Это Куби c Кривая Безье и, как известно, Куб c Кривая Безье имеет четыре контрольные точки , как показано рисунок ниже: [P0, P1, P2, P3]. Согласно примеру кода Python, который я обнаружил, он дает начальную и конечную точки, и это меня смутило, и я не знаю, как он указал другие точки. I'd like to convert it from Cubic Bezier curve to quadratic Bezier curve. PS. Кривая Безье Quadrati c имеет три контрольных точки [P0, P1, P2]. Любая помощь, пожалуйста? Как я поставил точки, чтобы преобразовать его в Quadrati c Кривая Безье?
Причина преобразования в Quadrati c, мне нужно попробовать это с проблемами планирования пути.
Куби c Кривая Безье
Куби c Безье
Python код Безье
"""
Path planning with Bezier curve.
author: Atsushi Sakai(@Atsushi_twi)
"""
import matplotlib.pyplot as plt
import numpy as np
import scipy.special
show_animation = True
def calc_4points_bezier_path(sx, sy, syaw, ex, ey, eyaw, offset):
"""
Compute control points and path given start and end position.
:param sx: (float) x-coordinate of the starting point
:param sy: (float) y-coordinate of the starting point
:param syaw: (float) yaw angle at start
:param ex: (float) x-coordinate of the ending point
:param ey: (float) y-coordinate of the ending point
:param eyaw: (float) yaw angle at the end
:param offset: (float)
:return: (numpy array, numpy array)
"""
dist = np.sqrt((sx - ex) ** 2 + (sy - ey) ** 2) / offset
control_points = np.array(
[[sx, sy],
[sx + dist * np.cos(syaw), sy + dist * np.sin(syaw)],
[ex - dist * np.cos(eyaw), ey - dist * np.sin(eyaw)],
[ex, ey]])
path = calc_bezier_path(control_points, n_points=100)
return path, control_points
def calc_bezier_path(control_points, n_points=100):
"""
Compute bezier path (trajectory) given control points.
:param control_points: (numpy array)
:param n_points: (int) number of points in the trajectory
:return: (numpy array)
"""
traj = []
for t in np.linspace(0, 1, n_points):
traj.append(bezier(t, control_points))
return np.array(traj)
def bernstein_poly(n, i, t):
"""
Bernstein polynom.
:param n: (int) polynom degree
:param i: (int)
:param t: (float)
:return: (float)
"""
return scipy.special.comb(n, i) * t ** i * (1 - t) ** (n - i)
def bezier(t, control_points):
"""
Return one point on the bezier curve.
:param t: (float) number in [0, 1]
:param control_points: (numpy array)
:return: (numpy array) Coordinates of the point
"""
n = len(control_points) - 1
return np.sum([bernstein_poly(n, i, t) * control_points[i] for i in range(n + 1)], axis=0)
def bezier_derivatives_control_points(control_points, n_derivatives):
"""
Compute control points of the successive derivatives of a given bezier curve.
A derivative of a bezier curve is a bezier curve.
See https://pomax.github.io/bezierinfo/#derivatives
for detailed explanations
:param control_points: (numpy array)
:param n_derivatives: (int)
e.g., n_derivatives=2 -> compute control points for first and second derivatives
:return: ([numpy array])
"""
w = {0: control_points}
for i in range(n_derivatives):
n = len(w[i])
w[i + 1] = np.array([(n - 1) * (w[i][j + 1] - w[i][j])
for j in range(n - 1)])
return w
def curvature(dx, dy, ddx, ddy):
"""
Compute curvature at one point given first and second derivatives.
:param dx: (float) First derivative along x axis
:param dy: (float)
:param ddx: (float) Second derivative along x axis
:param ddy: (float)
:return: (float)
"""
return (dx * ddy - dy * ddx) / (dx ** 2 + dy ** 2) ** (3 / 2)
def plot_arrow(x, y, yaw, length=1.0, width=0.5, fc="r", ec="k"): # pragma: no cover
"""Plot arrow."""
if not isinstance(x, float):
for (ix, iy, iyaw) in zip(x, y, yaw):
plot_arrow(ix, iy, iyaw)
else:
plt.arrow(x, y, length * np.cos(yaw), length * np.sin(yaw),
fc=fc, ec=ec, head_width=width, head_length=width)
plt.plot(x, y)
def main():
"""Plot an example bezier curve."""
start_x = 10.0 # [m]
start_y = 1.0 # [m]
start_yaw = np.radians(180.0) # [rad]
end_x = -0.0 # [m]
end_y = -3.0 # [m]
end_yaw = np.radians(-45.0) # [rad]
offset = 3.0
path, control_points = calc_4points_bezier_path(
start_x, start_y, start_yaw, end_x, end_y, end_yaw, offset)
# Note: alternatively, instead of specifying start and end position
# you can directly define n control points and compute the path:
#control_points = np.array([[5., 1.], [-2.78, 1.], [-11.5, -4.5], [-6., -8.]])
#path = calc_bezier_path(control_points, n_points=100)
# Display the tangent, normal and radius of cruvature at a given point
t = 0.86 # Number in [0, 1]
x_target, y_target = bezier(t, control_points)
derivatives_cp = bezier_derivatives_control_points(control_points, 2)
point = bezier(t, control_points)
dt = bezier(t, derivatives_cp[1])
ddt = bezier(t, derivatives_cp[2])
# Radius of curvature
radius = 1 / curvature(dt[0], dt[1], ddt[0], ddt[1])
# Normalize derivative
dt /= np.linalg.norm(dt, 2)
tangent = np.array([point, point + dt])
normal = np.array([point, point + [- dt[1], dt[0]]])
curvature_center = point + np.array([- dt[1], dt[0]]) * radius
circle = plt.Circle(tuple(curvature_center), radius,
color=(0, 0.8, 0.8), fill=False, linewidth=1)
assert path.T[0][0] == start_x, "path is invalid"
assert path.T[1][0] == start_y, "path is invalid"
assert path.T[0][-1] == end_x, "path is invalid"
assert path.T[1][-1] == end_y, "path is invalid"
if show_animation: # pragma: no cover
fig, ax = plt.subplots()
ax.plot(path.T[0], path.T[1], label="Cubic Bezier Path")
ax.plot(control_points.T[0], control_points.T[1],
'--o', label="Control Points")
ax.plot(x_target, y_target)
ax.plot(tangent[:, 0], tangent[:, 1], label="Tangent")
ax.plot(normal[:, 0], normal[:, 1], label="Normal")
ax.add_artist(circle)
plot_arrow(start_x, start_y, start_yaw)
plot_arrow(end_x, end_y, end_yaw)
plt.xlabel('X')
plt.ylabel('Y')
ax.legend()
ax.axis("equal")
ax.grid(True)
for i, p in enumerate(control_points):
ax.annotate(f'P{i:d}', xy=p, xytext=(0,5), textcoords='offset points', ha='center')
plt.show()
if __name__ == '__main__':
main()