curves

curves

Curve classes for the cxregions package.

This module contains all curve-related classes including the base JuliaCurve class and specific curve types like Line, Circle, Segment, Ray, and Arc.

Attributes

Name	Description
unitcircle	Unit circle centered at the origin.

Classes

Name	Description
Arc	A circular arc in the complex plane.
Circle	A circle in the complex plane.
ClosedCurve	A closed parametric curve in the complex plane.
Curve	A general parametric curve in the complex plane.
JuliaCurve	Base class for wrapping Julia curve objects from ComplexRegions.jl.
Line	An infinite straight line in the complex plane.
Ray	A semi-infinite ray in the complex plane.
Segment	A line segment in the complex plane.

Arc

curves.Arc(a, b=None, c=None)

A circular arc in the complex plane.

An arc is a portion of a circle between two points.

Parameters

Name	Type	Description	Default
a	complex, Circle, or juliacall.AnyValue	Start point, circle, or Julia Arc object to wrap	required
b	complex	End point (if a is start point)	None
c	complex	Center point (if constructing from start, end, center)	None

Attributes

Name	Type	Description
circle	Circle or Segment	The underlying circle (or degenerate segment)
start	float	Starting parameter on the circle
delta	float	Parameter range of the arc

Examples

>>> # Quarter circle arc from 1 to i around origin
>>> arc = Arc(1, 1j, 0)

Methods

Name	Description
inv	Compute the inversion of the arc with respect to the origin.

inv

curves.Arc.inv()

Compute the inversion of the arc with respect to the origin.

Returns

Name	Type	Description
	Arc, Ray, or Segment	The type depends on the geometry after inversion

Circle

curves.Circle(a, b=None, c=None, ccw=True)

A circle in the complex plane.

Circles can be constructed in several ways: - From center and radius - From three points on the circle - From a Julia Circle object

Parameters

Name	Type	Description	Default
a	complex or juliacall.AnyValue	Center point, first point on circle, or Julia Circle object	required
b	float or complex	Radius (if a is center) or second point on circle	None
c	complex	Third point on circle (if constructing from three points)	None
ccw	bool	Whether the circle has counterclockwise orientation, default True	True

Attributes

Name	Type	Description
center	complex	Center of the circle
radius	float	Radius of the circle
ccw	bool	Whether the circle has counterclockwise orientation

Examples

>>> # Unit circle at origin
>>> circle1 = Circle(0, 1)
>>> # Circle from center and radius with clockwise orientation
>>> circle2 = Circle(1+1j, 2, ccw=False)
>>> # Circle through three points
>>> circle3 = Circle(1, 1j, -1)

Methods

Name	Description
inv	Compute the inversion of the circle with respect to the origin.
isfinite	Check if the circle is finite.
ispositive	Check if the circle has positive (counterclockwise) orientation.

inv

curves.Circle.inv()

Compute the inversion of the circle with respect to the origin.

Returns

Name	Type	Description
	Circle or Line	Circle if the original circle doesn’t pass through the origin, Line if it does

isfinite

curves.Circle.isfinite()

Check if the circle is finite.

Returns

Name	Type	Description
	bool	Always returns True for circles

ispositive

curves.Circle.ispositive()

Check if the circle has positive (counterclockwise) orientation.

Returns

Name	Type	Description
	bool	True if counterclockwise, False if clockwise

ClosedCurve

curves.ClosedCurve(point, tangent=None, domain=(0.0, 1.0))

A closed parametric curve in the complex plane.

This class represents a closed curve that forms a Jordan curve, enabling computation of winding numbers and interior/exterior tests.

Parameters

Name	Type	Description	Default
point	callable or juliacall.AnyValue	Either a function that maps parameter values to complex points, or a Julia closed curve object to wrap	required
tangent	callable	Function that maps parameter values to tangent vectors	None
domain	tuple of float	Parameter domain as (start, end), default is (0.0, 1.0)	(0.0, 1.0)

Examples

>>> # Create a closed curve (unit circle)
>>> point = lambda t: np.exp(2j * np.pi * t)
>>> tangent = lambda t: 2j * np.pi * np.exp(2j * np.pi * t)
>>> curve = ClosedCurve(point, tangent)

Methods

Name	Description
hasinside	Check if point z is in the interior of the closed curve.
winding	Compute the winding number of the curve around point z.

hasinside

curves.ClosedCurve.hasinside(z)

Check if point z is in the interior of the closed curve.

Parameters

Name	Type	Description	Default
z	complex	Point to test	required

Returns

Name	Type	Description
	bool	True if z is in the interior, False otherwise

winding

curves.ClosedCurve.winding(z)

Compute the winding number of the curve around point z.

Parameters

Name	Type	Description	Default
z	complex	Point around which to compute winding number	required

Returns

Name	Type	Description
	int	Winding number (positive for counterclockwise orientation)

Curve

curves.Curve(point, tangent=None, domain=(0.0, 1.0))

A general parametric curve in the complex plane.

This class represents a curve defined by a point function and optional tangent function over a parameter domain.

Parameters

Name	Type	Description	Default
point	callable or juliacall.AnyValue	Either a function that maps parameter values to complex points, or a Julia curve object to wrap	required
tangent	callable	Function that maps parameter values to tangent vectors	None
domain	tuple of float	Parameter domain as (start, end), default is (0.0, 1.0)	(0.0, 1.0)

Examples

>>> # Create a curve from a function
>>> curve = Curve(lambda t: t + 1j*t**2, lambda t: 1 + 2j*t)

Methods

Name	Description
inv	Compute the inversion of the curve with respect to the origin.

inv

curves.Curve.inv()

Compute the inversion of the curve with respect to the origin.

Returns

Name	Type	Description
	Curve	Inverted curve

JuliaCurve

curves.JuliaCurve(julia_obj)

Base class for wrapping Julia curve objects from ComplexRegions.jl.

This class provides a Python interface to Julia curve objects, handling the conversion between Python and Julia types and providing access to geometric operations on curves.

Parameters

Name	Type	Description	Default
julia_obj	juliacall.AnyValue	A Julia curve object from ComplexRegions.jl	required

Attributes

Name	Type	Description
julia	juliacall.AnyValue	The underlying Julia curve object

Methods

Name	Description
arclength	Compute the arc length of the curve.
arg	Find the parameter value corresponding to point z on the curve.
closest	Find the closest point on the curve to z.
conj	Return the complex conjugate of the curve.
dist	Compute the distance from point z to the curve.
get	Get a field from the underlying Julia object.
intersect	Find intersection points with another curve.
inv	Compute the inversion of the curve with respect to the origin.
isapprox	Check if this curve is approximately equal to another.
isfinite	Check if the curve has finite length.
isleft	Check if point z is to the left of the curve.
ispositive	Check if the curve has positive orientation.
isreal	Check if the curve lies on the real axis.
isright	Check if point z is to the right of the curve.
normal	Compute the normal at parameter t.
point	Evaluate the curve at parameter value t.
reflect	Reflect point z across the curve.
reverse	Return the curve with reversed orientation.
tangent	Compute the tangent at parameter t.
unittangent	Compute the unit tangent at parameter t.

arclength

curves.JuliaCurve.arclength()

Compute the arc length of the curve.

Returns

Name	Type	Description
	float	Total arc length of the curve

arg

curves.JuliaCurve.arg(z)

Find the parameter value corresponding to point z on the curve.

Parameters

Name	Type	Description	Default
z	complex	Point to locate on the curve	required

Returns

Name	Type	Description
	float or None	Parameter value if z is on the curve, None otherwise

closest

curves.JuliaCurve.closest(z)

Find the closest point on the curve to z.

Parameters

Name	Type	Description	Default
z	complex	Reference point	required

Returns

Name	Type	Description
	complex	Closest point on the curve to z

conj

curves.JuliaCurve.conj()

Return the complex conjugate of the curve.

Returns

Name	Type	Description
	JuliaCurve	Complex conjugate of this curve

dist

curves.JuliaCurve.dist(z)

Compute the distance from point z to the curve.

Parameters

Name	Type	Description	Default
z	complex	Reference point	required

Returns

Name	Type	Description
	float	Distance from z to the closest point on the curve

get

curves.JuliaCurve.get(field)

Get a field from the underlying Julia object.

Parameters

Name	Type	Description	Default
field	str	Name of the field to retrieve	required

Returns

Name	Type	Description
	Any	The value of the requested field

intersect

curves.JuliaCurve.intersect(other)

Find intersection points with another curve.

Parameters

Name	Type	Description	Default
other	JuliaCurve	Another curve to intersect with	required

Returns

Name	Type	Description
	numpy.ndarray or JuliaCurve	Array of intersection points, or a curve if the intersection is a continuous curve segment

inv

curves.JuliaCurve.inv()

Compute the inversion of the curve with respect to the origin.

Returns

Name	Type	Description
	juliacall.AnyValue	Julia object representing the inverted curve

Notes

This method returns a raw Julia object. Subclasses should override this method to return properly wrapped Python objects.

isapprox

curves.JuliaCurve.isapprox(other)

Check if this curve is approximately equal to another.

Parameters

Name	Type	Description	Default
other	JuliaCurve	Another curve to compare with	required

Returns

Name	Type	Description
	bool	True if curves are approximately equal, False otherwise

isfinite

curves.JuliaCurve.isfinite()

Check if the curve has finite length.

Returns

Name	Type	Description
	bool	True if the curve is finite, False otherwise

isleft

curves.JuliaCurve.isleft(z)

Check if point z is to the left of the curve.

Parameters

Name	Type	Description	Default
z	complex	Point to test	required

Returns

Name	Type	Description
	bool	True if z is to the left of the curve, False otherwise

ispositive

curves.JuliaCurve.ispositive()

Check if the curve has positive orientation.

Returns

Name	Type	Description
	bool	True if positively oriented, False otherwise

isreal

curves.JuliaCurve.isreal()

Check if the curve lies on the real axis.

Returns

Name	Type	Description
	bool	True if the curve is real, False otherwise

isright

curves.JuliaCurve.isright(z)

Check if point z is to the right of the curve.

Parameters

Name	Type	Description	Default
z	complex	Point to test	required

Returns

Name	Type	Description
	bool	True if z is to the right of the curve, False otherwise

normal

curves.JuliaCurve.normal(t)

Compute the normal at parameter t.

Parameters

Name	Type	Description	Default
t	float	Parameter value	required

Returns

Name	Type	Description
	complex	Normal at parameter t

point

curves.JuliaCurve.point(t)

Evaluate the curve at parameter value t.

Parameters

Name	Type	Description	Default
t	float	Parameter value, typically in [0, 1]	required

Returns

Name	Type	Description
	complex	Point on the curve at parameter t

reflect

curves.JuliaCurve.reflect(z)

Reflect point z across the curve.

Parameters

Name	Type	Description	Default
z	complex	Point to reflect	required

Returns

Name	Type	Description
	complex	Reflected point

reverse

curves.JuliaCurve.reverse()

Return the curve with reversed orientation.

Returns

Name	Type	Description
	JuliaCurve	Curve with reversed orientation

tangent

curves.JuliaCurve.tangent(t=0.0)

Compute the tangent at parameter t.

Parameters

Name	Type	Description	Default
t	float	Parameter value, default is 0	0.0

Returns

Name	Type	Description
	complex	Tangent at parameter t

unittangent

curves.JuliaCurve.unittangent(t=0.0)

Compute the unit tangent at parameter t.

Parameters

Name	Type	Description	Default
t	float	Parameter value, default is 0	0.0

Returns

Name	Type	Description
	complex	Unit tangent at parameter t

Line

curves.Line(a, b=None, direction=None)

An infinite straight line in the complex plane.

A line can be constructed from two points, or from a point and a direction. Lines have infinite arc length and are always considered to have positive orientation.

Parameters

Name	Type	Description	Default
a	complex or juliacall.AnyValue	Either a point on the line, or a Julia Line object to wrap	required
b	complex	Second point on the line (if constructing from two points)	None
direction	complex	Direction vector of the line (if constructing from point and direction)	None

Attributes

Name	Type	Description
base	complex	A base point on the line
direction	complex	Direction vector of the line

Examples

>>> # Line through two points
>>> line1 = Line(0, 1+1j)
>>> # Line through origin with given direction
>>> line2 = Line(0, direction=1+1j)

Methods

Name	Description
angle	Get the angle of the line’s direction vector.
arclength	Return infinite arc length for lines.
inv	Compute the inversion of the line with respect to the origin.
isfinite	Check if the line is finite.
ispositive	Check if the line has positive orientation.
slope	Get the slope of the line.

angle

curves.Line.angle()

Get the angle of the line’s direction vector.

Returns

Name	Type	Description
	float	Angle in radians

arclength

curves.Line.arclength()

Return infinite arc length for lines.

Returns

Name	Type	Description
	float	Always returns numpy.inf

inv

curves.Line.inv()

Compute the inversion of the line with respect to the origin.

Returns

Name	Type	Description
	Circle or Line	Circle if the line doesn’t pass through the origin, Line otherwise

isfinite

curves.Line.isfinite()

Check if the line is finite.

Returns

Name	Type	Description
	bool	Always returns False for lines

ispositive

curves.Line.ispositive()

Check if the line has positive orientation.

Returns

Name	Type	Description
	bool	Always returns True for lines

slope

curves.Line.slope()

Get the slope of the line.

Returns

Name	Type	Description
	float	Slope of the line (dy/dx)

Ray

curves.Ray(base, angle=None)

A semi-infinite ray in the complex plane.

A ray starts at a base point and extends infinitely in a given direction.

Parameters

Name	Type	Description	Default
base	complex or juliacall.AnyValue	Starting point of the ray or Julia Ray object to wrap	required
angle	float	Angle of the ray direction in radians	None

Attributes

Name	Type	Description
base	complex	Starting point of the ray
angle	float	Angle of the ray direction

Examples

>>> # Ray from origin at 45 degrees
>>> ray = Ray(0, np.pi/4)

Segment

curves.Segment(a, b=None)

A line segment in the complex plane.

A segment is a finite portion of a line connecting two endpoints.

Parameters

Name	Type	Description	Default
a	complex or juliacall.AnyValue	First endpoint or Julia Segment object to wrap	required
b	complex	Second endpoint (if a is first endpoint)	None

Attributes

Name	Type	Description
first	complex	First endpoint of the segment
last	complex	Last endpoint of the segment

Examples

>>> # Segment from origin to (1,1)
>>> seg = Segment(0, 1+1j)

Methods

Name	Description
inv	Compute the inversion of the segment with respect to the origin.

inv

curves.Segment.inv()

Compute the inversion of the segment with respect to the origin.

Returns

Name	Type	Description
	Arc, Ray, or Segment	The type depends on the geometry after inversion

Functions

Name	Description
wrap_jl_curve	Wrap a Julia curve object in the appropriate Python class.

wrap_jl_curve

curves.wrap_jl_curve(jul)

Wrap a Julia curve object in the appropriate Python class.

Parameters

Name	Type	Description	Default
jul	juliacall.AnyValue	A Julia curve object from ComplexRegions.jl	required

Returns

Name	Type	Description
	Curve	The appropriate Python curve wrapper (Circle, Arc, Line, Segment, Ray, etc.)

Raises

Name	Type	Description
	ValueError	If the argument is not a Julia object or not a recognized curve type