also: rotate method which moves the center.

>>> 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)

See also

rotation

rotate: Creates a copy of the RegularPolygon rotated about a Point

vertices¶

The vertices of the RegularPolygon.

Returns: vertices : list

Each vertex is a Point.

Examples

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

See also

sympy.geometry.point.Point

class sympy.geometry.polygon.Triangle[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.geometry 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))

See also

sympy.geometry.point.Point, Polygon

Attributes

vertices
altitudes
orthocenter
circumcenter
circumradius
circumcircle
inradius
incircle
exradii
medians
medial
nine_point_circle

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.geometry 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.geometry 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

circumcenter¶

The circumcenter of the triangle

The circumcenter is the center of the circumcircle.

Returns: circumcenter : Point

Examples

>>> from sympy.geometry 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)

See also

sympy.geometry.point.Point

circumcircle¶

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

Returns: circumcircle : Circle

Examples

>>> from sympy.geometry 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)

See also

sympy.geometry.ellipse.Circle

circumradius¶

The radius of the circumcircle of the triangle.

Returns: circumradius : number of Basic instance

Examples

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

See also

sympy.geometry.ellipse.Circle.radius

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.geometry 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))

exradii¶

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: exradii : dict

Examples

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

>>> from sympy.geometry import Point, Triangle, Segment2D, Point2D
>>> p1, p2, p3 = Point(0, 0), Point(6, 0), Point(0, 2)
>>> t = Triangle(p1, p2, p3)
>>> t.exradii[t.sides[2]]
-2 + sqrt(10)

References

[1] http://mathworld.wolfram.com/Exradius.html [2] http://mathworld.wolfram.com/Excircles.html

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.geometry 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)

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.geometry 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))

See also

sympy.geometry.ellipse.Circle

inradius¶

The radius of the incircle.

Returns: inradius : number of Basic instance

Examples

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

is_equilateral()[source]¶

Are all the sides the same length?

Returns: is_equilateral : boolean

Examples

>>> from sympy.geometry 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.geometry 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.geometry 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.geometry 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.geometry 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

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.geometry 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))

See also

sympy.geometry.line.Segment.midpoint

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.geometry 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))

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.geometry 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)

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.geometry 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)

See also

sympy.geometry.point.Point

vertices¶

The triangle’s vertices

Returns: vertices : tuple

Each element in the tuple is a Point

Examples

>>> from sympy.geometry 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))

See also

sympy.geometry.point.Point

Polygons¶

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