This repository was archived by the owner on Apr 2, 2026. It is now read-only.
Repository navigation
Expand file tree
/
Copy pathmindist.go
More file actions
238 lines (219 loc) · 5.07 KB
/
Copy pathmindist.go
File metadata and controls
238 lines (219 loc) · 5.07 KB
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
package curve
import (
"math"
"honnef.co/go/stuff/container/maybe"
)
//! Minimum distance between two Bézier curves
//!
//! This implements the algorithm in "Computing the minimum distance between
//! two Bézier curves", Chen et al., *Journal of Computational and Applied
//! Mathematics* 229(2009), 294-301
func minDistParam(
bez1, bez2 []Vec2,
u, v [2]float64,
epsilon float64,
bestAlpha float64,
) [3]float64 {
if len(bez1) == 0 || len(bez2) == 0 {
panic("called with empty path")
}
n := len(bez1) - 1
m := len(bez2) - 1
umin, umax := u[0], u[1]
vmin, vmax := v[0], v[1]
umid := (umin + umax) / 2.0
vmid := (vmin + vmax) / 2.0
svalues := [4][3]float64{
{s(umin, vmin, bez1, bez2), umin, vmin},
{s(umin, vmax, bez1, bez2), umin, vmax},
{s(umax, vmin, bez1, bez2), umax, vmin},
{s(umax, vmax, bez1, bez2), umax, vmax},
}
alpha := svalues[0][0]
for _, sval := range svalues {
alpha = min(alpha, sval[0])
}
if alpha > bestAlpha {
return [3]float64{alpha, umid, vmid}
}
if math.Abs(umax-umin) < epsilon || math.Abs(vmax-vmin) < epsilon {
return [3]float64{alpha, umid, vmid}
}
// Property one: D(r>k) > alpha
isOutside := true
var minDrk maybe.Option[float64]
var minIj maybe.Option[[2]int]
for r := range 2 * n {
for k := range 2 * m {
dRk := dRk(r, k, bez1, bez2)
if dRk < alpha {
isOutside = false
}
if v, ok := minDrk.Get(); !ok || dRk < v {
minDrk = maybe.Some(dRk)
minIj = maybe.Some([2]int{r, k})
}
}
}
if isOutside {
return [3]float64{alpha, umid, vmid}
}
// Property two: boundary check
atBoundary0OnBez1 := true
atBoundary1OnBez1 := true
atBoundary0OnBez2 := true
atBoundary1OnBez2 := true
for i := range 2 * n {
for j := range 2 * m {
dij := dRk(i, j, bez1, bez2)
dkj := dRk(0, j, bez1, bez2)
if dij < dkj {
atBoundary0OnBez1 = false
}
dkj = dRk(2*n, j, bez1, bez2)
if dij < dkj {
atBoundary1OnBez1 = false
}
dkj = dRk(i, 0, bez1, bez2)
if dij < dkj {
atBoundary0OnBez2 = false
}
dkj = dRk(i, 2*n, bez1, bez2)
if dij < dkj {
atBoundary1OnBez2 = false
}
}
}
if atBoundary0OnBez1 && atBoundary0OnBez2 {
return svalues[0]
}
if atBoundary0OnBez1 && atBoundary1OnBez2 {
return svalues[1]
}
if atBoundary1OnBez1 && atBoundary0OnBez2 {
return svalues[2]
}
if atBoundary1OnBez1 && atBoundary1OnBez2 {
return svalues[3]
}
minI, minJ := minIj.Unwrap()[0], minIj.Unwrap()[1]
newUmid := umin + (umax-umin)*(float64(minI)/float64(2*n))
newVmid := vmin + (vmax-vmin)*(float64(minJ)/float64(2*m))
// Subdivide
results := [4][3]float64{
minDistParam(
bez1,
bez2,
[2]float64{umin, newUmid},
[2]float64{vmin, newVmid},
epsilon,
alpha,
),
minDistParam(
bez1,
bez2,
[2]float64{umin, newUmid},
[2]float64{newVmid, vmax},
epsilon,
alpha,
),
minDistParam(
bez1,
bez2,
[2]float64{newUmid, umax},
[2]float64{vmin, newVmid},
epsilon,
alpha,
),
minDistParam(
bez1,
bez2,
[2]float64{newUmid, umax},
[2]float64{newVmid, vmax},
epsilon,
alpha,
),
}
out := results[0]
for _, res := range results[1:] {
if math.IsNaN(res[0]) || res[0] < out[0] {
out = res
}
}
return out
}
func s(u, v float64, bez1, bez2 []Vec2) float64 {
n := len(bez1) - 1
m := len(bez2) - 1
summand := 0.0
for r := range 2*n + 1 {
for k := range 2*m + 1 {
summand +=
dRk(r, k, bez1, bez2) * basisFunction(2*n, r, u) * basisFunction(2*m, k, v)
}
}
return summand
}
func cRk(r, k int, bez1, bez2 []Vec2) float64 {
var left Vec2
{
n := len(bez1) - 1
upsilon := min(r, n)
theta := r - min(n, r)
items := bez1[:min(upsilon+1, len(bez1))]
skip := min(theta, len(items))
items = items[skip:]
for i, item := range items {
i += theta
left = left.Add(item.Mul(float64(choose(n, i)*choose(n, r-i)) / float64(choose(2*n, r))))
}
}
var right Vec2
{
m := len(bez2) - 1
varsigma := min(k, m)
sigma := k - min(m, k)
items := bez2[:min(varsigma+1, len(bez2))]
skip := min(sigma, len(items))
items = items[skip:]
for j, item := range items {
j += skip
right = right.Add(item.Mul(float64(choose(m, j)*choose(m, k-j)) / float64(choose(2*m, k))))
}
}
return left.Dot(right)
}
func aR(r int, p []Vec2) float64 {
n := len(p) - 1
upsilon := min(r, n)
theta := r - min(n, r)
var sum float64
for i := theta; i <= upsilon; i++ {
dot := p[i].Dot(p[r-i]) // These are bounds checked by the sum limits
factor := float64(choose(n, i)*choose(n, r-i)) / float64(choose(2*n, r))
sum += dot * factor
}
return sum
}
func dRk(r, k int, bez1, bez2 []Vec2) float64 {
// In the paper, B_k is used for the second factor, but it's the same thing
return aR(r, bez1) + aR(k, bez2) - 2.0*cRk(r, k, bez1, bez2)
}
// Bezier basis function
func basisFunction(n, i int, u float64) float64 {
return float64(choose(n, i)) * math.Pow(1.0-u, float64(n-i)) * math.Pow(u, float64(i))
}
// Binomial co-efficient, but returning zeros for values outside of domain
func choose(n, k int) uint32 {
if k > n {
return 0
}
p := 1
bound := n - k
for i := 1; i <= bound; i++ {
p *= n
p /= i
n -= 1
}
return uint32(p)
}