phyz 0.3.0

Multi-physics differentiable simulation engine
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

//! Quaternion utilities for 3D rotations.
//!
//! Convention: q = [w; x; y; z] where w is scalar, (x,y,z) is vector part.

use crate::{Mat3, Vec3};

/// A unit quaternion representing a 3D rotation.
#[derive(Debug, Clone, Copy)]
pub struct Quat {
    /// Scalar part (w).
    pub w: f64,
    /// Vector part (x, y, z).
    pub v: Vec3,
}

impl Quat {
    /// Create a new quaternion from scalar and vector parts.
    pub fn new(w: f64, x: f64, y: f64, z: f64) -> Self {
        Self {
            w,
            v: Vec3::new(x, y, z),
        }
    }

    /// Identity quaternion (no rotation).
    pub fn identity() -> Self {
        Self {
            w: 1.0,
            v: Vec3::zero(),
        }
    }

    /// Create quaternion from axis-angle representation.
    /// axis should be a unit vector, angle in radians.
    pub fn from_axis_angle(axis: &Vec3, angle: f64) -> Self {
        let half_angle = angle * 0.5;
        let (s, c) = half_angle.sin_cos();
        Self { w: c, v: *axis * s }
    }

    /// Normalize this quaternion to unit length.
    pub fn normalize(&self) -> Self {
        let norm = (self.w * self.w + self.v.norm_sq()).sqrt();
        if norm < 1e-12 {
            return Self::identity();
        }
        Self {
            w: self.w / norm,
            v: self.v / norm,
        }
    }

    /// Quaternion multiplication: self * other.
    pub fn mul(&self, other: &Quat) -> Quat {
        Quat {
            w: self.w * other.w - self.v.dot(other.v),
            v: self.v.cross(other.v) + other.v * self.w + self.v * other.w,
        }
    }

    /// Conjugate of the quaternion (inverse for unit quaternions).
    pub fn conjugate(&self) -> Quat {
        Quat {
            w: self.w,
            v: -self.v,
        }
    }

    /// Convert quaternion to 3x3 rotation matrix.
    pub fn to_matrix(&self) -> Mat3 {
        let w = self.w;
        let x = self.v.x;
        let y = self.v.y;
        let z = self.v.z;

        let x2 = x * x;
        let y2 = y * y;
        let z2 = z * z;
        let xy = x * y;
        let xz = x * z;
        let yz = y * z;
        let wx = w * x;
        let wy = w * y;
        let wz = w * z;

        Mat3::new(
            1.0 - 2.0 * (y2 + z2),
            2.0 * (xy - wz),
            2.0 * (xz + wy),
            2.0 * (xy + wz),
            1.0 - 2.0 * (x2 + z2),
            2.0 * (yz - wx),
            2.0 * (xz - wy),
            2.0 * (yz + wx),
            1.0 - 2.0 * (x2 + y2),
        )
    }

    /// Convert rotation matrix to quaternion.
    /// Reference: Shepperd's method (stable for all rotation matrices).
    pub fn from_matrix(m: &Mat3) -> Quat {
        let trace = m[(0, 0)] + m[(1, 1)] + m[(2, 2)];

        if trace > 0.0 {
            let s = (trace + 1.0).sqrt() * 2.0; // s = 4*w
            Quat {
                w: 0.25 * s,
                v: Vec3::new(
                    (m[(2, 1)] - m[(1, 2)]) / s,
                    (m[(0, 2)] - m[(2, 0)]) / s,
                    (m[(1, 0)] - m[(0, 1)]) / s,
                ),
            }
        } else if m[(0, 0)] > m[(1, 1)] && m[(0, 0)] > m[(2, 2)] {
            let s = (1.0 + m[(0, 0)] - m[(1, 1)] - m[(2, 2)]).sqrt() * 2.0; // s = 4*x
            Quat {
                w: (m[(2, 1)] - m[(1, 2)]) / s,
                v: Vec3::new(
                    0.25 * s,
                    (m[(0, 1)] + m[(1, 0)]) / s,
                    (m[(0, 2)] + m[(2, 0)]) / s,
                ),
            }
        } else if m[(1, 1)] > m[(2, 2)] {
            let s = (1.0 + m[(1, 1)] - m[(0, 0)] - m[(2, 2)]).sqrt() * 2.0; // s = 4*y
            Quat {
                w: (m[(0, 2)] - m[(2, 0)]) / s,
                v: Vec3::new(
                    (m[(0, 1)] + m[(1, 0)]) / s,
                    0.25 * s,
                    (m[(1, 2)] + m[(2, 1)]) / s,
                ),
            }
        } else {
            let s = (1.0 + m[(2, 2)] - m[(0, 0)] - m[(1, 1)]).sqrt() * 2.0; // s = 4*z
            Quat {
                w: (m[(1, 0)] - m[(0, 1)]) / s,
                v: Vec3::new(
                    (m[(0, 2)] + m[(2, 0)]) / s,
                    (m[(1, 2)] + m[(2, 1)]) / s,
                    0.25 * s,
                ),
            }
        }
    }

    /// Exponential map: exp(theta u) where theta u is axis-angle representation.
    /// Converts axis-angle to quaternion via q = [cos(theta/2), sin(theta/2) * u].
    /// For small angles, uses first-order approximation.
    pub fn exp(w: &Vec3) -> Quat {
        let theta = w.norm();
        if theta < 1e-10 {
            // First-order approximation for small angles
            Quat {
                w: 1.0,
                v: *w * 0.5,
            }
            .normalize()
        } else {
            let half_theta = theta * 0.5;
            Quat {
                w: half_theta.cos(),
                v: *w * (half_theta.sin() / theta),
            }
        }
    }

    /// Logarithmic map: log(q) returns the axis-angle vector theta u such that q = exp(theta u).
    pub fn log(&self) -> Vec3 {
        let v_norm = self.v.norm();
        if v_norm < 1e-10 {
            return Vec3::zero();
        }
        // For quaternion q = [cos(theta/2), sin(theta/2) * u], we have:
        // theta = 2 * atan2(sin(theta/2), cos(theta/2)) = 2 * atan2(|v|, w)
        let angle = 2.0 * v_norm.atan2(self.w);
        self.v * (angle / v_norm)
    }
}

#[cfg(test)]
mod tests {
    use super::*;

    const EPS: f64 = 1e-10;

    #[test]
    fn test_identity() {
        let q = Quat::identity();
        assert_eq!(q.w, 1.0);
        assert_eq!(q.v, Vec3::zero());
    }

    #[test]
    fn test_axis_angle() {
        let axis = Vec3::new(0.0, 0.0, 1.0);
        let angle = std::f64::consts::FRAC_PI_2; // 90 degrees
        let q = Quat::from_axis_angle(&axis, angle);

        let expected_w = (angle / 2.0).cos();
        let expected_z = (angle / 2.0).sin();

        assert!((q.w - expected_w).abs() < EPS);
        assert!((q.v.z - expected_z).abs() < EPS);
    }

    #[test]
    fn test_normalize() {
        let q = Quat::new(1.0, 2.0, 3.0, 4.0);
        let normalized = q.normalize();
        let norm = (normalized.w * normalized.w + normalized.v.norm_sq()).sqrt();
        assert!((norm - 1.0).abs() < EPS);
    }

    #[test]
    fn test_multiplication() {
        // 90 degree rotation about Z, then 90 degree rotation about Z
        let axis = Vec3::new(0.0, 0.0, 1.0);
        let q1 = Quat::from_axis_angle(&axis, std::f64::consts::FRAC_PI_2);
        let q2 = Quat::from_axis_angle(&axis, std::f64::consts::FRAC_PI_2);
        let result = q1.mul(&q2);

        // Should equal 180 degree rotation about Z
        let expected = Quat::from_axis_angle(&axis, std::f64::consts::PI);

        assert!((result.w - expected.w).abs() < EPS);
        assert!((result.v.x - expected.v.x).abs() < EPS);
        assert!((result.v.y - expected.v.y).abs() < EPS);
        assert!((result.v.z - expected.v.z).abs() < EPS);
    }

    #[test]
    fn test_to_matrix() {
        let axis = Vec3::new(0.0, 0.0, 1.0);
        let angle = std::f64::consts::FRAC_PI_2;
        let q = Quat::from_axis_angle(&axis, angle);
        let m = q.to_matrix();

        // 90 degree rotation about Z should map X to Y
        let x = Vec3::new(1.0, 0.0, 0.0);
        let y = m * x;
        assert!((y.x - 0.0).abs() < EPS);
        assert!((y.y - 1.0).abs() < EPS);
        assert!((y.z - 0.0).abs() < EPS);
    }

    #[test]
    fn test_matrix_roundtrip() {
        let axis = Vec3::new(1.0, 2.0, 3.0).normalize();
        let angle = 0.7;
        let q = Quat::from_axis_angle(&axis, angle);
        let m = q.to_matrix();
        let q2 = Quat::from_matrix(&m);

        // Quaternions q and -q represent the same rotation
        let same = (q.w - q2.w).abs() < EPS && (q.v - q2.v).norm() < EPS;
        let negated = (q.w + q2.w).abs() < EPS && (q.v + q2.v).norm() < EPS;

        assert!(same || negated);
    }

    #[test]
    fn test_exp_log() {
        let w = Vec3::new(0.1, 0.2, 0.3);
        let q = Quat::exp(&w);
        let w2 = q.log();
        assert!((w.x - w2.x).abs() < EPS);
        assert!((w.y - w2.y).abs() < EPS);
        assert!((w.z - w2.z).abs() < EPS);
    }

    #[test]
    fn test_conjugate() {
        let q = Quat::new(0.5, 0.5, 0.5, 0.5).normalize();
        let conj = q.conjugate();
        let result = q.mul(&conj);
        assert!((result.w - 1.0).abs() < EPS);
        assert!(result.v.norm() < EPS);
    }
}