{{em}在这里举例说明了{em}在一些给定的维数之间的导数的用法。在import numpy as np

from scipy import interpolate

def sampleCubicSplinesWithDerivative(points, tangents, resolution):

'''

Compute and sample the cubic splines for a set of input points with

optional information about the tangent (direction AND magnitude). The

splines are parametrized along the traverse line (piecewise linear), with

the resolution being the step size of the parametrization parameter.

The resulting samples have NOT an equidistant spacing.

Arguments: points: a list of n-dimensional points

tangents: a list of tangents

resolution: parametrization step size

Returns: samples

Notes: Lists points and tangents must have equal length. In case a tangent

is not specified for a point, just pass None. For example:

points = [[0,0], [1,1], [2,0]]

tangents = [[1,1], None, [1,-1]]

'''

resolution = float(resolution)

points = np.asarray(points)

nPoints, dim = points.shape

# Parametrization parameter s.

dp = np.diff(points, axis=0) # difference between points

dp = np.linalg.norm(dp, axis=1) # distance between points

d = np.cumsum(dp) # cumsum along the segments

d = np.hstack([[0],d]) # add distance from first point

l = d[-1] # length of point sequence

nSamples = int(l/resolution) # number of samples

s,r = np.linspace(0,l,nSamples,retstep=True) # sample parameter and step

# Bring points and (optional) tangent information into correct format.

assert(len(points) == len(tangents))

data = np.empty([nPoints, dim], dtype=object)

for i,p in enumerate(points):

t = tangents[i]

# Either tangent is None or has the same

# number of dimensions as the point p.

assert(t is None or len(t)==dim)

fuse = list(zip(p,t) if t is not None else zip(p,))

data[i,:] = fuse

# Compute splines per dimension separately.

samples = np.zeros([nSamples, dim])

for i in range(dim):

poly = interpolate.BPoly.from_derivatives(d, data[:,i])

samples[:,i] = poly(s)

return samples

为了演示此函数的用法，我们指定点和切线。该示例进一步演示了切线的“幅值”更改时的效果。在

^{pr2}$

结果如下：

有三点需要注意：以下内容也适用于两个以上的维度。在

样本之间的间距不是固定的。实现等距采样的一种简单方法是在返回的samples之间进行线性插值，正如在this post中所讨论的那样。在

切线的指定是可选的，但是BPoly.from_derivatives不能确保在此位置的样条曲线之间的平滑过渡。例如，如果上述示例中的tangents[1]设置为None，sampleCubicSplinesWithDerivative(points, tangents, resolution)，则结果如下：