{{em}在這里舉例說明了{em}在一些給定的維數之間的導數的用法。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

為了演示此函數的用法，我們指定點和切線。該示例進一步演示了切線的“幅值”更改時的效果。在

^{pr2}$

結果如下：

有三點需要注意：以下內容也適用于兩個以上的維度。在

樣本之間的間距不是固定的。實現等距采樣的一種簡單方法是在返回的samples之間進行線性插值，正如在this post中所討論的那樣。在

切線的指定是可選的，但是BPoly.from_derivatives不能確保在此位置的樣條曲線之間的平滑過渡。例如，如果上述示例中的tangents[1]設置為None，sampleCubicSplinesWithDerivative(points, tangents, resolution)，則結果如下：

總結

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