ISCWSA
Probabilities of Collision or Crossing
Empirical Distribution
Author: T. Torkildsen
December 2014
Nomenclature
Scaling factor
-
Scaling of standard ellipse (ellipsoid)
Flattening
-
Flattening of ellipse; (major-minor)/major
Probability
-
Probability of drilling on wrong side of limit line (plane)
Radius dist.
-
Distance from center to ellipse along Direction
Pedal distance -
Distance from center to the ellipse’s pedal curve (surface) along Direction
Direction
Direction relative to major axis
-
Distribution of Empirical data is approximated by a Student-t distribution with 6 degrees of freedom.
Empirical distributed data
Scaling factor
Probability, Pedal distance
2
4.6E-02
Probabilities, Radius distance
Flattening
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 / 90
4.6E-02
4.6E-02
4.6E-02
4.6E-02
4.6E-02
4.6E-02
4.6E-02
4.6E-02
4.6E-02
4.6E-02
4.6E-02
Direction
15 / 75
30 / 60
4.6E-02
4.6E-02
4.6E-02
4.7E-02
4.7E-02
4.9E-02
4.8E-02
5.3E-02
5.1E-02
6.0E-02
5.5E-02
7.2E-02
6.3E-02
9.5E-02
8.1E-02
1.4E-01
1.2E-01
2.1E-01
2.4E-01
3.3E-01
5.0E-01
5.0E-01
Inverse probability, Pedal distance
45
4.6E-02
4.7E-02
4.9E-02
5.5E-02
6.4E-02
8.0E-02
1.1E-01
1.6E-01
2.4E-01
3.5E-01
5.0E-01
22
Inverse probabilities, Radius distance
Flattening
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 / 90
22
22
22
22
22
22
22
22
22
22
22
Direction
15 / 75
30 / 60
22
22
22
21
21
21
21
19
20
17
18
14
16
11
12
7
8
5
4
3
2
2
45
22
21
20
18
16
12
9
6
4
3
2
Empirical distributed data
Scaling factor
Probability, Pedal distance
2.5
2.3E-02
Probabilities, Radius distance
Flattening
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 / 90
2.3E-02
2.3E-02
2.3E-02
2.3E-02
2.3E-02
2.3E-02
2.3E-02
2.3E-02
2.3E-02
2.3E-02
2.3E-02
Direction
15 / 75
30 / 60
2.3E-02
2.3E-02
2.3E-02
2.4E-02
2.4E-02
2.5E-02
2.5E-02
2.7E-02
2.6E-02
3.2E-02
2.9E-02
4.0E-02
3.4E-02
5.7E-02
4.7E-02
9.0E-02
8.0E-02
1.6E-01
1.9E-01
3.0E-01
5.0E-01
5.0E-01
Inverse probability, Pedal distance
45
2.3E-02
2.4E-02
2.5E-02
2.9E-02
3.5E-02
4.6E-02
6.8E-02
1.1E-01
1.9E-01
3.2E-01
5.0E-01
43
Inverse probabilities, Radius distance
Flattening
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 / 90
43
43
43
43
43
43
43
43
43
43
43
Direction
15 / 75
30 / 60
43
43
43
42
42
40
41
37
38
31
35
25
29
18
21
11
12
6
5
3
2
2
45
43
42
40
35
29
22
15
9
5
3
2
Empirical distributed data
Scaling factor
Probability, Pedal distance
3
1.2E-02
Probabilities, Radius distance
Flattening
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 / 90
1.2E-02
1.2E-02
1.2E-02
1.2E-02
1.2E-02
1.2E-02
1.2E-02
1.2E-02
1.2E-02
1.2E-02
1.2E-02
Direction
15 / 75
30 / 60
1.2E-02
1.2E-02
1.2E-02
1.2E-02
1.2E-02
1.3E-02
1.3E-02
1.4E-02
1.4E-02
1.7E-02
1.5E-02
2.3E-02
1.9E-02
3.4E-02
2.7E-02
6.0E-02
5.2E-02
1.2E-01
1.5E-01
2.6E-01
5.0E-01
5.0E-01
Inverse probability, Pedal distance
45
1.2E-02
1.2E-02
1.3E-02
1.5E-02
1.9E-02
2.7E-02
4.2E-02
7.5E-02
1.5E-01
2.9E-01
5.0E-01
83
Inverse probabilities, Radius distance
Flattening
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 / 90
83
83
83
83
83
83
83
83
83
83
83
Direction
15 / 75
30 / 60
83
83
83
82
81
78
78
70
73
58
65
44
53
29
37
17
19
8
7
4
2
2
45
83
82
76
66
52
38
24
13
7
3
2
Empirical distributed data
Scaling factor
Probability, Pedal distance
3.5
6.4E-03
Probabilities, Radius distance
Flattening
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 / 90
6.4E-03
6.4E-03
6.4E-03
6.4E-03
6.4E-03
6.4E-03
6.4E-03
6.4E-03
6.4E-03
6.4E-03
6.4E-03
Direction
15 / 75
30 / 60
6.4E-03
6.4E-03
6.5E-03
6.5E-03
6.6E-03
6.9E-03
6.9E-03
7.8E-03
7.4E-03
9.6E-03
8.4E-03
1.3E-02
1.1E-02
2.1E-02
1.6E-02
3.9E-02
3.3E-02
9.0E-02
1.2E-01
2.3E-01
5.0E-01
5.0E-01
Inverse probability, Pedal distance
45
6.4E-03
6.6E-03
7.1E-03
8.3E-03
1.1E-02
1.6E-02
2.6E-02
5.1E-02
1.1E-01
2.6E-01
5.0E-01
156
Inverse probabilities, Radius distance
Flattening
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 / 90
156
156
156
156
156
156
156
156
156
156
156
Direction
15 / 75
30 / 60
156
156
155
153
152
144
146
128
135
105
118
77
95
48
63
26
30
11
8
4
2
2
45
156
152
140
120
93
64
38
20
9
4
2
Empirical distributed data
Scaling factor
Probability, Pedal distance
4
3.6E-03
Probabilities, Radius distance
Flattening
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 / 90
3.6E-03
3.6E-03
3.6E-03
3.6E-03
3.6E-03
3.6E-03
3.6E-03
3.6E-03
3.6E-03
3.6E-03
3.6E-03
Direction
15 / 75
30 / 60
3.6E-03
3.6E-03
3.6E-03
3.6E-03
3.7E-03
3.9E-03
3.8E-03
4.4E-03
4.2E-03
5.5E-03
4.8E-03
7.7E-03
6.1E-03
1.3E-02
9.5E-03
2.6E-02
2.1E-02
6.7E-02
9.2E-02
2.0E-01
5.0E-01
5.0E-01
Inverse probability, Pedal distance
45
3.6E-03
3.7E-03
4.0E-03
4.7E-03
6.2E-03
9.3E-03
1.6E-02
3.5E-02
8.7E-02
2.3E-01
5.0E-01
281
Inverse probabilities, Radius distance
Flattening
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 / 90
281
281
281
281
281
281
281
281
281
281
281
Direction
15 / 75
30 / 60
281
281
279
276
273
258
261
227
241
183
209
130
164
79
106
39
47
15
11
5
2
2
45
281
274
251
212
162
108
61
29
11
4
2
Empirical distributed data
Scaling factor
Probability, Pedal distance
4.5
2.1E-03
Probabilities, Radius distance
Flattening
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 / 90
2.1E-03
2.1E-03
2.1E-03
2.1E-03
2.1E-03
2.1E-03
2.1E-03
2.1E-03
2.1E-03
2.1E-03
2.1E-03
Direction
15 / 75
30 / 60
2.1E-03
2.1E-03
2.1E-03
2.1E-03
2.1E-03
2.2E-03
2.2E-03
2.6E-03
2.4E-03
3.2E-03
2.8E-03
4.6E-03
3.6E-03
7.9E-03
5.8E-03
1.7E-02
1.4E-02
4.9E-02
7.1E-02
1.7E-01
5.0E-01
5.0E-01
Inverse probability, Pedal distance
45
2.1E-03
2.1E-03
2.3E-03
2.8E-03
3.7E-03
5.7E-03
1.1E-02
2.4E-02
6.7E-02
2.0E-01
5.0E-01
487
Inverse probabilities, Radius distance
Flattening
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 / 90
487
487
487
487
487
487
487
487
487
487
487
Direction
15 / 75
30 / 60
487
487
484
478
473
446
451
389
414
309
357
216
276
126
173
58
71
20
14
6
2
2
45
487
475
433
363
272
176
95
42
15
5
2
Empirical distributed data
Scaling factor
Probability, Pedal distance
5
1.2E-03
Probabilities, Radius distance
Flattening
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 / 90
1.2E-03
1.2E-03
1.2E-03
1.2E-03
1.2E-03
1.2E-03
1.2E-03
1.2E-03
1.2E-03
1.2E-03
1.2E-03
Direction
15 / 75
30 / 60
1.2E-03
1.2E-03
1.2E-03
1.3E-03
1.3E-03
1.3E-03
1.3E-03
1.5E-03
1.5E-03
2.0E-03
1.7E-03
2.9E-03
2.2E-03
5.0E-03
3.6E-03
1.2E-02
9.3E-03
3.7E-02
5.5E-02
1.5E-01
5.0E-01
5.0E-01
Inverse probability, Pedal distance
45
1.2E-03
1.3E-03
1.4E-03
1.7E-03
2.3E-03
3.6E-03
6.8E-03
1.7E-02
5.1E-02
1.8E-01
5.0E-01
816
Inverse probabilities, Radius distance
Flattening
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 / 90
816
816
816
816
816
816
816
816
816
816
816
Direction
15 / 75
30 / 60
816
816
810
799
790
744
752
645
688
508
590
349
451
198
276
86
108
27
18
7
2
2
45
816
793
722
600
443
281
146
60
19
6
2
Empirical distributed data
Scaling factor
Probability, Pedal distance
5.5
7.6E-04
Probabilities, Radius distance
Flattening
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 / 90
7.6E-04
7.6E-04
7.6E-04
7.6E-04
7.6E-04
7.6E-04
7.6E-04
7.6E-04
7.6E-04
7.6E-04
7.6E-04
Direction
15 / 75
30 / 60
7.6E-04
7.6E-04
7.6E-04
7.7E-04
7.8E-04
8.3E-04
8.2E-04
9.6E-04
9.0E-04
1.2E-03
1.1E-03
1.8E-03
1.4E-03
3.3E-03
2.3E-03
7.9E-03
6.3E-03
2.7E-02
4.3E-02
1.3E-01
5.0E-01
5.0E-01
Inverse probability, Pedal distance
45
7.6E-04
7.8E-04
8.6E-04
1.0E-03
1.4E-03
2.3E-03
4.5E-03
1.2E-02
3.9E-02
1.6E-01
5.0E-01
1320
Inverse probabilities, Radius distance
Flattening
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 / 90
1320
1320
1320
1320
1320
1320
1320
1320
1320
1320
1320
Direction
15 / 75
30 / 60
1320
1320
1311
1292
1278
1200
1215
1037
1108
810
945
549
715
304
430
127
160
37
24
8
2
2
45
1320
1283
1164
962
703
438
221
86
25
6
2
Empirical distributed data
Scaling factor
Probability, Pedal distance
6
4.8E-04
Probabilities, Radius distance
Flattening
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 / 90
4.8E-04
4.8E-04
4.8E-04
4.8E-04
4.8E-04
4.8E-04
4.8E-04
4.8E-04
4.8E-04
4.8E-04
4.8E-04
Direction
15 / 75
30 / 60
4.8E-04
4.8E-04
4.9E-04
4.9E-04
5.0E-04
5.3E-04
5.3E-04
6.2E-04
5.8E-04
8.0E-04
6.8E-04
1.2E-03
9.0E-04
2.2E-03
1.5E-03
5.5E-03
4.3E-03
2.0E-02
3.3E-02
1.1E-01
5.0E-01
5.0E-01
Inverse probability, Pedal distance
45
4.8E-04
5.0E-04
5.5E-04
6.7E-04
9.2E-04
1.5E-03
3.0E-03
8.2E-03
3.0E-02
1.4E-01
5.0E-01
2074
Inverse probabilities, Radius distance
Flattening
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 / 90
2074
2074
2074
2074
2074
2074
2074
2074
2074
2074
2074
Direction
15 / 75
30 / 60
2074
2074
2058
2029
2006
1881
1904
1619
1733
1256
1472
842
1106
457
654
183
234
49
30
9
2
2
45
2074
2014
1822
1499
1087
667
328
122
33
7
2
Empirical distributed data
Scaling factor
Probability, Pedal distance
6.5
3.2E-04
Probabilities, Radius distance
Flattening
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 / 90
3.2E-04
3.2E-04
3.2E-04
3.2E-04
3.2E-04
3.2E-04
3.2E-04
3.2E-04
3.2E-04
3.2E-04
3.2E-04
Direction
15 / 75
30 / 60
3.2E-04
3.2E-04
3.2E-04
3.2E-04
3.3E-04
3.5E-04
3.4E-04
4.1E-04
3.8E-04
5.3E-04
4.5E-04
7.9E-04
6.0E-04
1.5E-03
1.0E-03
3.8E-03
3.0E-03
1.5E-02
2.5E-02
9.5E-02
5.0E-01
5.0E-01
Inverse probability, Pedal distance
45
3.2E-04
3.3E-04
3.6E-04
4.4E-04
6.1E-04
1.0E-03
2.1E-03
5.8E-03
2.3E-02
1.2E-01
5.0E-01
3168
Inverse probabilities, Radius distance
Flattening
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 / 90
3168
3168
3168
3168
3168
3168
3168
3168
3168
3168
3168
Direction
15 / 75
30 / 60
3168
3168
3145
3099
3064
2869
2904
2462
2639
1900
2235
1261
1668
673
973
262
337
66
39
11
2
2
45
3168
3076
2778
2276
1639
993
478
171
43
8
2
Empirical distributed data
Scaling factor
Probability, Pedal distance
7
2.1E-04
Probabilities, Radius distance
Flattening
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 / 90
2.1E-04
2.1E-04
2.1E-04
2.1E-04
2.1E-04
2.1E-04
2.1E-04
2.1E-04
2.1E-04
2.1E-04
2.1E-04
Direction
15 / 75
30 / 60
2.1E-04
2.1E-04
2.1E-04
2.2E-04
2.2E-04
2.3E-04
2.3E-04
2.7E-04
2.5E-04
3.6E-04
3.0E-04
5.4E-04
4.1E-04
1.0E-03
7.0E-04
2.7E-03
2.1E-03
1.1E-02
2.0E-02
8.1E-02
5.0E-01
5.0E-01
Inverse probability, Pedal distance
45
2.1E-04
2.2E-04
2.4E-04
3.0E-04
4.1E-04
6.9E-04
1.5E-03
4.2E-03
1.8E-02
1.1E-01
5.0E-01
4723
Inverse probabilities, Radius distance
Flattening
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 / 90
4723
4723
4723
4723
4723
4723
4723
4723
4723
4723
4723
Direction
15 / 75
30 / 60
4723
4723
4687
4617
4564
4269
4323
3655
3922
2809
3313
1850
2460
973
1419
368
478
87
51
12
2
2
45
4723
4583
4132
3375
2415
1448
686
237
56
9
2