Welcome to planegeometry’s documentation!¶

Indices and tables¶

Members functions¶

class planegeometry.geometry.Affine(A, b)¶

A class for affine transformations.

An affine transformation is initialized by a 2x2 matrix (a linear transformation), and a length two vector (the ‘intercept’, an array-like object).

compose(transfo, left=True)¶

Compose the reference affine map with another affine map.

Parameters:

transfo – an Affine object
left – Boolean, whether to compose at left or at right (i.e. returns f1 o f0 or f0 o f1)

Returns:

An Affine object.

classmethod from_ellipse_to_ellipse(ell1, ell2)¶

Affine transformation mapping a given ellipse to a given ellipse.

Parameters:

ell1 – Ellipse or Circle object
ell2 – Ellipse or Circle object

Returns:

An Affine object representing the transformation which maps ell1 to ell2.

classmethod from_mapping_three_points(P1, P2, P3, Q1, Q2, Q3)¶

Affine transformation mapping three given points to three given points.

Parameters:

P1 – a point
P2 – another point
P3 – another point; the three points must be non-collinear
Q1 – a point
Q2 – another point
Q3 – another point; the three points must be non-collinear

Returns:

An Affine object representing the transformation which maps Pi to Qi for each i=1,2,3.

get3x3matrix()¶: Get the 3x3 matrix corresponding to the affine transformation.

inverse()¶: The inverse affine transformation if it exists.

transform(m)¶

Transform a point or several points by the affine map.

Parameters:: m – a point or a two-column matrix of points, one point per row
Returns:: A matrix or a vector.

transform_ellipse(ell)¶

Transform an ellipse by the reference affine transformation (only for an invertible affine map).

Parameters:: ell – an Ellipse object or a Circle object
Returns:: An Ellipse object.

transform_line(line)¶

Transform a line by the affine map.

Returns:: A Line object.

class planegeometry.geometry.Arc(center, radius, alpha1, alpha2, degrees=True)¶

Arc class.

A circular arc is initialized by its center (array-like object of length two), a radius, a starting angle and an ending angle. They are respectively named center, radius, alpha1 and alpha2.

ending_point()¶: Ending point of the arc.

path(n_points=100)¶

Path that forms the arc.

Parameters:: n_points – number of points of the path
Returns:: A matrix with two columns and n_points rows.

starting_point()¶: Starting point of the arc.

class planegeometry.geometry.Circle(center, radius)¶

A class for circles. A circle is given by its center and its radius.

angle(circ2)¶

Angle between the reference circle and a given circle, if they intersect.

Parameters:: circ2 – a Circle object
Returns:: The angle in radians.

as_ellipse()¶

Converts the circle to an Ellipse object.

Returns:: An Ellipse object.

contains(P)¶

Check whether a point is contained in the reference circle.

Parameters:: P – a point
Returns:: A Boolean value.

includes(P)¶

Check whether a point belongs to the reference circle.

Parameters:: P – a point
Returns:: A Boolean value.

intersection_with_circle(circ2)¶

Intersection(s) of the reference circle with another circle.

Parameters:: circ2 – a Circle object
Returns:: None (no intersection), a point (the two circles are tangent), or two points.

intersection_with_line(line)¶

Intersection(s) of the reference circle with a line.

Parameters:: line – a Line object
Returns:: None (no intersection), a point (the line is tangent to the circle), or two points.

is_equal(circ2)¶

Check whether the reference circle is equal to another circle.

Parameters:: circ2 – a Circle object
Returns:: A Boolean value.

is_orthogonal(circ2)¶

Check whether the reference circle is orthogonal to a given circle.

Parameters:: circ2 – a Circle object
Returns:: A Boolean value.

orthogonal_through_two_points_in_circle(P1, P2, arc=False)¶

Orthogonal circle passing through two points within the reference circle.

Parameters:

P1 – a point in the interior of the reference circle
P2 – another point in the interior of the reference circle
arc – Boolean, whether to return the arc joining the two points instead of the circle

Returns:

A Circle object or an Arc object, or a Line object if the two points are on a diameter.

orthogonal_through_two_points_on_circle(alpha1, alpha2, arc=False)¶

Orthogonal circle passing through two points on the reference circle.

Parameters:

alpha1 – an angle defining a point on the reference circle
alpha2 – another angle defining a point on the reference circle
arc – Boolean, whether to return only the arc at the interior of the reference circle

Returns:

A Circle object if arc=False, an Arc object if arc=True, or a `Line object: the diameter of the reference circle defined by the two points in case when the two angles differ by pi.

point_from_angle(alpha, degrees=True)¶

Get a point on the reference circle from its polar angle.

Parameters:

alpha – a number, the angle
degrees – Boolean, whether the angle is given in degrees

Returns:

The point on the circle with polar angle alpha.

power(M)¶

Power of a point with respect to the reference circle.

Parameters:: M – a point (array-like of length two)
Returns:: A number, the power of M with respect to the circle.

radical_axis(circ2)¶

Radical axis of two circles.

Parameters:: circ2 – a Circle object
Returns:: A Line object, the radical axis of the reference circle and circ2.

radical_center(circ2)¶

Radical center of two circles.

Parameters:: circ2 – a Circle object
Returns:: The radical center of the reference circle and circ2.

random_points(n_points, where='in')¶

Random points in the circle or on the circle.

Parameters:

n_points – desired number of points
where – either “in” or “on”

Returns:

A matrix with n_points rows and two columns; each row is a random point inside the circle if where=”in” or on the boundary of the circle if where=”on”.

class planegeometry.geometry.Ellipse(center, rmajor, rminor, alpha, degrees=True)¶

Ellipse class.

An ellipse is initialized by its center (array-like object of length two), its major radius, its minor radius, and the angle alpha between the x-axis and the major axis.

classmethod LownerJohnEllipse(ipoints, bpoints=None)¶

Minimum area ellipse containing a set of points (ellipse hull).

Parameters:

ipoints – an array of shape (n,2) containing points as row vectors, which are inside the desired ellipse; it must have at least three distinct rows
bpoints – an array of shape (n,2) containing points as row vectors, which are on the boundary of the desired ellipse; could be None (the default)

Returns:

An Ellipse object, the Löwner-John ellipse.

Author:

Dor Shaviv

contains(P)¶

Check whether a point is contained in the ellipse.

Parameters:: P – a point
Returns:: A Boolean value.

diameter(t, conjugate=False)¶

Diameter of the ellipse.

Parameters:

t – eccentric angle; for t=0, the diameter is the major axis
conjugate – Boolean, whether to return the conjugate diameter as well

Returns:

A Line object or a list of two Line objects if conjugate=True.

equation()¶

The coefficients of the implicit equation of the ellipse, Ax² + Bxy + Cy² + Dx + Ey + F = 0.

Returns:: A dictionary giving the values of the coefficients.

classmethod equation_from_five_points(P1, P2, P3, P4, P5)¶

The implicit equation of the ellipse is Ax² + Bxy + Cy² + Dx + Ey + F = 0. This function returns A, B, C, D, E and F.

Parameters:

P1 – a point
P2 – another point
P3 – another point
P4 – another point
P5 – another point

Returns:

A dictionary giving A, B, C, D, E and F.

classmethod from_boundary3(points)¶

Compute the smallest ellipse that passes through 3 boundary points.

Parameters:: points – an array of shape (3,2) containing points as row vectors, which are on the boundary of the desired ellipse.
Returns:: An Ellipse object.
Author:: Dor Shaviv

classmethod from_boundary4(points)¶

Compute the smallest ellipse that passes through 4 boundary points, based on the algorithm by: B. W. Silverman and D. M. Titterington, “Minimum covering ellipses,” SIAM Journal on Scientific and Statistical Computing 1, no. 4 (1980): 401-409.

Parameters:: points – an array of shape (4,2) containing points as row vectors, which are on the boundary of the desired ellipse.
Returns:: An Ellipse object.
Author:: Dor Shaviv

classmethod from_center_and_matrix(center, S)¶

Ellipse from its center and its shape matrix.

Parameters:

center – a point
S – a 2x2 symmetric matrix

Returns:

An Ellipse object.

classmethod from_equation(A, B, C, D, E, F)¶

Ellipse from its implicit equation.

Parameters:

A – coefficient of the implicit equation of the ellipse
B – coefficient of the implicit equation of the ellipse
C – coefficient of the implicit equation of the ellipse
D – coefficient of the implicit equation of the ellipse
E – coefficient of the implicit equation of the ellipse
F – coefficient of the implicit equation of the ellipse

Returns:

An Ellipse object.

classmethod from_five_points(P1, P2, P3, P4, P5)¶

Ellipse from five points on this ellipse.

Parameters:

P1 – a point
P2 – another point
P3 – another point
P4 – another point
P5 – another point

Returns:

An Ellipse object.

includes(P)¶

Check whether a point belongs to the ellipse.

Parameters:: P – a point
Returns:: A Boolean value.

intersection_with_line(line)¶

Intersection of the ellipse and a line.

Parameters:: line – a Line object
Returns:: The intersection, it can be None, one point, or a list of two points.

is_equal(ell2)¶

Check whether the reference ellipse equals another ellipse.

Parameters:: ell2 – an Ellipse object
Returns:: A Boolean value.

normal(t)¶

Normal unit vector to the ellipse.

Parameters:: t – a number, the eccentric angle in radians of the point of the ellipse at which we want the normal unit vector
Returns:: The normal unit vector to the ellipse at the point given by eccentric angle t.

path(n_points=100)¶

Path that forms the ellipse.

Parameters:: n_points – number of points of the path
Returns:: A matrix with two columns and n_points rows.

point_from_angle(theta, degrees=True)¶

Point on the ellipse with given eccentric angle.

Parameters:

theta – eccentric angle
degrees – Boolean, whether theta is given in degrees

Returns:

A point on the ellipse.

random_points(n_points, where='in')¶

Random points in/on the ellipse.

Parameters:

n_points – desired number of points
where – either “in” or “on”

Returns:

A matrix with n_points rows and two columns; each row is a random point inside the ellipse if where=”in” or on the boundary of the ellipse if where=”on”.

shape_matrix()¶: The 2x2 symmetric matrix S associated to the reference ellipse. The equation of the ellipse is (M-O)’ S (M-O) = 1.

theta2t(theta, degrees=True)¶

Convert angle to eccentric angle.

Parameters:

theta – angle between the major axis and the half-line starting at the center of the ellipse and passing through the point of interest on the ellipse
degrees – Boolean, whether theta is given in degrees

Returns:

The eccentric angle of the point of interest on the ellipse, in radians.

class planegeometry.geometry.Homothety(center, scale)¶

A homothety is given by a center and a scale factor.

as_affine()¶: Converting to an Affine object.

get3x3matrix()¶: Get the augmented matrix of the homothety.

transform(M)¶

Transform one or more points.

Parameters:: M – a point or a matrix of points
Returns:: A point or a matrix of points.

transform_circle(circ)¶

Transform a circle by the homothety.

Parameters:: circ – a Circle object
Returns:: A Circle object.

class planegeometry.geometry.Inversion(pole, power)¶

Inversion class.

An inversion is initialized by its pole and its power (a number, possibly negative).

compose(iota2, left=True)¶

Compose the reference inversion with another inversion. The result is a Möbius transformation.

Parameters:

iota2 – an Inversion object
left – Boolean, whether to compose at left or at right (i.e. returns iota2 o iota1 or iota1 o iota2)

Returns:

A Mobius object.

classmethod from_fixing_three_circles(circ1, circ2, circ3)¶

Inversion fixing three circles.

Parameters:

circ1 – a Circle object
circ2 – a Circle object
circ3 – a Circle object

Returns:

An Inversion object representing an inversion which leaves each of the three circles invariant.

classmethod from_fixing_two_circles(circ1, circ2)¶

Inversion fixing two circles.

Parameters:

circ1 – a Circle object
circ2 – a Circle object

Returns:

An Inversion object representing an inversion which leaves each of the two circles invariant.

classmethod from_swapping_two_circles(circ1, circ2, positive=True)¶

Inversion swapping two circles.

Parameters:

circ1 – a Circle object
circ2 – a Circle object
positive – Boolean, whether the sign of the desired inversion power must be positive or negative

Returns:

An Inversion object, which maps circ1 to circ2 and circ2 to circ1, except in the case when circ1 and circ2 are congruent and tangent: in this case a Reflection object is returned (a reflection is an inversion on a line).

invert_gcircle(gcircle)¶

Invert a generalized circle, that is, a circle or a line.

Params gcircle`:: a Circle object or a Line object
Returns:: A Circle object or a Line object.

classmethod on_circle(circ)¶

An inversion on a circle is the inversion whose pole is the center of the circle and whose power is the squared radius of the circle.

Parameters:: circ – Circle object
Returns:: An Inversion object.

class planegeometry.geometry.Line(A, B, extendA=True, extendB=True)¶

A class for lines. A line is initialized by two points it passes through, and for each of these points a Boolean value to indicate whether the line should be extended besides this point.

direction_offset()¶

Direction and offset of the line. The equation of the line is cos(direction)x + sin(direction)y = offset.

Returns:: The direction and the offset in a dictionary.

distance(M)¶

Distance from a point to the reference line.

Parameters:: M – a point
Returns:: A number.

includes(M, strict=False, checkCollinear=True)¶

Check whether a point belongs to the line.

Parameters:

M – the point for which we want to test whether it belongs to the line
strict – Boolean, whether to take into account extendA and extendB
checkCollinear – Boolean, whether to check the collinearity of A, B, and M; set to False only if you use strict=True and you are sure that M is on the line (AB)

Returns:

A Boolean value.

intersection_with_circle(circ)¶

Intersection(s) of the line with a circle.

Parameters:: circ – a Circle object
Returns:: None, a point, or a list of two points.

intersection_with_ellipse(ell)¶

Intersection(s) of the line with an ellipse.

Parameters:: ell – an Ellipse object
Returns:: None, a point, or a list of two points.

intersection_with_line(line2, strict=False)¶

Intersection(s) of the reference line with another line.

Parameters:: line2 – a Line object
Returns:: None (the lines are parallel), a point, or a Line object (the two lines are equal).

invert(iota)¶

Invert the reference line.

Parameters:: iota – an Inversion object
Returns:: A Line object or a Circle object.

is_equal(line2)¶

Check whether the reference line is equal to another line.

Parameters:: line2 – a Line object
Returns:: A Boolean value.

is_parallel(line2)¶

Check whether the reference line is parallel to another line.

Parameters:: line2 – a Line object
Returns:: A Boolean value.

is_perpendicular(line2)¶

Check whether the reference line is perpendicular to another line.

Parameters:: line2 – a Line object
Returns:: A Boolean value.

perpendicular(M, extendH=False, extendM=True)¶

Perpendicular line passing through a given point.

Parameters:

M – the point through which the perpendicular passes
extendH – Boolean, whether to extend the perpendicular line beyond the meeting point
extendM – Boolean, whether to extend the perpendicular line beyond the point M

Returns:

A Line object; its two points are the meeting point and the point M.

projection(M)¶

Orthogonal projection of a point to the reference line.

Parameters:: M – a point
Returns:: A point on the reference line.

reflection(M)¶

Reflection of a point with respect to the reference line.

Parameters:: M – a point
Returns:: A point.

rotate(alpha, O, degrees=True)¶

Rotate the reference line.

Parameters:

alpha – angle of rotation
O – center of rotation
degrees – Boolean, whether alpha is given in degrees

Returns:

A Line object.

translate(v)¶

Translate the reference line.

Parameters:: v – the vector of translation
Returns:: A Line object.

class planegeometry.geometry.Mobius(M)¶

A class for Möbius transformations.

A Möbius transformation is initialized by a complex 2x2 matrix with a non-zero determinant.

compose(M1, left=True)¶

Compose the reference Möbius transformation with another Möbius transformation.

Parameters:

M1 – a Mobius object
left – Boolean, whether to compose at left or at right (i.e. returns M1 o M0 or M0 o M1)

Returns:

A Mobius object.

fixed_points()¶

Fixed points of the Möbius transformation.

Returns:: One point, or a list of two points, or a message in the case when the transformation is the identity map.

classmethod from_mapping_one_circle(circ1, circ2)¶

Returns a Möbius transformation which maps a given circle to another given circle.

Parameters:

circ1 – a `Circle`object
circ2 – a `Circle`object

classmethod from_mapping_three_points(P1, P2, P3, Q1, Q2, Q3)¶

Möbius transformation mapping three given points to three given points.

Parameters:

P1 – a point, inf allowed
P2 – another point, inf allowed
P3 – another point, inf allowed
Q1 – a point, inf allowed
Q2 – another point, inf allowed
Q3 – another point, inf allowed

Returns:

A Mobius object, representing the Möbius transformation which sends Pi to Qi for each i=1,2,3.

gpower(t)¶

Generalized power of the Möbius transformation.

Parameters:: t – a float, possibly negative
Returns:: A Mobius object corresponding to the Möbius transformation raised to the power t.

inverse()¶

Inverse of the Möbius transformation.

Returns:: A Möbius transformation.

power(k)¶

Power of the Möbius transformation.

Parameters:: k – an integer, possibly negative
Returns:: A Mobius object corresponding to the Möbius transformation raised to the power k.

transform(P)¶

Transform a point by the Möbius transformation.

Parameters:: P – a point (array-like of length two) or inf
Returns:: The image of P by the Möbius transformation (can be inf).

transform_circle(circ)¶

Transform a circle by the Möbius transformation.

Parameters:: circ – a Circle object
Returns:: A Circle object or a Line object.

transform_line(line)¶

Transform a line by the Möbius transformation.

Parameters:: line – a Line object
Returns:: A Circle object or a Line object.

class planegeometry.geometry.Projection(D, Delta)¶

A class for projections. A projection on a line is given by the line of projection D and the directrix line Delta.

For an orthogonal projection, you can also use the projection method of the Line class.

as_affine()¶: Convert the projection to an Affine object.

get3x3matrix()¶

Augmented matrix of the projection.

Returns:: A 3x3 matrix.

project(M)¶

Projection of a point.

Parameters:: M – a point
Returns:: A point on D, the projection of M.

transform(M)¶: An alias of project

class planegeometry.geometry.Reflection(line)¶

A class for reflections.

A reflection is initialized by a line.

as_affine()¶: Convert the reflection to an Affine object.

get3x3matrix()¶: Augmented matrix of the reflection.

reflect(P)¶: An alias of transform.

reflect_circle(circ)¶

Reflect a circle.

Parameters:: circ – a Circle object
Returns:: A Circle object.

reflect_line(line)¶

Reflect a line.

Parameters:: line – a Line object
Returns:: A Line object.

transform(P)¶

Transform a point by the refection.

Parameters:: P – a point, inf allowed
Returns:: The image of P.

transform_circle(circ)¶: An alias of reflect_circle.

transform_line(line)¶: An alias of reflect_line.

class planegeometry.geometry.Rotation(center, theta, degrees=True)¶

A class for rotations. A rotation is given by its center and its angle.

Parameters:

center – a point
theta – a number, the angle of the rotation
degrees – Boolean, whether the angle is given in degrees

as_affine()¶: Convert the rotation to an Affine object.

get3x3matrix()¶: Augmented matrix of the rotation.

rotate(M)¶

Rotate a point.

Parameters:: M – a point or a matrix of points
Returns:: A point or a matrix of points.

rotate_circle(circ)¶

Rotate a circle.

Parameters:: circ – a Circle object
Returns:: A Circle object.

rotate_ellipse(ell)¶

Rotate an ellipse.

Parameters:: ell – an Ellipse object
Returns:: An Ellipse object.

rotate_line(line)¶

Rotate a line.

Parameters:: line – a Line object
Returns:: A Line object.

transform(M)¶: An alias of rotate.

transform_circle(circ)¶: An alias of rotate_circle.

transform_ellipse(ell)¶: An alias of rotate_ellipse.

transform_line(line)¶: An alias of rotate_line.

class planegeometry.geometry.Scaling(center, direction, scale)¶

A class representing a (non-uniform) scaling. A non-uniform scaling is given by a center, a direction vector, and a scale factor.

Example:

>>> Q = np.array([1,1]); w = np.array([1,3]); s = 2
>>> scaling = Scaling(Q, w, s)
>>> # the center is mapped to itself:
>>> scaling.transform(Q)
>>> # any vector `u` parallel to the direction vector is mapped to `s*u`:
>>> u = 3*w
>>> np.equal(s*u, S.transform(u) - S.transform((0,0)))
>>> # any vector perpendicular to the direction vector is mapped to itself:
>>> wt = 3 * np.array([-w[1], w[0]])
>>> np.equal(wt, S.transform(wt) - S.transform((0,0)))

as_affine()¶: Converting to an Affine object.

get3x3matrix()¶: Get the augmented matrix of the scaling.

scale_circle(circ)¶

Scale a circle. The result is an ellipse.

Parameters:: circ – a Circle object
Returns:: An Ellipse object.

transform(M)¶

Transform one or more points.

Parameters:: M – a point or a matrix of points
Returns:: A point or a matrix of points.

class planegeometry.geometry.ScalingXY(center, sx, sy)¶

A class for axis-scalings. An axis-scaling is given by a center, and two scale factors sx and sy, one for the x-axis and one for the y-axis.

as_affine()¶: Converting to an Affine object.

get3x3matrix()¶: Get the augmented matrix of the axes-scaling.

transform(M)¶

Transform one or more points.

Parameters:: M – a point or a matrix of points
Returns:: A point or a matrix of points.

class planegeometry.geometry.Shear(vertex, vector, ratio, angle, degrees=True)¶

A class for shear transformations. A shear is given by a vertex, two perpendicular vectors, and an angle.

Example:

>>> P = np.array([0,0]); w = np.array([1,0]); ratio = 1; angle = 45
>>> shear = Shear(P, w, ratio, angle)
>>> wt = ratio * np.array([-w[1], w[0]])
>>> Q = P + w; R = Q + wt; S = P + wt
>>> A = shear.transform(P); B = shear.transform(Q)
>>> C = shear.transform(R); D = shear.transform(S)
>>> import matplotlib.pyplot as plt
>>> figure, axes = plt.subplots(figsize=(10, 10))
>>> axes.set_aspect(1)
>>> unit_square = plt.Polygon([P,Q,R,S], fill=False)
>>> axes.add_artist(unit_square)
>>> image = plt.Polygon([A,B,C,D], fill=False, color="red")
>>> axes.add_artist(image)
>>> plt.xlim(0, 1)
>>> plt.ylim(0, 2)
>>> plt.show()

as_affine()¶: Convert the shear to an Affine object.

get3x3matrix()¶: Get the augmented matrix of the shear.

transform(M)¶

Transform one or more points.

Parameters:: M – a point or a matrix of points
Returns:: A point or a matrix of points.

class planegeometry.geometry.Triangle(A, B, C)¶

Triangle class.

A triangle is initialized by its three vertices, some array-like objects of length two.

property a¶: Length of the side BC.

property angleA¶: The angle at the vertex A in radians.

property angleB¶: The angle at the vertex B in radians.

property angleC¶: The angle at the vertex C in radians.

property b¶: Length of the side AC.

property c¶: Length of the side AB.

contains(M)¶: Check whether a point lies inside the reference triangle.

property edges¶: Edge lengths of the triangle.

equal_detour_point()¶

Equal detour point of the triangle, also known as the X(176) triangle center.

Returns:: A pair, the equal detour point and the detour.

excircles()¶

The excircles of the triangle.

Returns:: A dictionary of three Circle objects.

property flatness¶: Flatness, a number between 0 and 1; a triangle is flat when its flatness is 1.

incircle()¶: The incircle of the triangle.

property is_acute¶: Check whether the triangle is acute.

malfatti_circles()¶

The Malfatti circles of the triangle.

Returns:: Three circles and three tangency points.

property orientation¶: Orientation of the triangle; 1 for counterclockwise, -1 for clockwise, 0 for collinear.

orthic_triangle()¶

Orthic triangle. Its vertices are the feet of the altitudes of the reference triangle.

Returns:: A Triangle object.

random_points(n_points, where='in')¶

Random points inside the triangle or on the boundary of the triangle.

Parameters:

n_points – desired number of points
where – either “in” or “on”

Returns:

A matrix with n_points rows and two columns; each row is a random point inside the triangle if where=”in” or on the boundary of the triangle if where=”on”.

steiner_ellipse()¶

The Steiner ellipse (or circumellipse) of the reference triangle. This is the ellipse passing through the three vertices of the triangle and centered at the centroid of the triangle.

Returns:: An Ellipse object.

steiner_inellipse()¶

The Steiner inellipse (or midpoint ellipse) of the reference triangle. This is the ellipse tangent to the sides of the triangle at their midpoints, and centered at the centroid of the triangle.

Returns:: An Ellipse object.

planegeometry.geometry.circleAB(A, B)¶

Circle with diameter AB.

Parameters:: A,B – two distinct points
Returns:: A Circle object.

planegeometry.geometry.circleOA(O, A)¶

Circle with center O and passing through A.

Parameters:: O,A – two distinct points
Returns:: A Circle object.

planegeometry.geometry.cross_ratio(A, B, C, D)¶

Cross ratio of four points.

Parameters:

A – a point
B – another point
C – another point
D – another point

Returns:

A complex number. It is real if and only if the four points lie on a generalized circle (that is a circle or a line).

planegeometry.geometry.intersection_circle_circle(circ1, circ2)¶

Intersection(s) of two circles.

Parameters:

circ1 – a Circle object
circ2 – a Circle object

Returns:

A Circle object if the two circles are equal, None if the two circles do not intersect, a point if the two circles are tangent, or a list of two points.

planegeometry.geometry.intersection_circle_line(circ, line)¶

Intersection(s) of a circle and a line.

Parameters:

circ – a Circle object
line – a Line object

Returns:

None if the intersection is empty, otherwise either one point (the line is tangent to the circle) or a list of two points.

planegeometry.geometry.intersection_ellipse_line(ell, line)¶

Intersection(s) of an ellipse and a line.

Parameters:

ell – an Ellipse object
line – a Line object

Returns:

None if the intersection is empty, otherwise either one point (the line is tangent to the ellipse) or a list of two points.

planegeometry.geometry.mid_circles(circ1, circ2)¶

Return the mid-circle(s) of two circles. A mid-circle of two circles is a generalized circle (i.e. a circle or a line) such that the Inversion.fro on this circle swaps the two circles. The case of a line appears only when the two circles have equal radii.

Parameters:

circ1 – a Circle object
circ2 – a Circle object

Returns:

A Circle object, or a Line object, or a list of two such objects.

planegeometry.geometry.radical_center(circ1, circ2, circ3)¶

Radical center of three circles.

Parameters:

circ1 – a Circle object
circ2 – a Circle object
circ3 – a Circle object

Returns:

A point, the radical center of the three circles.

planegeometry.geometry.unimodular_matrices(n)¶

Generates unimodular matrices.

Parameters:: n – integer, the maximum size of entries of matrices, at least 1
Returns:: List of unimodular matrices.

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