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.
If the Polygon has intersecting sides.
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) Segment(Point2D(0, 0), Point2D(1, 0)) >>> Polygon(p1, p2, p5) Segment(Point2D(0, 0), Point2D(3, 0))
While the sides of a polygon are not allowed to cross implicitly, they can do so explicitly. For example, a polygon shaped like a Z with the top left connecting to the bottom right of the Z must have the point in the middle of the Z explicitly given:
>>> mid = Point(1, 1) >>> Polygon((0, 2), (2, 2), mid, (0, 0), (2, 0), mid).area 0 >>> Polygon((0, 2), (2, 2), mid, (2, 0), (0, 0), mid).area -2
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.
See also
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)
-
area¶ The area of the polygon.
See also
Notes
The area calculation can be positive or negative based on the orientation of the 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.area 3
-
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 complex.
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.
References
[1] http://paulbourke.net/geometry/polygonmesh/#insidepoly
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
-
intersection(o)[source]¶ The intersection of two polygons.
The intersection may be empty and can contain individual Points and complete Line Segments.
Parameters: other: Polygon
Returns: intersection : list
The list of Segments and Points
Examples
>>> from sympy import Point, Polygon >>> 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(2/3, 0), Point2D(9/5, 1/5), Point2D(7/3, 1), Point2D(1/3, 1)]
-
is_convex()[source]¶ Is the polygon convex?
A polygon is convex if all its interior angles are less than 180 degrees.
Returns: is_convex : boolean
True if this polygon is convex, False otherwise.
See also
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
-
perimeter¶ The perimeter of the polygon.
Returns: perimeter : number or Basic instance See also
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]
-
sides¶ The line segments that form the sides of the polygon.
Returns: sides : list of sides
Each side is a Segment.
Notes
The Segments that represent the sides are an undirected line segment so cannot be used to tell the orientation of the polygon.
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 [Segment(Point2D(0, 0), Point2D(1, 0)), Segment(Point2D(1, 0), Point2D(5, 1)), Segment(Point2D(0, 1), Point2D(5, 1)), Segment(Point2D(0, 0), Point2D(0, 1))]
-
vertices¶ The vertices of the polygon.
Returns: vertices : tuple of Points See also
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.args[0] Point2D(0, 0)
-
-
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.
See also
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)
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 See also
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 See also
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.
>>> from sympy import RegularPolygon, Line
>>> RegularPolygon((0, 0), 1, 4).reflect(Line((0, 1), slope=-2)) RegularPolygon(Point2D(4/5, 2/5), -1, 4, acos(3/5))
-
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)
-
rotation¶ CCW angle by which the RegularPolygon is rotated
Returns: rotation : number or instance of Basic Examples
>>> from sympy import pi >>> from sympy.geometry import RegularPolygon, Point >>> RegularPolygon(Point(0, 0), 3, 4, pi).rotation pi
-
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)
-
vertices¶ The vertices of the RegularPolygon.
Returns: vertices : list
Each vertex is a Point.
See also
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)]
-
-
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.
See also
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))
Attributes
vertices altitudes orthocenter circumcenter circumradius circumcircle inradius incircle 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] Segment(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(0, sqrt(2) - 1), Point(1, 0)) True
-
circumcenter¶ The circumcenter of the triangle
The circumcenter is the center of the circumcircle.
Returns: circumcenter : Point See also
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)
-
circumcircle¶ The circle which passes through the three vertices of the triangle.
Returns: circumcircle : Circle See also
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)
-
circumradius¶ The radius of the circumcircle of the triangle.
Returns: circumradius : number of Basic instance See also
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)
-
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 Line(Point2D(0, 0), Point2D(1/2, 1/2))
-
incenter¶ The center of the incircle.
The incircle is the circle which lies inside the triangle and touches all three sides.
Returns: incenter : Point See also
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(-sqrt(2)/2 + 1, -sqrt(2)/2 + 1)
-
incircle¶ The incircle of the triangle.
The incircle is the circle which lies inside the triangle and touches all three sides.
Returns: incircle : Circle See also
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(-sqrt(2) + 2, -sqrt(2) + 2), -sqrt(2) + 2)
-
inradius¶ The radius of the incircle.
Returns: inradius : number of Basic instance See also
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 See also
sympy.geometry.entity.GeometryEntity.is_similar,RegularPolygon,is_isosceles,is_right,is_scaleneExamples
>>> 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 See also
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 See also
sympy.geometry.line.LinearEntity.is_perpendicular,is_equilateral,is_isosceles,is_scaleneExamples
>>> 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 See also
Examples
>>> from sympy.geometry import Triangle, Point >>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(1, 4)) >>> t1.is_scalene() True
-
is_similar(t1, 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 See also
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))
-
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] Segment(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 See also
sympy.geometry.line.Segment.midpoint,sympy.geometry.polygon.Triangle.medial,sympy.geometry.polygon.Triangle.orthocenterExamples
>>> 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 See also
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)
-
vertices¶ The triangle’s vertices
Returns: vertices : tuple
Each element in the tuple is a Point
See also
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))
-
