105 lines
3.5 KiB
Python
105 lines
3.5 KiB
Python
import numpy
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from numpy.typing import ArrayLike, NDArray
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from numpy import pi
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try:
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from numpy import trapezoid
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except ImportError:
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from numpy import trapz as trapezoid
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def bezier(
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nodes: ArrayLike,
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tt: ArrayLike,
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weights: ArrayLike | None = None,
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) -> NDArray[numpy.float64]:
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"""
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Sample a Bezier curve with the provided control points at the parametrized positions `tt`.
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Using the calculation method from arXiv:1803.06843, Chudy and Woźny.
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Args:
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nodes: `[[x0, y0], ...]` control points for the Bezier curve
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tt: Parametrized positions at which to sample the curve (1D array with values in the interval [0, 1])
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weights: Control point weights; if provided, length should be the same as number of control points.
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Default 1 for all control points.
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Returns:
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`[[x0, y0], [x1, y1], ...]` corresponding to `[tt0, tt1, ...]`
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"""
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nodes = numpy.asarray(nodes)
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tt = numpy.asarray(tt)
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nn = nodes.shape[0]
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weights = numpy.ones(nn) if weights is None else numpy.asarray(weights)
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with numpy.errstate(divide='ignore'):
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umul = (tt / (1 - tt)).clip(max=1)
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udiv = ((1 - tt) / tt).clip(max=1)
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hh = numpy.ones((tt.size,))
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qq = nodes[None, 0, :] * hh[:, None]
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for kk in range(1, nn):
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hh *= umul * (nn - kk) * weights[kk]
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hh /= kk * udiv * weights[kk - 1] + hh
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qq *= 1.0 - hh[:, None]
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qq += hh[:, None] * nodes[None, kk, :]
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return qq
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def euler_bend(
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switchover_angle: float,
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num_points: int = 200,
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) -> NDArray[numpy.float64]:
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"""
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Generate a 90 degree Euler bend (AKA Clothoid bend or Cornu spiral).
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Args:
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switchover_angle: After this angle, the bend will transition into a circular arc
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(and transition back to an Euler spiral on the far side). If this is set to
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`>= pi / 4`, no circular arc will be added.
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num_points: Number of points in the curve
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Returns:
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`[[x0, y0], ...]` for the curve
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"""
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ll_max = numpy.sqrt(2 * switchover_angle) # total length of (one) spiral portion
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ll_tot = 2 * ll_max + (pi / 2 - 2 * switchover_angle)
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num_points_spiral = numpy.floor(ll_max / ll_tot * num_points).astype(int)
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num_points_arc = num_points - 2 * num_points_spiral
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def gen_spiral(ll_max: float):
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xx = []
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yy = []
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for ll in numpy.linspace(0, ll_max, num_points_spiral):
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qq = numpy.linspace(0, ll, 1000) # integrate to current arclength
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xx.append(trapezoid( numpy.cos(qq * qq / 2), qq))
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yy.append(trapezoid(-numpy.sin(qq * qq / 2), qq))
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xy_part = numpy.stack((xx, yy), axis=1)
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return xy_part
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xy_spiral = gen_spiral(ll_max)
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xy_parts = [xy_spiral]
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if switchover_angle < pi / 4:
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# Build a circular segment to join the two euler portions
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rmin = 1.0 / ll_max
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half_angle = pi / 4 - switchover_angle
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qq = numpy.linspace(half_angle * 2, 0, num_points_arc + 1) + switchover_angle
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xc = rmin * numpy.cos(qq)
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yc = rmin * numpy.sin(qq) + xy_spiral[-1, 1]
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xc += xy_spiral[-1, 0] - xc[0]
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yc += xy_spiral[-1, 1] - yc[0]
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xy_parts.append(numpy.stack((xc[1:], yc[1:]), axis=1))
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endpoint_xy = xy_parts[-1][-1, :]
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second_spiral = xy_spiral[::-1, ::-1] + endpoint_xy - xy_spiral[-1, ::-1]
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xy_parts.append(second_spiral)
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xy = numpy.concatenate(xy_parts)
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# Remove any 2x-duplicate points
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xy = xy[(numpy.roll(xy, 1, axis=0) != xy).any(axis=1)]
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return xy
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