raygeo.geo.shape.bezier

Cubic bezier curve queries and conversions.

Provides point evaluation at a parameter t, splitting into two halves, bounding rectangle computation, flattening to line segments (both fixed-step and adaptive subdivision), rectangle clipping, flatness testing, perpendicular distance measurement, and conversion from cubic to quadratic form.

Functions

clip_bezier_with_rect()

clip_bezier_with_rect(
    p0: types.Point,
    p1: types.Point,
    p2: types.Point,
    p3: types.Point,
    rect: types.Rect,
) -> list[tuple[types.Point, types.Point, types.Point, types.Point]]

Clip a cubic bezier with a rectangle.

Parameter	Type	Description
p0	types.Point	Start control point (x, y).
p1	types.Point	First control point (x, y).
p2	types.Point	Second control point (x, y).
p3	types.Point	End control point (x, y).
rect	types.Rect	Clipping rectangle (x_min, y_min, x_max, y_max).
Returns	list[tuple[types.Point, types.Point, types.Point, types.Point]]	List of bezier segments inside the rectangle.
Complexity		O(n)

convert_cubic_bezier_to_quadratic()

convert_cubic_bezier_to_quadratic(
    p0: types.Point,
    p1: types.Point,
    p2: types.Point,
    p3: types.Point,
) -> tuple[types.Point, types.Point, types.Point]

Convert a cubic bezier to a quadratic bezier.

Parameter	Type	Description
p0	types.Point	Start control point (x, y).
p1	types.Point	First control point (x, y).
p2	types.Point	Second control point (x, y).
p3	types.Point	End control point (x, y).
Returns	tuple[types.Point, types.Point, types.Point]	Quadratic bezier (p0, p1, p2).
Complexity		O(1)

fit_cubic_bezier()

fit_cubic_bezier(
    points: list[tuple[float, float]],
) -> Optional[types.CubicBezier]

Fit a cubic Bezier curve to a sequence of points (least-squares).

Parameter	Type	Description
points	list[tuple[float, float]]	List of (x, y) points.
Returns	Optional[types.CubicBezier]	(p0, c1, c2, p3) or None if fewer than 2 points.

Cubic Bezier curves fitted to sample points — sine, cosine hump, quarter-circle, and shallow wave

flatten_bezier()

flatten_bezier(
    p0: types.Point3D,
    p1: types.Point3D,
    p2: types.Point3D,
    p3: types.Point3D,
    tolerance: float,
    max_subdivisions: int,
    pts: list,
) -> None

Flatten a bezier curve into points.

Parameter	Type	Description
p0	types.Point3D	Start control point (x, y, z).
p1	types.Point3D	First control point (x, y, z).
p2	types.Point3D	Second control point (x, y, z).
p3	types.Point3D	End control point (x, y, z).
tolerance	float	Flattening tolerance.
max_subdivisions	int	Maximum recursion depth.
pts	list	Output list to append points to.
Returns	None	List of (x, y) points along the flattened bezier.
Complexity		O(n)

Bezier flattening: adaptive subdivision at varied tolerances

get_bezier_bounds()

get_bezier_bounds(
    p0: types.Point,
    p1: types.Point,
    p2: types.Point,
    p3: types.Point,
) -> types.Rect

Get the bounding rectangle of a cubic bezier.

Parameter	Type	Description
p0	types.Point	Start control point (x, y).
p1	types.Point	First control point (x, y).
p2	types.Point	Second control point (x, y).
p3	types.Point	End control point (x, y).
Returns	types.Rect	Bounding rectangle as (x_min, y_min, x_max, y_max).
Complexity		O(1)

get_bezier_flatness_sq()

get_bezier_flatness_sq(
    a: types.Point3D,
    b: types.Point3D,
    c: types.Point3D,
    d: types.Point3D,
) -> float

Compute the flatness squared of a cubic bezier.

Parameter	Type	Description
a	types.Point3D	Start point (x, y, z).
b	types.Point3D	First control point (x, y, z).
c	types.Point3D	Second control point (x, y, z).
d	types.Point3D	End point (x, y, z).
Returns	float	Flatness squared value.
Complexity		O(1)

get_bezier_length()

get_bezier_length(
    p0: types.Point,
    c1: types.Point,
    c2: types.Point,
    p1: types.Point,
) -> float

Compute the arc length of a cubic Bezier curve.

Parameter	Type	Description
p0	types.Point	Start point (x, y).
c1	types.Point	First control point (x, y).
c2	types.Point	Second control point (x, y).
p1	types.Point	End point (x, y).
Returns	float	Arc length.
Complexity		O(n)

get_bezier_point_at()

get_bezier_point_at(
    p0: types.Point,
    p1: types.Point,
    p2: types.Point,
    p3: types.Point,
    t: float,
) -> types.Point

Get a point on a cubic bezier at parameter t.

Parameter	Type	Description
p0	types.Point	Start control point (x, y).
p1	types.Point	First control point (x, y).
p2	types.Point	Second control point (x, y).
p3	types.Point	End control point (x, y).
t	float	Parameter value (0..1).
Returns	types.Point	Point on the bezier curve (x, y).
Complexity		O(1)

Bezier point evaluation at parameter t

get_bezier_rect_intersections()

get_bezier_rect_intersections(
    p0: types.Point,
    p1: types.Point,
    p2: types.Point,
    p3: types.Point,
    rect: types.Rect,
) -> list[float]

Get intersection t-values of a bezier with a rectangle.

Parameter	Type	Description
p0	types.Point	Start control point (x, y).
p1	types.Point	First control point (x, y).
p2	types.Point	Second control point (x, y).
p3	types.Point	End control point (x, y).
rect	types.Rect	Rectangle (x_min, y_min, x_max, y_max).
Returns	list[float]	List of t-values where the bezier intersects.
Complexity		O(n)

get_perpendicular_dist_sq()

get_perpendicular_dist_sq(
    pt: types.Point3D,
    origin: types.Point3D,
    vx: float,
    vy: float,
    vz: float = 0,
    norm_sq: float = 0,
) -> float

Compute the perpendicular distance squared.

Parameter	Type	Description
pt	types.Point3D	Point to measure from.
origin	types.Point3D	Origin of the line.
vx	float	X component of line direction.
vy	float	Y component of line direction.
vz	float = 0	Z component of line direction.
norm_sq	float = 0	Precomputed squared norm (optional).
Returns	float	Perpendicular distance squared.
Complexity		O(1)

is_bezier_flat()

is_bezier_flat(
    p0: types.Point,
    p1: types.Point,
    p2: types.Point,
    p3: types.Point,
    tolerance_sq: float,
) -> bool

Test whether a cubic bezier curve is flat enough to approximate with a line segment.

Uses a chord-distance flatness test. For non-degenerate curves (p0 != p3) it checks whether both control points lie within tolerance_sq of the chord line. For degenerate curves (p0 approx p3) it checks the control point distances from the start point.

Parameter	Type	Description
p0	types.Point	Start control point (x, y).
p1	types.Point	First control point (x, y).
p2	types.Point	Second control point (x, y).
p3	types.Point	End control point (x, y).
tolerance_sq	float	Squared tolerance for flatness.
Returns	bool	True if the curve is flat enough.
Complexity		O(1)

is_bezier_inside_polygons()

is_bezier_inside_polygons(
    p0: types.Point,
    p1: types.Point,
    p2: types.Point,
    p3: types.Point,
    polygons: Any,
) -> bool

Check if a bezier curve is inside a set of polygons.

Parameter	Type	Description
p0	types.Point	Start control point (x, y).
p1	types.Point	First control point (x, y).
p2	types.Point	Second control point (x, y).
p3	types.Point	End control point (x, y).
polygons	Any	List of polygons to check against.
Returns	bool	True if the bezier is inside all polygons.
Complexity		O(n * m)

linearize_bezier()

linearize_bezier(
    p0: types.Point3D,
    p1: types.Point3D,
    p2: types.Point3D,
    p3: types.Point3D,
    num_steps: int,
) -> list[tuple[types.Point3D, types.Point3D]]

Linearize a bezier into line segments.

Parameter	Type	Description
p0	types.Point3D	Start control point (x, y, z).
p1	types.Point3D	First control point (x, y, z).
p2	types.Point3D	Second control point (x, y, z).
p3	types.Point3D	End control point (x, y, z).
num_steps	int	Number of linearization steps.
Returns	list[tuple[types.Point3D, types.Point3D]]	List of (p1, p2) segment pairs.
Complexity		O(n)

linearize_bezier_adaptive()

linearize_bezier_adaptive(
    p0: types.Point,
    p1: types.Point,
    p2: types.Point,
    p3: types.Point,
    tolerance_sq: float,
    max_subdivisions: int = 20,
) -> types.Polygon

Adaptively linearize a bezier curve.

Parameter	Type	Description
p0	types.Point	Start control point (x, y).
p1	types.Point	First control point (x, y).
p2	types.Point	Second control point (x, y).
p3	types.Point	End control point (x, y).
tolerance_sq	float	Squared tolerance for subdivision.
max_subdivisions	int = 20	Maximum recursion depth.
Returns	types.Polygon	List of linearized points (x, y).
Complexity		O(n)

linearize_bezier_segment()

linearize_bezier_segment(
    p0: types.Point3D,
    p1: types.Point3D,
    p2: types.Point3D,
    p3: types.Point3D,
    tolerance: float = 0.1,
) -> list[types.Point3D]

Linearize a single bezier segment.

Parameter	Type	Description
p0	types.Point3D	Start control point (x, y, z).
p1	types.Point3D	First control point (x, y, z).
p2	types.Point3D	Second control point (x, y, z).
p3	types.Point3D	End control point (x, y, z).
tolerance	float = 0.1	Linearization tolerance.
Returns	list[types.Point3D]	List of linearized points (x, y, z).
Complexity		O(n)

nearest_tangent_circle_on_bezier()

nearest_tangent_circle_on_bezier(
    point: types.Point,
    bezier: types.CubicBezier,
    radius: float,
) -> Optional[tuple[types.Point, types.Point, float]]

Circle through point tangent to a cubic Bezier.

Parameter	Type	Description
point	types.Point	Point the circle must pass through.
bezier	types.CubicBezier	(p0, c1, c2, p3) control points.
radius	float	Circle radius.
Returns	Optional[tuple[types.Point, types.Point, float]]	(centre, tangent_point, t) or None.

split_bezier()

split_bezier(
    p0: types.Point,
    p1: types.Point,
    p2: types.Point,
    p3: types.Point,
    t: float,
) -> tuple[tuple[types.Point, types.Point, types.Point, types.Point], tuple[types.Point, types.Point, types.Point, types.Point]]

Split a cubic bezier at parameter t.

Parameter	Type	Description
p0	types.Point	Start control point (x, y).
p1	types.Point	First control point (x, y).
p2	types.Point	Second control point (x, y).
p3	types.Point	End control point (x, y).
t	float	Split parameter (0..1).
Returns	tuple[tuple[types.Point, types.Point, types.Point, types.Point], tuple[types.Point, types.Point, types.Point, types.Point]]	Two bezier curves (left, right).
Complexity		O(1)

Bezier split at parameter t