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115 lines
4.2 KiB
Python
115 lines
4.2 KiB
Python
# https://en.wikipedia.org/wiki/B%C3%A9zier_curve
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# https://www.tutorialspoint.com/computer_graphics/computer_graphics_curves.htm
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from __future__ import annotations
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from scipy.special import comb
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class BezierCurve:
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"""
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Bezier curve is a weighted sum of a set of control points.
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Generate Bezier curves from a given set of control points.
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This implementation works only for 2d coordinates in the xy plane.
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"""
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def __init__(self, list_of_points: list[tuple[float, float]]):
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"""
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list_of_points: Control points in the xy plane on which to interpolate. These
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points control the behavior (shape) of the Bezier curve.
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"""
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self.list_of_points = list_of_points
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# Degree determines the flexibility of the curve.
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# Degree = 1 will produce a straight line.
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self.degree = len(list_of_points) - 1
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def basis_function(self, t: float) -> list[float]:
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"""
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The basis function determines the weight of each control point at time t.
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t: time value between 0 and 1 inclusive at which to evaluate the basis of
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the curve.
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returns the x, y values of basis function at time t
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>>> curve = BezierCurve([(1,1), (1,2)])
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>>> [float(x) for x in curve.basis_function(0)]
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[1.0, 0.0]
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>>> [float(x) for x in curve.basis_function(1)]
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[0.0, 1.0]
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"""
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assert 0 <= t <= 1, "Time t must be between 0 and 1."
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output_values: list[float] = []
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for i in range(len(self.list_of_points)):
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# basis function for each i
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output_values.append(
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comb(self.degree, i) * ((1 - t) ** (self.degree - i)) * (t**i)
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)
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# the basis must sum up to 1 for it to produce a valid Bezier curve.
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assert round(sum(output_values), 5) == 1
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return output_values
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def bezier_curve_function(self, t: float) -> tuple[float, float]:
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"""
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The function to produce the values of the Bezier curve at time t.
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t: the value of time t at which to evaluate the Bezier function
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Returns the x, y coordinates of the Bezier curve at time t.
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The first point in the curve is when t = 0.
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The last point in the curve is when t = 1.
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>>> curve = BezierCurve([(1,1), (1,2)])
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>>> tuple(float(x) for x in curve.bezier_curve_function(0))
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(1.0, 1.0)
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>>> tuple(float(x) for x in curve.bezier_curve_function(1))
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(1.0, 2.0)
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"""
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assert 0 <= t <= 1, "Time t must be between 0 and 1."
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basis_function = self.basis_function(t)
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x = 0.0
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y = 0.0
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for i in range(len(self.list_of_points)):
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# For all points, sum up the product of i-th basis function and i-th point.
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x += basis_function[i] * self.list_of_points[i][0]
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y += basis_function[i] * self.list_of_points[i][1]
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return (x, y)
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def plot_curve(self, step_size: float = 0.01):
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"""
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Plots the Bezier curve using matplotlib plotting capabilities.
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step_size: defines the step(s) at which to evaluate the Bezier curve.
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The smaller the step size, the finer the curve produced.
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"""
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from matplotlib import pyplot as plt
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to_plot_x: list[float] = [] # x coordinates of points to plot
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to_plot_y: list[float] = [] # y coordinates of points to plot
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t = 0.0
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while t <= 1:
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value = self.bezier_curve_function(t)
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to_plot_x.append(value[0])
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to_plot_y.append(value[1])
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t += step_size
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x = [i[0] for i in self.list_of_points]
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y = [i[1] for i in self.list_of_points]
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plt.plot(
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to_plot_x,
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to_plot_y,
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color="blue",
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label="Curve of Degree " + str(self.degree),
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)
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plt.scatter(x, y, color="red", label="Control Points")
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plt.legend()
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plt.show()
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if __name__ == "__main__":
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import doctest
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doctest.testmod()
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BezierCurve([(1, 2), (3, 5)]).plot_curve() # degree 1
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BezierCurve([(0, 0), (5, 5), (5, 0)]).plot_curve() # degree 2
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BezierCurve([(0, 0), (5, 5), (5, 0), (2.5, -2.5)]).plot_curve() # degree 3
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