Inversions

Circle inversion transformations.

Circle inversion is a transformation that maps points inside a circle to points outside (and vice versa). Points on the circle of inversion map to themselves.

API Reference

geolet.primitives.inversion.api.InvertPoint

InvertPoint(point: PointT, *, respect_to: CircleT, label: str = '', label_dir: str = 'NE', color: str = 'black') -> InvertedPoint

Invert a point with respect to a circle.

Circle inversion is a transformation that maps points inside the circle to points outside (and vice versa), with the formula: P' = O + (r^2 / |P - O|^2) * (P - O)

Points on the circle map to themselves. The center of inversion maps to infinity (not representable).

Parameters:

Name	Type	Description	Default
point	PointT	The point to invert.	required
respect_to	CircleT	The circle of inversion (CircleWithRadius with center and radius).	required
label	str	Display name for the inverted point. Omit for hidden.	''
label_dir	str	Direction to place label relative to point.	'NE'
color	str	Asymptote color name.	'black'

Returns:

Type	Description
InvertedPoint	An inverted point drawable.

Example

O = Point("O", 0, 0) c = CircleWithRadius(O, 2) P = Point("P", 1, 0) P_inv = InvertPoint(P, respect_to=c, label="P'") # Maps to (4, 0)

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geolet.primitives.inversion.api.InvertLine

InvertLine(line: LineT | SegmentT, *, respect_to: CircleT, label: str = '', label_dir: str = 'N', label_pos: float = 0.0, color: str = 'black', style: str = 'solid', width: float = 1.0) -> CircleFromInvertedLine

Invert a line with respect to a circle.

Circle inversion of a line that does NOT pass through the center produces a circle passing through the center of inversion.

The returned object has a center_point() method to get a drawable center.

Note: This function handles the general case where the line does not pass through the center. For lines passing through the center of inversion (which map to themselves), use InvertLineThroughCenter instead.

Parameters:

Name	Type	Description	Default
line	LineT | SegmentT	The line to invert (Line or Segment). Must not pass through the center of the inversion circle.	required
respect_to	CircleT	The circle of inversion (CircleWithRadius with center and radius).	required
label	str	Display label for the resulting circle (empty string for no label).	''
label_dir	str	Direction for label placement (N, NE, E, SE, S, SW, W, NW).	'N'
label_pos	float	Relative position along path for label (0.0 = start, 1.0 = end).	0.0
color	str	Asymptote color name.	'black'
style	str	Line style. One of: solid, dashed, dotted.	'solid'
width	float	Line width in Asymptote units.	1.0

Returns:

Type	Description
CircleFromInvertedLine	A CircleFromInvertedLine with a center_point() method for the center.

Example

O = Point("O", 0, 0) c = CircleWithRadius(O, 2) l = Line(Point("P", 1, -2), Point("Q", 1, 2)) # Vertical line x=1 inverted = InvertLine(l, respect_to=c) # Circle through O O_prime = inverted.center_point(label="O'", color="red")

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geolet.primitives.inversion.api.InvertCircle

InvertCircle(target: CircleT, *, respect_to: CircleT, label: str = '', label_dir: str = 'N', label_pos: float = 0.0, color: str = 'black', style: str = 'solid', width: float = 1.0) -> InvertedCircle

Invert a circle with respect to another circle.

Circle inversion of a circle that does NOT pass through the center produces another circle.

Note: This function handles the general case where the target circle does not pass through the center of inversion. For circles passing through the center (which map to lines), use InvertCircleThroughCenter.

Parameters:

Name	Type	Description	Default
target	CircleT	The circle to invert (CircleWithRadius). Must not pass through the center of the inversion circle.	required
respect_to	CircleT	The circle of inversion (CircleWithRadius with center and radius).	required
label	str	Display label for the resulting circle (empty string for no label).	''
label_dir	str	Direction for label placement (N, NE, E, SE, S, SW, W, NW).	'N'
label_pos	float	Relative position along path for label (0.0 = start, 1.0 = end).	0.0
color	str	Asymptote color name.	'black'
style	str	Line style. One of: solid, dashed, dotted.	'solid'
width	float	Line width in Asymptote units.	1.0

Returns:

Type	Description
InvertedCircle	An inverted circle drawable.

Example

O = Point("O", 0, 0) inv_circle = CircleWithRadius(O, 3) c = CircleWithRadius(Point("C", 2, 0), 1) c_inv = InvertCircle(c, respect_to=inv_circle) # Another circle

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geolet.primitives.inversion.api.InvertLineThroughCenter

InvertLineThroughCenter(line: LineT | SegmentT, *, respect_to: CircleT, label: str = '', label_dir: str = 'N', label_pos: float = 0.0, color: str = 'black', style: str = 'solid', width: float = 1.0) -> InvertedLineThroughCenter

Invert a line that passes through the center of inversion.

A line through the center of inversion maps to itself under inversion. This function includes an Asymptote assertion that verifies the line actually passes through the center at render time.

Parameters:

Name	Type	Description	Default
line	LineT | SegmentT	The line to invert (Line or Segment). Must pass through the center of the inversion circle.	required
respect_to	CircleT	The circle of inversion (CircleWithRadius).	required
label	str	Display label for the line (empty string for no label).	''
label_dir	str	Direction for label placement (N, NE, E, SE, S, SW, W, NW).	'N'
label_pos	float	Relative position along path for label (0.0 = start, 1.0 = end).	0.0
color	str	Asymptote color name.	'black'
style	str	Line style. One of: solid, dashed, dotted.	'solid'
width	float	Line width in Asymptote units.	1.0

Returns:

Type	Description
InvertedLineThroughCenter	An inverted line drawable (same as original line).

Example

O = Point("O", 0, 0) c = CircleWithRadius(O, 2) l = Line(Point("P", -1, 0), Point("Q", 1, 0)) # Line through O l_inv = InvertLineThroughCenter(l, respect_to=c) # Same line

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geolet.primitives.inversion.api.InvertCircleThroughCenter

InvertCircleThroughCenter(target: CircleT, *, respect_to: CircleT, label: str = '', label_dir: str = 'N', label_pos: float = 0.0, color: str = 'black', style: str = 'solid', width: float = 1.0) -> InvertedCircleThroughCenter

Invert a circle that passes through the center of inversion.

A circle through the center of inversion maps to a line under inversion. The resulting line is perpendicular to the line connecting the centers and passes through the inverted point opposite to the center.

This function includes an Asymptote assertion that verifies the circle actually passes through the center at render time.

Parameters:

Name	Type	Description	Default
target	CircleT	The circle to invert (CircleWithRadius). Must pass through the center of the inversion circle.	required
respect_to	CircleT	The circle of inversion (CircleWithRadius).	required
label	str	Display label for the resulting line (empty string for no label).	''
label_dir	str	Direction for label placement (N, NE, E, SE, S, SW, W, NW).	'N'
label_pos	float	Relative position along path for label (0.0 = start, 1.0 = end).	0.0
color	str	Asymptote color name.	'black'
style	str	Line style. One of: solid, dashed, dotted.	'solid'
width	float	Line width in Asymptote units.	1.0

Returns:

Type	Description
InvertedCircleThroughCenter	An inverted circle drawable (draws as a line).

Example

O = Point("O", 0, 0) inv_circle = CircleWithRadius(O, 2)

Circle centered at (1, 0) with radius 1 passes through O

c = CircleWithRadius(Point("C", 1, 0), 1) c_inv = InvertCircleThroughCenter(c, respect_to=inv_circle)

Examples

Invert a Point

from geolet import Point, CircleWithRadius, Segment, InvertPoint, autofigure


@autofigure
def invert_point():
    O = Point("O", 0, 0, label_dir="SW")
    inv_circle = CircleWithRadius(O, 2, color="blue")

    # Point inside the circle
    P = Point("P", 1, 0, label_dir="S")

    # Its inverse is outside (at distance r²/d from center)
    P_inv = InvertPoint(P, respect_to=inv_circle, label="P'", label_dir="S")

    Segment(O, P_inv, style="dashed", color="gray")

Invert a Line

A line not passing through the center inverts to a circle through the center:

from geolet import Point, Line, CircleWithRadius, InvertLine, autofigure


@autofigure
def invert_line():
    O = Point("O", 0, 0, label_dir="SW")
    inv_circle = CircleWithRadius(O, 2, color="blue")

    # Vertical line x = 1
    P = Point("", 1, -2)
    Q = Point("", 1, 2)
    line = Line(P, Q, color="red")

    # Inverts to a circle through O
    InvertLine(line, respect_to=inv_circle, color="green")

Invert a Circle

A circle not passing through the center inverts to another circle:

from geolet import Point, CircleWithRadius, InvertCircle, autofigure


@autofigure
def invert_circle():
    O = Point("O", 0, 0, label_dir="SW")
    inv_circle = CircleWithRadius(O, 3, color="blue", style="dashed")

    # Circle not passing through O
    C = Point("C", 2, 0, label_dir="S")
    target = CircleWithRadius(C, 0.5, color="red")

    # Inverts to another circle
    InvertCircle(target, respect_to=inv_circle, color="green")

Circle Through Center Inverts to Line

from geolet import Point, CircleWithRadius, InvertCircleThroughCenter, autofigure


@autofigure
def circle_to_line():
    O = Point("O", 0, 0, label_dir="SW")
    inv_circle = CircleWithRadius(O, 2, color="blue", style="dashed")

    # Circle passing through O
    C = Point("C", 1, 0, label_dir="S")
    target = CircleWithRadius(C, 1, color="red")  # Touches O

    # Inverts to a line
    InvertCircleThroughCenter(target, respect_to=inv_circle, color="green")