Python iles与旧的飞机自动测量系统相比有很多缺陷）数据不稳定，需要一些平滑处理（这就是我的原始代码，顺便说一句，这是我的第一个商业产品：）。。。很高兴能帮上忙。。。 x_Coordinates = [1, 0.9, 0.7, 0.5, 0.3,

iles与旧的飞机自动测量系统相比有很多缺陷）数据不稳定，需要一些平滑处理（这就是我的原始代码，顺便说一句，这是我的第一个商业产品：）。。。很高兴能帮上忙。。。

x_Coordinates = [1,   0.9,   0.7,   0.5,   0.3,   0.1, 0.0, ...] 
y_Coordinates = [0, -0.02, -0.06, -0.08, -0.10, -0.05, 0.0, ...]

(1.0, 0.95, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.25, 0.2, 0.15, 0.1, 0.075, 0.05, 0.025, 0.0125, 0.005, 0.0, 0.005, 0.0125, 0.025, 0.05, 0.075, 0.1, 0.15, 0.2, 0.25, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 1.0)
(0.00095, 0.00605, 0.01086, 0.01967, 0.02748, 0.03423, 0.03971, 0.04352, 0.04501, 0.04456, 0.04303, 0.04009, 0.03512, 0.0315, 0.02666, 0.01961, 0.0142, 0.0089, 0.0, -0.0089, -0.0142, -0.01961, -0.02666, -0.0315, -0.03512, -0.04009, -0.04303, -0.04456, -0.04501, -0.04352, -0.03971, -0.03423, -0.02748, -0.01967, -0.01086, -0.00605, -0.00095)

dx(i)=x(i)-x(i-1)

    List<double> pnt;   // outline 2D pnts = {x0,y0,x1,y1,x2,y2,...}
    List<double> axis;  // axis line 2D pnts = {x0,y0,x1,y1,x2,y2,...}
    int ix0,ix1;        // edge points

void compute()
    {
    int i,i0,i1;
    double d,dd;
    double *p0,*p1,*p2;
    double x0,x1,y0,y1;
    List<double> axis0,axis1;

    // find leading edge point
    ix0=0; ix1=0;
    for (p0=pnt.dat,p1=p0+2,p2=p1+2,i=2;i<pnt.num;i+=2,p0=p1,p1=p2,p2+=2)
     if ((p1[0]-p0[0])*(p2[0]-p1[0])<=0.0) { ix1=i; break; }
    // find axis0: midpoint of i0=(0-ix1) i1=find closest from (ix1,pnt.num)
    for (i0=2,i1=pnt.num-2;i0<ix1-2;i0+=2)
        {
        x0=pnt[i0+0];
        y0=pnt[i0+1];
        for (d=-1.0,i=i1;i>ix1+2;i-=2)
            {
            x1=pnt[i+0];
            y1=pnt[i+1];
            dd=((x1-x0)*(x1-x0))+((y1-y0)*(y1-y0));
            if ((d<0.0)||(dd<=d)) { i1=i; d=dd; }
            }
        if (d>=0.0)
            {
            x1=pnt[i1+0];
            y1=pnt[i1+1];
            axis0.add(0.5*(x0+x1));
            axis0.add(0.5*(y0+y1));
            }
        }
    // find axis1: midpoint of i0=(ix1,pnt.num) i1=find closest from (0,ix1)
    for (i1=2,i0=pnt.num-2;i0>ix1+2;i0-=2)
        {
        x0=pnt[i0+0];
        y0=pnt[i0+1];
        for (d=-1.0,i=i1;i<ix1-2;i+=2)
            {
            x1=pnt[i+0];
            y1=pnt[i+1];
            dd=((x1-x0)*(x1-x0))+((y1-y0)*(y1-y0));
            if ((d<0.0)||(dd<=d)) { i1=i; d=dd; }
            }
        if (d>=0.0)
            {
            x1=pnt[i1+0];
            y1=pnt[i1+1];
            axis1.add(0.5*(x0+x1));
            axis1.add(0.5*(y0+y1));
            }
        }
    // find axis: midpoint of i0=<0-axis0.num) i1=find closest from <0-axis1.num)
    axis.add(pnt[ix0+0]);
    axis.add(pnt[ix0+1]);
    for (i0=0,i1=0;i0<axis0.num;i0+=2)
        {
        x0=axis0[i0+0];
        y0=axis0[i0+1];
        for (d=-1.0,i=i1;i<axis1.num;i+=2)
            {
            x1=axis1[i+0];
            y1=axis1[i+1];
            dd=((x1-x0)*(x1-x0))+((y1-y0)*(y1-y0));
            if ((d<0.0)||(dd<=d)) { i1=i; d=dd; }
            }
        if (d>=0.0)
            {
            x1=axis1[i1+0];
            y1=axis1[i1+1];
            axis.add(0.5*(x0+x1));
            axis.add(0.5*(y0+y1));
            }
        }
    axis.add(pnt[ix1+0]);
    axis.add(pnt[ix1+1]);
    }

void compute()
    {
    int i,i0,i1,ii,n=4;
    double d,dd;
    double x0,x1,y0,y1;
    List<double> axis0,axis1;
    ix0=0; ix1=0;

    // find axis0: midpoint of i0=(0-ix1) i1=find closest from (ix1,pnt.num)
    for (i0=0,i1=pnt.num-2;i0+n<i1;i0+=2)
        {
        x0=pnt[i0+0];
        y0=pnt[i0+1];
        i=i1+n; if (i>pnt.num-2) i=pnt.num-2; ii=i1;
        for (d=-1.0;i>i0+n;i-=2)
            {
            x1=pnt[i+0];
            y1=pnt[i+1];
            dd=((x1-x0)*(x1-x0))+((y1-y0)*(y1-y0));
            if ((d<0.0)||((dd<=d)&&(dd>1e-10))) { i1=i; d=dd; }
            if ((d>=0.0)&&(dd>d)) break;
            }
        if (d>=0.0)
            {
            if (i1-i0<=n+2) { i1=ii; break; } // stop if non valid closest point found
            x1=pnt[i1+0];
            y1=pnt[i1+1];
            axis0.add(0.5*(x0+x1));
            axis0.add(0.5*(y0+y1));
            }
        }
    // find leading edge point (the farest point from last found axis point)
    x0=axis0[axis0.num-2];
    y0=axis0[axis0.num-1];
    for (d=0.0,i=i0;i<=i1;i+=2)
        {
        x1=pnt[i+0];
        y1=pnt[i+1];
        dd=((x1-x0)*(x1-x0))+((y1-y0)*(y1-y0));
        if (dd>d) { ix1=i; d=dd; }
        }
    axis0.add(pnt[ix1+0]);
    axis0.add(pnt[ix1+1]);

    // find axis1: midpoint of i0=(ix1,pnt.num) i1=find closest from (0,ix1)
    for (i1=0,i0=pnt.num-2;i0+n>i1;i0-=2)
        {
        x0=pnt[i0+0];
        y0=pnt[i0+1];
        i=i1-n; if (i<0) i=0; ii=i1;
        for (d=-1.0;i<i0-n;i+=2)
            {
            x1=pnt[i+0];
            y1=pnt[i+1];
            dd=((x1-x0)*(x1-x0))+((y1-y0)*(y1-y0));
            if ((d<0.0)||((dd<=d)&&(dd>1e-10))) { i1=i; d=dd; }
            if ((d>=0.0)&&(dd>d)) break;
            }
        if (d>=0.0)
            {
            if (i0-i1<=n+2) { i1=ii; break; } // stop if non valid closest point found
            x1=pnt[i1+0];
            y1=pnt[i1+1];
            axis1.add(0.5*(x0+x1));
            axis1.add(0.5*(y0+y1));
            }
        }
    // find leading edge point (the farest point from last found axis point)
    x0=axis1[axis1.num-2];
    y0=axis1[axis1.num-1];
    for (d=0.0,i=i1;i<=i0;i+=2)
        {
        x1=pnt[i+0];
        y1=pnt[i+1];
        dd=((x1-x0)*(x1-x0))+((y1-y0)*(y1-y0));
        if (dd>d) { ix1=i; d=dd; }
        }
    axis1.add(pnt[ix1+0]);
    axis1.add(pnt[ix1+1]);

    // find axis: midpoint of i0=<0-axis0.num) i1=find closest from <0-axis1.num)
    for (i0=0,i1=0;i0<axis0.num;i0+=2)
        {
        x0=axis0[i0+0];
        y0=axis0[i0+1];
        for (d=-1.0,i=i1;i<axis1.num;i+=2)
            {
            x1=axis1[i+0];
            y1=axis1[i+1];
            dd=((x1-x0)*(x1-x0))+((y1-y0)*(y1-y0));
            if ((d<0.0)||(dd<=d)) { i1=i; d=dd; }
            }
        if (d>=0.0)
            {
            x1=axis1[i1+0];
            y1=axis1[i1+1];
            axis.add(0.5*(x0+x1));
            axis.add(0.5*(y0+y1));
            }
        }
    }

1.000000 0.000000
0.990000 0.006719
0.980000 0.013307
0.970000 0.019757
0.960000 0.026064
0.950000 0.032223
0.940000 0.038228
0.930000 0.044075
0.920000 0.049759
0.910000 0.055276
0.900000 0.060623
0.890000 0.065795
0.880000 0.070790
0.870000 0.075604
0.860000 0.080234
0.850000 0.084678
0.840000 0.088935
0.830000 0.093001
0.820000 0.096876
0.810000 0.100558
0.800000 0.104046
0.790000 0.107339
0.780000 0.110438
0.770000 0.113342
0.760000 0.116051
0.750000 0.118566
0.740000 0.120887
0.730000 0.123016
0.720000 0.124954
0.710000 0.126702
0.700000 0.128262
0.690000 0.129637
0.680000 0.130829
0.670000 0.131839
0.660000 0.132672
0.650000 0.133331
0.640000 0.133818
0.630000 0.134137
0.620000 0.134292
0.610000 0.134287
0.600000 0.134127
0.590000 0.133815
0.580000 0.133356
0.570000 0.132755
0.560000 0.132016
0.550000 0.131146
0.540000 0.130148
0.530000 0.129030
0.520000 0.127795
0.510000 0.126450
0.500000 0.125000
0.490000 0.123452
0.480000 0.121811
0.470000 0.120083
0.460000 0.118275
0.450000 0.116392
0.440000 0.114441
0.430000 0.112429
0.420000 0.110361
0.410000 0.108244
0.400000 0.106085
0.390000 0.103889
0.380000 0.101663
0.370000 0.099414
0.360000 0.097148
0.350000 0.094870
0.340000 0.092589
0.330000 0.090309
0.320000 0.088037
0.310000 0.085779
0.300000 0.083541
0.290000 0.081329
0.280000 0.079149
0.270000 0.077006
0.260000 0.074906
0.250000 0.072855
0.240000 0.070858
0.230000 0.068920
0.220000 0.067047
0.210000 0.065242
0.113262 0.047023
0.110002 0.042718
0.106385 0.038580
0.102428 0.034615
0.098146 0.030832
0.093556 0.027239
0.088673 0.023844
0.083516 0.020652
0.078101 0.017670
0.072448 0.014904
0.066574 0.012361
0.060499 0.010044
0.054241 0.007958
0.047820 0.006108
0.041256 0.004497
0.034569 0.003129
0.027779 0.002005
0.020907 0.001129
0.013972 0.000502
0.006997 0.000126
0.000000 0.000000
0.000000 0.000000
-0.003997 0.000126
-0.007972 0.000502
-0.011907 0.001129
-0.015779 0.002005
-0.019569 0.003129
-0.023256 0.004497
-0.026820 0.006108
-0.030241 0.007958
-0.033499 0.010044
-0.036574 0.012361
-0.039448 0.014904
-0.042101 0.017670
-0.044516 0.020652
-0.046673 0.023844
-0.048556 0.027239
-0.050146 0.030832
-0.051428 0.034615
-0.052385 0.038580
-0.053002 0.042718
-0.053262 0.047023
-0.053153 0.051484
-0.052659 0.056093
-0.051768 0.060841
-0.050467 0.065717
-0.048744 0.070711
-0.046588 0.075813
-0.043988 0.081012
-0.040935 0.086297
-0.037420 0.091658
-0.033435 0.097082
-0.028972 0.102558
-0.024025 0.108074
-0.018589 0.113618
-0.012657 0.119178
-0.006228 0.124741
0.000704 0.130295
0.008139 0.135828
0.016079 0.141326
0.024525 0.146777
0.033475 0.152169
0.042930 0.157488
0.052885 0.162722
0.063339 0.167858
0.074287 0.172883
0.085723 0.177784
0.097643 0.182549
0.110038 0.187166
0.122902 0.191621
0.136226 0.195903
0.150000 0.200000
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0.178856 0.207590
0.193914 0.211059
0.209376 0.214297
0.225227 0.217291
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0.274968 0.224711
0.292223 0.226629
0.309787 0.228254
0.327641 0.229575
0.345766 0.230585
0.364142 0.231274
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0.401566 0.231662
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0.439740 0.230679
0.459054 0.229657
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0.498015 0.226525
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0.596231 0.212132
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0.654700 0.198876
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0.978646 0.024614
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1.000000 0.000000

def spline(self, x, y, points=200, degree=2, evaluate=False):
    """Interpolate spline through given points

    Args:
        spline (int, optional): Number of points on the spline
        degree (int, optional): Degree of the spline
        evaluate (bool, optional): If True, evaluate spline just at
                                   the coordinates of the knots
    """

    # interpolate B-spline through data points
    # returns knots of control polygon
    # tck ... tuple (t,c,k) containing the vector of knots,
    # the B-spline coefficients, and the degree of the spline.
    # u ... array of the parameters for each knot
    # NOTE: s=0.0 is important as no smoothing should be done on the spline
    # after interpolating it
    tck, u = interpolate.splprep([x, y], s=0.0, k=degree)

    # number of points on interpolated B-spline (parameter t)
    t = np.linspace(0.0, 1.0, points)

    # if True, evaluate spline just at the coordinates of the knots
    if evaluate:
        t = u

    # evaluate B-spline at given parameters
    # der=0: returns point coordinates
    coo = interpolate.splev(t, tck, der=0)

    # evaluate 1st derivative at given parameters
    der1 = interpolate.splev(t, tck, der=1)

    # evaluate 2nd derivative at given parameters
    der2 = interpolate.splev(t, tck, der=2)

    spline_data = [coo, u, t, der1, der2, tck]

    return spline_data

    def getCurvature(spline_data):
    """Curvature and radius of curvature of a parametric curve

    der1 is dx/dt and dy/dt at each point
    der2 is d2x/dt2 and d2y/dt2 at each point

    Returns:
        float: Tuple of numpy arrays carrying gradient of the curve,
               the curvature, radiusses of curvature circles and
               curvature circle centers for each point of the curve
    """

    coo = spline_data[0]
    der1 = spline_data[3]
    der2 = spline_data[4]

    xd = der1[0]
    yd = der1[1]
    x2d = der2[0]
    y2d = der2[1]
    n = xd**2 + yd**2
    d = xd*y2d - yd*x2d

    # gradient dy/dx = dy/du / dx/du
    gradient = der1[1] / der1[0]

    # radius of curvature
    R = n**(3./2.) / abs(d)

    # curvature
    C = d / n**(3./2.)

    # coordinates of curvature-circle center points
    xc = coo[0] - R * yd / np.sqrt(n)
    yc = coo[1] + R * xd / np.sqrt(n)

    return [gradient, C, R, xc, yc]