del-geo 0.1.24

2D/3D geometry utility codes
Documentation 
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
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322

//! methods for 3D edge (line segment)

use nalgebra::clamp;
use num_traits::AsPrimitive;

pub fn nearest_point3_<T>(
    point_pos: &[T;3],
    edge_pos0: &[T;3],
    edge_pos1: &[T;3]) -> [T; 3]
    where T: num_traits::Float + 'static + Copy + PartialOrd,
          f64: num_traits::AsPrimitive<T>
{
    use crate::vec3;
    let d = [
        edge_pos1[0] - edge_pos0[0],
        edge_pos1[1] - edge_pos0[1],
        edge_pos1[2] - edge_pos0[2]];
    let t = {
        if vec3::dot_(&d, &d) > 1.0e-20_f64.as_() {
            let ps = [
                edge_pos0[0] - point_pos[0],
                edge_pos0[1] - point_pos[1],
                edge_pos0[2] - point_pos[2]];
            let a = vec3::dot_(&d, &d);
            let b = vec3::dot_(&d, &ps);
            let mut r: T = -b / a;
            r = if r < 0_f64.as_() { 0_f64.as_() } else { r };
            r = if r > 1_f64.as_() { 1_f64.as_() } else { r };
            r
        } else {
            0.5_f64.as_()
        }
    };
    [edge_pos0[0] + t * d[0],
        edge_pos0[1] + t * d[1],
        edge_pos0[2] + t * d[2]]
}

// above: w/o nalgebra
/* -------------------------- */
// below: w/ nalgebra

/// return  distance and parameter
pub fn nearest_to_point3<T>(
    edge_pos0: &nalgebra::Vector3<T>,
    edge_pos1: &nalgebra::Vector3<T>,
    point_pos: &nalgebra::Vector3<T>) -> (T, T)
    where T: nalgebra::RealField + 'static + Copy + PartialOrd,
          f64: num_traits::AsPrimitive<T>
{
    let d = edge_pos1 - edge_pos0;
    let dsq = d.norm_squared();
    let t = if dsq != T::zero() {
        let r: T = -d.dot(&(edge_pos0 - point_pos)) / dsq;
        clamp(r, T::zero(), T::one())
    } else {
        0.5_f64.as_()
    };
    let distance = (edge_pos0 + d.scale(t) - point_pos).norm();
    (distance, t)
}


/// the two edges need to be co-planar
pub fn intersection_edge3_when_coplanar<T>(
    p0: &nalgebra::Vector3<T>,
    p1: &nalgebra::Vector3<T>,
    q0: &nalgebra::Vector3<T>,
    q1: &nalgebra::Vector3<T>) -> Option<(T, T, T, T)>
    where T: nalgebra::RealField + Copy + 'static,
          f64: num_traits::AsPrimitive<T>
{
    let n = {
        let n0 = (p1 - p0).cross(&(q0 - p0));
        let n1 = (p1 - p0).cross(&(q1 - p0));
        if n0.norm_squared() < n1.norm_squared() { n1 } else { n0 }
    };
    let p2 = p0 + n;
    let rq1 = crate::tet::volume(p0, p1, &p2, q0);
    let rq0 = crate::tet::volume(p0, p1, &p2, q1);
    let rp1 = crate::tet::volume(q0, q1, &p2, p0);
    let rp0 = crate::tet::volume(q0, q1, &p2, p1);
    if (rp0 - rp1).abs() <= T::zero() { return None; }
    if (rq0 - rq1).abs() <= T::zero() { return None; }
    let t = T::one() / (rp0 - rp1);
    let (rp0, rp1) = (rp0.scale(t), -rp1.scale(t));
    let t = T::one() / (rq0 - rq1);
    let (rq0, rq1) = (rq0.scale(t), -rq1.scale(t));
    Some((rp0, rp1, rq0, rq1))
}

pub fn nearest_to_line3(
    edge_start: &nalgebra::Vector3<f64>,
    edge_end: &nalgebra::Vector3<f64>,
    line_origin: &nalgebra::Vector3<f64>,
    line_direction: &nalgebra::Vector3<f64>) -> (nalgebra::Vector3<f64>, nalgebra::Vector3<f64>)
{
    let (scale, scaled_ratio_edge, _, scaled_nearest_edge, scaled_nearest_line)
        = crate::line3::nearest_to_line3(
        edge_start, &(edge_end - edge_start),
        line_origin, line_direction);
    if scale.abs() < 1.0e-10 { // pararell
        let nearest_edge = (edge_start + edge_end) * 0.5;
        let (nearest_line, _) = crate::line::nearest_to_point(
            &nearest_edge, line_origin, line_direction);
        return (nearest_edge, nearest_line);
    }
    let ratio_edge = scaled_ratio_edge / scale;
    if ratio_edge > 0_f64 && ratio_edge < 1_f64 { // nearst point is inside the segment
        let nearest_edge = scaled_nearest_edge / scale;
        let nearest_line = scaled_nearest_line / scale;
        return (nearest_edge, nearest_line);
    }
    //
    let (p1, _) = crate::line::nearest_to_point(edge_start, line_origin, line_direction);
    let (p2, _) = crate::line::nearest_to_point(edge_end, line_origin, line_direction);
    let dist1 = (p1 - edge_start).norm();
    let dist2 = (p2 - edge_end).norm();
    if dist1 < dist2 {
        let nearest_edge = edge_start;
        let nearest_line = p1;
        return (*nearest_edge, nearest_line);
    }
    let nearest_edge = edge_end;
    let nearest_line = p2;
    (*nearest_edge, nearest_line)
}

pub fn wdw_integral_of_inverse_distance_cubic(
    q: &nalgebra::Vector3::<f64>,
    p0: &nalgebra::Vector3::<f64>,
    p1: &nalgebra::Vector3::<f64>) -> (f64, nalgebra::Vector3::<f64>) {
    let len = (p1 - p0).norm();
    let lsinv = 1.0 / (len * len);
    // dist^2 = er^2+2br+c
    let d = (p0 - p1).dot(&(q - p0)) * lsinv;
    let a = (q - p0).norm_squared() * lsinv - d * d;
    // dist^2 = e{ x^2 + a2}, x = r + d
    // \int 1/sqrt(x^2+a)^3 dx = x/(a\sqrt{a+x^2})
    let f = |x: f64| x / (a * (a + x * x).sqrt());
    let v = (f(d + 1.) - f(d)) * lsinv;
    //
    let dd = (p0 - p1) * lsinv;
    let da = (q - p0).scale(2_f64) * lsinv - 2.0 * d * dd;
    // these formula was calculated by WolframAlpha
    let dfdx = |x: f64| 1_f64 / (a + x * x).powf(1.5);
    let dfda = |x: f64| -(x * (3. * a + 2. * x * x)) / (2. * a * a * (a + x * x).powf(1.5));
    let dv = (dfdx(d + 1.) - dfdx(d)) * dd + (dfda(d + 1.) - dfda(d)) * da;
    (v, dv * lsinv)
}


pub fn nearest_to_edge3<T>(
    p0: &nalgebra::Vector3::<T>,
    p1: &nalgebra::Vector3::<T>,
    q0: &nalgebra::Vector3::<T>,
    q1: &nalgebra::Vector3::<T>) -> (T, T, T)
    where T: nalgebra::RealField + Copy + 'static,
          f64: AsPrimitive<T>
{
    let vp = p1 - p0;
    let vq = q1 - q0;
    assert!(vp.norm() > T::zero());
    assert!(vq.norm() > T::zero());
    if vp.cross(&vq).norm() < T::default_epsilon() { // handling parallel edge
        let pq0 = p0 - q0;
        let uvp = vp.normalize();
        // a vector vertical to vp and vq and in the plane of vp and vq
        let vert = pq0 - uvp.scale(pq0.dot(&uvp));
        let dist = vert.norm(); // distance betwen two edges
        let lp0 = p0.dot(&uvp);
        let lp1 = p1.dot(&uvp);
        let lq0 = q0.dot(&uvp);
        let lq1 = q1.dot(&uvp);
        let (lp_min, lp_max, p_min, p_max, rp_min, rp_max)
            = (lp0, lp1, p0, p1, T::zero(), T::one());
        assert!(lp_min < lp_max);
        let (lq_min, lq_max, q_min, q_max, rq_min, rq_max) =
            if lq0 < lq1 {
                (
                    lq0, lq1, q0, q1, T::zero(), T::one())
            } else {
                (lq1, lq0, q1, q0, T::one(), T::zero())
            };
        if lp_max < lq_min { return ((p_max - q_min).norm(), rp_max, rq_min); }
        if lq_max < lp_min { return ((q_max - p_min).norm(), rp_min, rq_max); }
        let lm_min = lp_min.max(lq_min);
        let lm_max = lp_max.min(lq_max);
        let lm = (lm_min + lm_max) * 0.5f64.as_();
        let ratio_p = (lm - lp0) / (lp1 - lp0);
        let ratio_q = (lm - lq0) / (lq1 - lq0);
        return (dist, ratio_p, ratio_q);
    }
    let (rp1,rq1) = { // line-line intersection
        let t0 = vp.dot(&vp);
        let t1 = vq.dot(&vq);
        let t2 = vp.dot(&vq);
        let t3 = vp.dot(&(q0 - p0));
        let t4 = vq.dot(&(q0 - p0));
        let det = t0 * t1 - t2 * t2;
        let invdet = T::one() / det;
        let rp1 = (t1 * t3 - t2 * t4) * invdet;
        let rq1 = (t2 * t3 - t0 * t4) * invdet;
        (rp1,rq1)
    };
    if T::zero() <= rp1 && rp1 <= T::one() && T::zero() <= rq1 && rq1 <= T::one() { // both in range
        let pc = p0 + vp.scale(rp1);
        let qc = q0 + vq.scale(rq1);
        return ((pc - qc).norm(), rp1, rq1);
    }
    if (T::zero() <= rp1 && rp1 <= T::one()) && (rq1<=T::zero() || T::one()<=rq1) { // p in range
        let rq1 = clamp(rq1, T::zero(), T::one());
        let qc = q0 + vq.scale(rq1);
        let (dist, rp1) = nearest_to_point3(p0, p1, &qc);
        return (dist, rp1, rq1);
    }
    if (T::zero() <= rq1 && rq1 <= T::one()) && (rp1<=T::zero() || T::one()<=rp1) { // q in range
        let rp1 = clamp(rp1, T::zero(), T::one());
        let pc = p0 + vp.scale(rp1);
        let (dist, rq1) = nearest_to_point3(q0, q1, &pc);
        return (dist, rp1, rq1);
    }
    // convex projection technique
    let rp1 = clamp(rp1, T::zero(), T::one());
    let pc = p0 + vp.scale(rp1);
    let (_dist, rq1) = nearest_to_point3(q0, q1, &pc);
    let qc = q0 + (q1-q0).scale(rq1);
    let (_dist, rp1) = nearest_to_point3(p0, p1, &qc);
    let pc = p0 + (p1-p0).scale(rp1);
    let (dist, rq1) = nearest_to_point3(q0, q1, &pc);
    (dist, rp1, rq1)
}

#[cfg(test)]
mod tests {
    fn numerical(
        q: &nalgebra::Vector3::<f64>,
        p0: &nalgebra::Vector3::<f64>,
        p1: &nalgebra::Vector3::<f64>,
        n: usize,
        p: usize) -> f64 {
        use num_traits::Pow;
        let len = (p1 - p0).norm();
        let mut ret = 0.;
        for i_seg in 0..n {
            let r0 = i_seg as f64 / n as f64;
            let r1 = (i_seg + 1) as f64 / n as f64;
            let pr0q = p0.scale(1. - r0) + p1.scale(r0) - q;
            let pr1q = p0.scale(1. - r1) + p1.scale(r1) - q;
            let dist0 = pr0q.norm();
            let dist1 = pr1q.norm();
            let v0 = 1. / dist0.pow(p as i32);
            let v1 = 1. / dist1.pow(p as i32);
            let v = (v0 + v1) * 0.5;
            ret += v;
        }
        ret *= len / (n as f64);
        ret
    }

    #[test]
    fn test_wdw_integral_of_inverse_distance_cubic() {
        for _i in 0..10000 {
            let p0 = crate::vec3::sample_unit_cube();
            let p1 = crate::vec3::sample_unit_cube();
            let q = crate::vec3::sample_unit_cube();
            let len = (p0 - p1).norm();
            let height = crate::tri3::height(&p0, &p1, &q);
            if height < 0.1 { continue; }
            if len < 0.1 { continue; }
            if (p0 - q).norm() < 0.1 { continue; }
            if (p1 - q).norm() < 0.1 { continue; }
            // dbg!(numerical(&q, &p0, &p1, 10, 3));
            // dbg!(numerical(&q, &p0, &p1, 100, 3));
            // dbg!(numerical(&q, &p0, &p1, 1000, 3));
            let (v0, dv0) = crate::edge3::wdw_integral_of_inverse_distance_cubic(&q, &p0, &p1);
            assert!((v0 - numerical(&q, &p0, &p1, 1000, 3)).abs() < 1.0e-4 * v0.abs());
            let eps = 1.0e-4_f64;
            let qex = nalgebra::Vector3::new(q.x + eps, q.y, q.z);
            let qey = nalgebra::Vector3::new(q.x, q.y + eps, q.z);
            let qez = nalgebra::Vector3::new(q.x, q.y, q.z + eps);
            let vx = (numerical(&qex, &p0, &p1, 1000, 3) - v0) / eps;
            let vy = (numerical(&qey, &p0, &p1, 1000, 3) - v0) / eps;
            let vz = (numerical(&qez, &p0, &p1, 1000, 3) - v0) / eps;
            let dv1 = nalgebra::Vector3::<f64>::new(vx, vy, vz);
            // dbg!(p0, p1, q);
            assert!((dv0 - dv1).norm() < 0.03 * (dv0.norm() + 1.0));
        }
    }

    #[test]
    fn test_distance() {
        let eps = 1.0e-4;
        for _i in 0..10000 {
            let p0 = crate::vec3::sample_unit_cube::<f64>();
            let p1 = crate::vec3::sample_unit_cube::<f64>();
            let q0 = crate::vec3::sample_unit_cube::<f64>();
            let q1 = crate::vec3::sample_unit_cube::<f64>();
            let (dist, rp, rq) = crate::edge3::nearest_to_edge3(&p0, &p1, &q0, &q1);
            //
            let vp = p1 - p0;
            //let pc0 = p0 + f64::clamp(rp - eps, 0.0, 1.0) * vp;
            let pc0 = p0 + f64::clamp(rp - eps, 0.0, 1.0) * vp;
            let pc1 = p0 + rp * vp;
            let pc2 = p0 + f64::clamp(rp + eps, 0.0, 1.0) * vp;
            //
            let vq = q1 - q0;
            let qc0 = q0 + f64::clamp(rq - eps, 0.0, 1.0) * vq;
            let qc1 = q0 + rq * vq;
            let qc2 = q0 + f64::clamp(rq + eps, 0.0, 1.0) * vq;
            assert!((dist - (pc1 - qc1).norm()).abs() < 1.0e-5);
            assert!(dist <= (pc0 - qc0).norm());
            assert!(dist <= (pc0 - qc1).norm());
            assert!(dist <= (pc0 - qc2).norm());
            assert!(dist <= (pc1 - qc0).norm());
            assert!(dist <= (pc1 - qc2).norm());
            assert!(dist <= (pc2 - qc0).norm());
            assert!(dist <= (pc2 - qc1).norm());
            assert!(dist <= (pc2 - qc2).norm());
        }
    }
}