f3l_core 0.3.0

3D Point Cloud Library
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

use num_traits::Float;

/// x^2 + `factor_0` * x + `factor_1` = 0
pub fn root2<T: Float>(factor: [T; 2]) -> [T; 2] {
    let inner = (factor[0].powi(2) - T::from(4f32).unwrap() * factor[1]).max(T::zero());
    let mut out = [
        T::from(0.5f32).unwrap() * (-factor[0] - inner.sqrt()),
        T::from(0.5f32).unwrap() * (-factor[0] + inner.sqrt()),
    ];
    if out[0] > out[1] {
        out.swap(0, 1);
    }
    out
}

/// Ref: [wiki_en](https://en.wikipedia.org/wiki/Cubic_equation)<br>
/// x^3 + `factor_0` * x^2 + `factor_1` * x^ + `factor_2` = 0
pub fn root3<T: Float>(factor: [T; 3]) -> [T; 3] {
    let [b, c, d] = factor;
    if d == T::zero() {
        let qua = root2([b, c]);
        let mut out = [T::zero(), qua[0], qua[1]];
        out.sort_by(|&a, b| a.partial_cmp(b).unwrap());
        return out;
    }

    let beta = c / T::from(3f32).unwrap() - b.powi(2) / T::from(9f32).unwrap();
    let alpha = -b.powi(3) / T::from(27f32).unwrap() - d / T::from(2f32).unwrap()
        + b * c / T::from(6f32).unwrap();

    let front = -b / T::from(3f32).unwrap();
    let mul_factor = T::from(2).unwrap() * (-beta).sqrt();

    let theta_factor = alpha / (-beta).powf(T::from(1.5).unwrap());
    let theta_factor = theta_factor.max(-T::one());
    let theta_factor = theta_factor.min(T::one());

    let x1 = front + mul_factor * (theta_factor.acos() / T::from(3f32).unwrap()).cos();
    let x2 = front
        + mul_factor
            * ((theta_factor.acos() + T::from(2f32 * std::f32::consts::PI).unwrap())
                / T::from(3f32).unwrap())
            .cos();
    let x3 = front
        + mul_factor
            * ((theta_factor.acos() - T::from(2f32 * std::f32::consts::PI).unwrap())
                / T::from(3f32).unwrap())
            .cos();

    let mut out = [x1, x2, x3];
    out.sort_by(|&a, b| a.partial_cmp(b).unwrap());
    out
}

/// Ref: [wiki_zh](https://zh.wikipedia.org/zh-tw/%E4%B8%89%E6%AC%A1%E6%96%B9%E7%A8%8B)
pub fn root3_eq2<T: Float>(factor: [T; 3]) -> [T; 3] {
    let [b, c, d] = factor;
    if d == T::zero() {
        let qua = root2([b, c]);
        let mut out = [T::zero(), qua[0], qua[1]];
        out.sort_by(|&a, b| a.partial_cmp(b).unwrap());
        return out;
    }

    let a_ = b.sqrt() - T::from(3f32).unwrap() * c;
    let b_ = b * c - T::from(9).unwrap() * d;
    let c_ = c.powi(2) - T::from(3f32).unwrap() * b * d;
    let tri = b_.powi(2) - T::from(4).unwrap() * a_ * c_;

    let mut out = if a_ == b_ {
        let ans = -c / b;
        return [ans, ans, ans];
    } else if tri > T::zero() {
        let top = tri.sqrt();
        let (y1, y2) = (
            (a_ * b + T::from(3f32).unwrap() * (-b_ - top) / T::from(2f32).unwrap())
                .powf(T::from(1. / 3f32).unwrap()),
            (a_ * b + T::from(3f32).unwrap() * (-b_ + top) / T::from(2f32).unwrap())
                .powf(T::from(1. / 3f32).unwrap()),
        );
        [
            (-b - (y1 + y2)) / T::from(3f32).unwrap(),
            (-T::from(2f32).unwrap() * b + (y1 + y2) - T::from(3f32.sqrt()).unwrap() * (y1 - y2))
                / T::from(6f32).unwrap(),
            (-T::from(2f32).unwrap() * b + (y1 + y2) + T::from(3f32.sqrt()).unwrap() * (y1 - y2))
                / T::from(6f32).unwrap(),
        ]
    } else if tri == T::zero() {
        let k = b_ / a_;
        [
            -b + k,
            -k * T::from(0.5f32).unwrap(),
            -k * T::from(0.5f32).unwrap(),
        ]
    } else {
        let t = (T::from(2f32).unwrap() * a_ * b - T::from(3f32).unwrap() * b_)
            / (T::from(2f32).unwrap() * a_ * a_.sqrt());
        let theta = t.min(T::one()).max(-T::one()).acos() / T::from(3f32).unwrap();
        let cs = theta.cos();
        let ss = theta.sin();
        let a_sqrt = a_.sqrt();
        [
            (-b - T::from(2f32).unwrap() * a_sqrt * cs) / T::from(3f32).unwrap(),
            (-b + a_sqrt * (cs - T::from(3f32.sqrt()).unwrap() * ss)) / T::from(3f32).unwrap(),
            (-b + a_sqrt * (cs + T::from(3f32.sqrt()).unwrap() * ss)) / T::from(3f32).unwrap(),
        ]
    };
    out.sort_by(|&a, b| a.partial_cmp(b).unwrap());
    out
}

#[test]
fn test_root2() {
    // x^2 - 5 * x + 4 = 0
    let factor = [-5f32, 4f32];
    let answer = [1f32, 4f32];

    let target = root2(factor);
    assert_eq!(answer, crate::round_slice_n(target, 4));
}

#[test]
fn test_root3() {
    // x^3 + 0 * x^2 -15 * x -4 = 0
    let factor = [0f32, -15., -4.];
    let answer = crate::round_slice_n([-3.732050807568877, -0.26794919243112064, 4f32], 4);

    let target1 = crate::round_slice_n(root3(factor), 4);
    let target2 = crate::round_slice_n(root3_eq2(factor), 4);

    assert_eq!(answer, target1);
    assert_eq!(answer, target2);
}

#[test]
fn test_root3_d0() {
    // x^3 + 0 * x^2 -15 * x -4 = 0
    let factor = [0f32, -15., 0.];
    let answer = [-3.873, 0f32, 3.873];

    let target = root3(factor);

    assert_eq!(
        crate::round_slice_n(answer, 4),
        crate::round_slice_n(target, 4)
    );
}

#[test]
fn test_root3_by_eigenvalues() {
    use crate::matrix3x3::compute_eigenvalues;
    use f3l_glam::glam::{Mat3, Vec3};

    let factor = [0f32, -15., -4.];

    // a x^3 + b x^2 + c x + d = 0
    // | 0  0  -d/a |
    // | 1  0  -c/a |
    // | 0  1  -b/a |

    let mat = Mat3::from_cols(
        Vec3::new(0., 1., 0.),
        Vec3::new(0., 0., 1.),
        Vec3::new(-factor[2], -factor[1], -factor[0]),
    );

    let mut answer = crate::round_slice_n([-3.732050807568877, -0.26794919243112064, 4f32], 4);
    let mut target = crate::round_slice_n(compute_eigenvalues::<f32>(mat), 4);

    answer.sort_by(|&a, b| a.partial_cmp(b).unwrap());
    target.sort_by(|&a, b| a.partial_cmp(b).unwrap());

    assert_eq!(answer, target);
}