笔者在工作中遇到了一个问题，我需要车辆平滑的行驶路径来建图。但是实际采集的路径数据不理想，免不了包含倒车等带“尖角”的路径。如何对其进行平滑呢？我想到了一个简单的方法。
思路是计算相邻点间向量的夹角，如果超过一定阈值就令中间的点取前后两个点的平均值。其本质是限制曲线的曲率。但是曲率不仅与角度有关，还与弧长有关，所以仅仅计算角度还不够，还要不断对点进行下采样，确保其弧长不至于太小。
Mathematica代码如下：
首先是采样函数（Sample），其按照一定的分辨率（resolution）对点列（pts）进行下采样，去除太密集的点。
Sample[pts_, resolution_] := Module[{ptsa = {}},
Do[
If[Length[ptsa] < 1,
AppendTo[ptsa, pt],
If[Norm[ptsa[[-1]] - pt] > resolution, AppendTo[ptsa, pt]]
];
, {pt, pts}];
ptsa]平滑的代码如下，就是两个循环，外层的循环可以用一个while替换。
pts = pts0;
maxangle = 5.0/180*Pi;
Do[
ptsa = Sample[pts, 0.5];
n = Length[ptsa];
pts = ptsa;
Do[
angle =
VectorAngle[pts[[i]] - pts[[i - 1]], pts[[i + 1]] - pts[[i]]];
If[angle > maxangle, pts[[i]] = Mean[{pts[[i + 1]], pts[[i - 1]]}]];
, {i, 2, n - 1}];
, {j, 100}]
Graphics[{AbsolutePointSize[1], Gray, Point[ptsa0], AbsolutePointSize[1.5], Red, Point[pts]}]我们用一些数据来验证一下，假设汽车的行驶路径的坐标点如下图中左侧的黑点。数据在最后给出。平滑后的坐标点见右图中的红点，已经重新采样了所以看起来更均匀。可以看到，车辆倒车形成的尖角都被抹平了。
车辆实际的路径坐标数据如下。
94.071 -41.253
93.987 -41.429
93.902 -41.602
93.768 -41.859
93.674 -42.028
93.569 -42.208
93.467 -42.373
93.368 -42.521
93.194 -42.77
93.081 -42.924
92.954 -43.089
92.833 -43.239
92.502 -43.619
92.422 -43.706
92.354 -43.78
92.203 -43.937
92.059 -44.081
91.84 -44.298
91.535 -44.59
91.364 -44.749
91.284 -44.822
91.122 -44.972
90.959 -45.12
90.711 -45.343
90.531 -45.505
90.363 -45.657
90.18 -45.822
89.928 -46.055
89.76 -46.214
89.593 -46.374
89.415 -46.548
89.174 -46.788
88.731 -47.252
88.652 -47.337
88.585 -47.410
88.518 -47.482
88.372 -47.643
88.248 -47.801
88.136 -47.949
87.797 -48.355
87.74 -48.43
87.686 -48.505
87.533 -48.725
87.422 -48.879
87.244 -49.123
87.138 -49.296
87.046 -49.438
86.895 -49.679
86.777 -49.85
86.628 -50.109
86.48 -50.395
86.428 -50.49
86.319 -50.695
86.218 -50.883
86.062 -51.196
85.926 -51.526
85.885 -51.63
85.464 -52.874
85.327 -53.412
85.291 -53.554
85.214 -53.868
85.184 -54.017
85.152 -54.19
85.128 -54.337
85.106 -54.483
85.066 -54.77
85.028 -55.051
85.011 -55.191
84.978 -55.467
84.948 -55.743
84.922 -56.018
84.882 -56.456
84.854 -56.734
84.829 -57.013
84.803 -57.435
84.773 -57.976
84.764 -58.143
84.757 -58.287
84.736 -58.723
84.722 -59.018
84.710 -59.315
84.694 -59.762
84.682 -60.06
84.663 -60.654
84.66 -60.777
84.651 -61.219
84.647 -61.512
84.645 -61.78
84.642 -62.071
84.641 -62.507
84.641 -62.772
84.642 -63.061
84.643 -63.492
84.65 -63.775
84.659 -64.057
84.674 -64.455
84.686 -64.736
84.697 -65.017
84.721 -65.701
84.724 -65.844
84.726 -65.965
84.733 -66.255
84.743 -66.691
84.75 -66.982
84.757 -67.275
84.768 -67.716
84.776 -68.011
84.783 -68.306
84.791 -68.749
84.796 -69.02
84.804 -69.34
84.813 -69.636
84.834 -70.375
84.838 -70.524
84.842 -70.674
84.850 -70.95
84.861 -71.407
84.872 -71.87
84.877 -72.025
84.887 -72.31
84.899 -72.648
84.909 -72.985
84.924 -73.48
84.929 -73.662
84.952 -74.338
84.956 -74.495
84.964 -74.834
84.973 -75.171
84.985 -75.506
85.001 -75.965
85.011 -76.29
85.025 -76.754
85.03 -76.921
85.041 -77.225
85.054 -77.519
85.071 -77.93
85.086 -78.319
85.091 -78.436
85.104 -78.775
85.113 -78.99
85.124 -79.196
85.134 -79.394
85.147 -79.692
85.172 -80.204
85.196 -80.598
85.2 -80.674
85.205 -80.748
85.209 -80.822
85.214 -80.894
85.218 -80.964
85.231 -81.167
85.235 -81.233
85.246 -81.421
85.254 -81.542
85.262 -81.663
85.274 -81.847
85.284 -81.967
85.294 -82.13
85.301 -82.242
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85.316 -82.495
85.33 -82.69
85.338 -82.846
85.348 -82.992
85.368 -83.294
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85.634 -87.277
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85.692 -88.091
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85.713 -88.389
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85.752 -89.078
85.759 -89.219
85.765 -89.42
85.768 -89.543
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85.768 -89.874
85.764 -90.066
85.761 -90.128
85.755 -90.252
85.741 -90.433
85.729 -90.551
85.716 -90.668
85.701 -90.775
85.661 -91.018
85.65 -91.077
85.626 -91.195
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85.438 -91.896
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69.38 -96.586
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61.063 -97.11
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60.4 -97.166
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