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 : sequence of Points
Optional parameters
==========
n : If > 0, an n-sided RegularPolygon is created. See below.
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 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)
See also
- 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
See also
- 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
Nonewhen 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
https://github.com/sympy/sympy/wiki/A-method-to-return-a-cut-section-of-any-polygon-geometry
- 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=Nonethen 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
https://skyciv.com/docs/tutorials/section-tutorials/calculating-the-statical-or-first-moment-of-area-of-beam-sections/?cc=BMD https://mechanicalc.com/reference/cross-sections
- 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
- 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
See also
- 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)
See also
- 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.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
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¶
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
- property area¶
Returns the area.
Examples
>>> from sympy.geometry 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.geometry 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.geometry import RegularPolygon, Point >>> rp = RegularPolygon(Point(0, 0), 5, 4) >>> rp.center Point2D(0, 0)
- property circumcenter¶
Alias for center.
Examples
>>> from sympy.geometry 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.geometry import RegularPolygon, Point >>> rp = RegularPolygon(Point(0, 0), 4, 8) >>> rp.circumcircle Circle(Point2D(0, 0), 4)
See also
- property 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
- property 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
- property 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))
See also
- property 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
- property 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
- 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.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
- property 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, 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)
- 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.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)
- property 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
- 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.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
Attributes
vertices
altitudes
orthocenter
circumcenter
circumradius
circumcircle
inradius
incircle
exradii
medians
medial
nine_point_circle
- 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.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
- property 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
- property 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
- property 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
- 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.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))
- 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.geometry 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
- property 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 >>> 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
- 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.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)
See also
- 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.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
- property 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
See also
- 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
See also
- 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
See also
- 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
- 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.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
- 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.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))
- 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.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)
- 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.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
- property 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
