337 lines
11 KiB
Python
337 lines
11 KiB
Python
# Copyright (c) 2010-2024 Manfred Moitzi
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# License: MIT License
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from __future__ import annotations
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from typing import (
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TYPE_CHECKING,
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Iterable,
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Iterator,
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Sequence,
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TypeVar,
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Generic,
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)
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import math
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# The pure Python implementation can't import from ._ctypes or ezdxf.math!
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from ._vector import Vec3, Vec2
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from ._matrix44 import Matrix44
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from ._construct import arc_angle_span_deg
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if TYPE_CHECKING:
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from ezdxf.math import UVec
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from ezdxf.math.ellipse import ConstructionEllipse
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__all__ = [
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"Bezier4P",
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"cubic_bezier_arc_parameters",
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"cubic_bezier_from_arc",
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"cubic_bezier_from_ellipse",
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]
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T = TypeVar("T", Vec2, Vec3)
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class Bezier4P(Generic[T]):
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"""Implements an optimized cubic `Bézier curve`_ for exact 4 control points.
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A `Bézier curve`_ is a parametric curve, parameter `t` goes from 0 to 1,
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where 0 is the first control point and 1 is the fourth control point.
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The class supports points of type :class:`Vec2` and :class:`Vec3` as input, the
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class instances are immutable.
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Args:
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defpoints: sequence of definition points as :class:`Vec2` or
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:class:`Vec3` compatible objects.
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"""
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__slots__ = ("_control_points", "_offset")
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def __init__(self, defpoints: Sequence[T]):
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if len(defpoints) != 4:
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raise ValueError("Four control points required.")
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point_type = defpoints[0].__class__
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if not point_type.__name__ in ("Vec2", "Vec3"): # Cython types!!!
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raise TypeError(f"invalid point type: {point_type.__name__}")
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# The start point is the curve offset
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offset: T = defpoints[0]
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self._offset: T = offset
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# moving the curve to the origin reduces floating point errors:
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self._control_points: tuple[T, ...] = tuple(p - offset for p in defpoints)
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@property
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def control_points(self) -> Sequence[T]:
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"""Control points as tuple of :class:`Vec3` or :class:`Vec2` objects."""
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# ezdxf optimization: p0 is always (0, 0, 0)
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p0, p1, p2, p3 = self._control_points
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offset = self._offset
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return offset, p1 + offset, p2 + offset, p3 + offset
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def tangent(self, t: float) -> T:
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"""Returns direction vector of tangent for location `t` at the
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Bèzier-curve.
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Args:
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t: curve position in the range ``[0, 1]``
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"""
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if not (0 <= t <= 1.0):
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raise ValueError("t not in range [0 to 1]")
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return self._get_curve_tangent(t)
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def point(self, t: float) -> T:
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"""Returns point for location `t` at the Bèzier-curve.
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Args:
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t: curve position in the range ``[0, 1]``
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"""
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if not (0 <= t <= 1.0):
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raise ValueError("t not in range [0 to 1]")
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return self._get_curve_point(t)
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def approximate(self, segments: int) -> Iterator[T]:
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"""Approximate `Bézier curve`_ by vertices, yields `segments` + 1
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vertices as ``(x, y[, z])`` tuples.
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Args:
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segments: count of segments for approximation
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"""
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if segments < 1:
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raise ValueError(segments)
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delta_t = 1.0 / segments
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cp = self.control_points
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yield cp[0]
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for segment in range(1, segments):
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yield self._get_curve_point(delta_t * segment)
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yield cp[3]
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def flattening(self, distance: float, segments: int = 4) -> Iterator[T]:
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"""Adaptive recursive flattening. The argument `segments` is the
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minimum count of approximation segments, if the distance from the center
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of the approximation segment to the curve is bigger than `distance` the
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segment will be subdivided.
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Args:
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distance: maximum distance from the center of the cubic (C3)
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curve to the center of the linear (C1) curve between two
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approximation points to determine if a segment should be
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subdivided.
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segments: minimum segment count
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"""
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stack: list[tuple[float, T]] = []
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dt: float = 1.0 / segments
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t0: float = 0.0
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t1: float
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cp = self.control_points
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start_point: T = cp[0]
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end_point: T
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yield start_point
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while t0 < 1.0:
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t1 = t0 + dt
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if math.isclose(t1, 1.0):
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end_point = cp[3]
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t1 = 1.0
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else:
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end_point = self._get_curve_point(t1)
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while True:
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mid_t: float = (t0 + t1) * 0.5
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mid_point: T = self._get_curve_point(mid_t)
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chk_point: T = start_point.lerp(end_point)
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d = chk_point.distance(mid_point)
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if d < distance:
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yield end_point
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t0 = t1
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start_point = end_point
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if stack:
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t1, end_point = stack.pop()
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else:
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break
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else:
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stack.append((t1, end_point))
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t1 = mid_t
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end_point = mid_point
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def _get_curve_point(self, t: float) -> T:
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# 1st control point (p0) is always (0, 0, 0)
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# => p0 * a is always (0, 0, 0)
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# add offset at last - it is maybe very large
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_, p1, p2, p3 = self._control_points
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t2 = t * t
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_1_minus_t = 1.0 - t
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# a = _1_minus_t_square * _1_minus_t
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b = 3.0 * _1_minus_t * _1_minus_t * t
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c = 3.0 * _1_minus_t * t2
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d = t2 * t
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return p1 * b + p2 * c + p3 * d + self._offset
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def _get_curve_tangent(self, t: float) -> T:
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# tangent vector is independent from offset location!
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# 1st control point (p0) is always (0, 0, 0)
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# => p0 * a is always (0, 0, 0)
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_, p1, p2, p3 = self._control_points
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t2 = t * t
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# a = -3.0 * (1.0 - t) ** 2
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b = 3.0 * (1.0 - 4.0 * t + 3.0 * t2)
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c = 3.0 * t * (2.0 - 3.0 * t)
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d = 3.0 * t2
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return p1 * b + p2 * c + p3 * d
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def approximated_length(self, segments: int = 128) -> float:
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"""Returns estimated length of Bèzier-curve as approximation by line
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`segments`.
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"""
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length = 0.0
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prev_point = None
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for point in self.approximate(segments):
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if prev_point is not None:
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length += prev_point.distance(point)
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prev_point = point
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return length
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def reverse(self) -> Bezier4P[T]:
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"""Returns a new Bèzier-curve with reversed control point order."""
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return Bezier4P(list(reversed(self.control_points)))
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def transform(self, m: Matrix44) -> Bezier4P[Vec3]:
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"""General transformation interface, returns a new :class:`Bezier4p`
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curve as a 3D curve.
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Args:
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m: 4x4 transformation :class:`Matrix44`
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"""
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defpoints = Vec3.generate(self.control_points)
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return Bezier4P(tuple(m.transform_vertices(defpoints)))
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def cubic_bezier_from_arc(
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center: UVec = (0, 0, 0),
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radius: float = 1,
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start_angle: float = 0,
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end_angle: float = 360,
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segments: int = 1,
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) -> Iterator[Bezier4P[Vec3]]:
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"""Returns an approximation for a circular 2D arc by multiple cubic
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Bézier-curves.
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Args:
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center: circle center as :class:`Vec3` compatible object
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radius: circle radius
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start_angle: start angle in degrees
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end_angle: end angle in degrees
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segments: count of Bèzier-curve segments, at least one segment for each
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quarter (90 deg), 1 for as few as possible.
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"""
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center_: Vec3 = Vec3(center)
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radius = float(radius)
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angle_span: float = arc_angle_span_deg(start_angle, end_angle)
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if abs(angle_span) < 1e-9:
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return
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s: float = start_angle
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start_angle = math.radians(s) % math.tau
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end_angle = math.radians(s + angle_span)
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while start_angle > end_angle:
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end_angle += math.tau
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for control_points in cubic_bezier_arc_parameters(start_angle, end_angle, segments):
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defpoints = [center_ + (p * radius) for p in control_points]
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yield Bezier4P(defpoints)
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PI_2: float = math.pi / 2.0
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def cubic_bezier_from_ellipse(
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ellipse: "ConstructionEllipse", segments: int = 1
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) -> Iterator[Bezier4P[Vec3]]:
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"""Returns an approximation for an elliptic arc by multiple cubic
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Bézier-curves.
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Args:
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ellipse: ellipse parameters as :class:`~ezdxf.math.ConstructionEllipse`
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object
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segments: count of Bèzier-curve segments, at least one segment for each
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quarter (π/2), 1 for as few as possible.
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"""
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param_span: float = ellipse.param_span
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if abs(param_span) < 1e-9:
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return
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start_angle: float = ellipse.start_param % math.tau
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end_angle: float = start_angle + param_span
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while start_angle > end_angle:
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end_angle += math.tau
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def transform(points: Iterable[Vec3]) -> Iterator[Vec3]:
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center = Vec3(ellipse.center)
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x_axis: Vec3 = ellipse.major_axis
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y_axis: Vec3 = ellipse.minor_axis
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for p in points:
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yield center + x_axis * p.x + y_axis * p.y
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for defpoints in cubic_bezier_arc_parameters(start_angle, end_angle, segments):
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yield Bezier4P(tuple(transform(defpoints)))
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# Circular arc to Bezier curve:
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# Source: https://stackoverflow.com/questions/1734745/how-to-create-circle-with-b%C3%A9zier-curves
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# Optimization: https://spencermortensen.com/articles/bezier-circle/
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# actual c = 0.5522847498307935 = 4.0/3.0*(sqrt(2)-1.0) and max. deviation of ~0.03%
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DEFAULT_TANGENT_FACTOR = 4.0 / 3.0 # 1.333333333333333333
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# optimal c = 0.551915024494 and max. deviation of ~0.02%
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OPTIMIZED_TANGENT_FACTOR = 1.3324407374108935
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# Not sure if this is the correct way to apply this optimization,
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# so i stick to the original version for now:
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TANGENT_FACTOR = DEFAULT_TANGENT_FACTOR
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def cubic_bezier_arc_parameters(
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start_angle: float, end_angle: float, segments: int = 1
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) -> Iterator[tuple[Vec3, Vec3, Vec3, Vec3]]:
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"""Yields cubic Bézier-curve parameters for a circular 2D arc with center
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at (0, 0) and a radius of 1 in the form of [start point, 1. control point,
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2. control point, end point].
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Args:
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start_angle: start angle in radians
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end_angle: end angle in radians (end_angle > start_angle!)
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segments: count of Bèzier-curve segments, at least one segment for each
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quarter (π/2)
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"""
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if segments < 1:
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raise ValueError("Invalid argument segments (>= 1).")
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delta_angle: float = end_angle - start_angle
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if delta_angle > 0:
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arc_count = max(math.ceil(delta_angle / math.pi * 2.0), segments)
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else:
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raise ValueError("Delta angle from start- to end angle has to be > 0.")
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segment_angle: float = delta_angle / arc_count
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tangent_length: float = TANGENT_FACTOR * math.tan(segment_angle / 4.0)
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angle: float = start_angle
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end_point: Vec3 = Vec3.from_angle(angle)
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for _ in range(arc_count):
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start_point = end_point
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angle += segment_angle
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end_point = Vec3.from_angle(angle)
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control_point_1 = start_point + (
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-start_point.y * tangent_length,
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start_point.x * tangent_length,
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)
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control_point_2 = end_point + (
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end_point.y * tangent_length,
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-end_point.x * tangent_length,
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)
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yield start_point, control_point_1, control_point_2, end_point
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