```  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
```
```use na::RealField;

use crate::math::{Isometry, Point, Translation, Vector};
use crate::utils::IsometryOps;

/// A continuous rigid motion.
///
/// This is a function, assumed to be continuous, that, given a parameter `t` returns a direct isometry.
/// Mathematically speaking this is a one-parameter curve on the space of direct isometries. This curve
/// should have a continuity of at least `C0`.
pub trait RigidMotion<N: RealField> {
/// Get a position at the time `t`.
fn position_at_time(&self, t: N) -> Isometry<N>;
}

impl<N: RealField> RigidMotion<N> for Isometry<N> {
fn position_at_time(&self, _: N) -> Isometry<N> {
*self
}
}

/// Interpolation between two isometries using LERP for the translation part and SLERP for the rotation.
pub struct InterpolatedRigidMotion<N: RealField> {
/// The transformation at `t = 0.0`.
pub start: Isometry<N>,
/// The transformation at `t = 1.0`.
pub end: Isometry<N>,
}

impl<N: RealField> InterpolatedRigidMotion<N> {
/// Initialize a lerp-slerp interpolation with the given start and end transformations.
///
/// The `start` is the transformation at the time `t = 0.0` and `end` is the transformation at
/// the time `t = 1.0`.
pub fn new(start: Isometry<N>, end: Isometry<N>) -> Self {
InterpolatedRigidMotion { start, end }
}
}

impl<N: RealField> RigidMotion<N> for InterpolatedRigidMotion<N> {
fn position_at_time(&self, t: N) -> Isometry<N> {
self.start.lerp_slerp(&self.end, t)
}
}

/// A linear motion from a starting isometry traveling at constant translational velocity.
pub struct ConstantLinearVelocityRigidMotion<N: RealField> {
/// The time at which this parametrization begins. Can be negative.
pub t0: N,
/// The starting isometry at `t = self.t0`.
pub start: Isometry<N>,
/// The translational velocity of this motion.
pub velocity: Vector<N>,
}

impl<N: RealField> ConstantLinearVelocityRigidMotion<N> {
/// Initialize a linear motion from a starting isometry and a translational velocity.
pub fn new(t0: N, start: Isometry<N>, velocity: Vector<N>) -> Self {
ConstantLinearVelocityRigidMotion {
t0,
start,
velocity,
}
}
}

impl<N: RealField> RigidMotion<N> for ConstantLinearVelocityRigidMotion<N> {
fn position_at_time(&self, t: N) -> Isometry<N> {
Isometry::from_parts(
(self.start.translation.vector + self.velocity * (t - self.t0)).into(),
self.start.rotation,
)
}
}

/// A linear motion from a starting isometry traveling at constant translational velocity.
#[derive(Debug)]
pub struct ConstantVelocityRigidMotion<N: RealField> {
/// The time at which this parametrization begins. Can be negative.
pub t0: N,
/// The starting isometry at `t = self.t0`.
pub start: Isometry<N>,
/// The local-space point at which the rotational part of this motion is applied.
pub local_center: Point<N>,
/// The translational velocity of this motion.
pub linvel: Vector<N>,
/// The angular velocity of this motion.
#[cfg(feature = "dim2")]
pub angvel: N,
/// The angular velocity of this motion.
#[cfg(feature = "dim3")]
pub angvel: Vector<N>,
}

impl<N: RealField> ConstantVelocityRigidMotion<N> {
/// Initialize a motion from a starting isometry and linear and angular velocities.
#[cfg(feature = "dim2")]
pub fn new(
t0: N,
start: Isometry<N>,
local_center: Point<N>,
linvel: Vector<N>,
angvel: N,
) -> Self {
ConstantVelocityRigidMotion {
t0,
start,
local_center,
linvel,
angvel,
}
}

/// Initialize a motion from a starting isometry and linear and angular velocities.
#[cfg(feature = "dim3")]
pub fn new(
t0: N,
start: Isometry<N>,
local_center: Point<N>,
linvel: Vector<N>,
angvel: Vector<N>,
) -> Self {
ConstantVelocityRigidMotion {
t0,
start,
local_center,
linvel,
angvel,
}
}
}

impl<N: RealField> RigidMotion<N> for ConstantVelocityRigidMotion<N> {
fn position_at_time(&self, t: N) -> Isometry<N> {
let scaled_linvel = self.linvel * (t - self.t0);
let scaled_angvel = self.angvel * (t - self.t0);

let center = self.start.rotation * self.local_center.coords;
let lhs = self.start.translation * Translation::from(center);
let rhs = Translation::from(-center) * self.start.rotation;

lhs * Isometry::new(scaled_linvel, scaled_angvel) * rhs
}
}

/*
* For composition.
*/

/// Trait for composing some rigid motions.
pub trait RigidMotionComposition<N: RealField>: RigidMotion<N> {
/// Prepend a translation to the rigid motion `self`.
fn prepend_translation(&self, translation: Vector<N>) -> PrependTranslation<N, Self> {
PrependTranslation {
motion: self,
translation,
}
}

/// Prepend a transformation to the rigid motion `self`.
fn prepend_transformation(
&self,
transformation: Isometry<N>,
) -> PrependTransformation<N, Self> {
PrependTransformation {
motion: self,
transformation,
}
}
}

impl<N: RealField, M: ?Sized + RigidMotion<N>> RigidMotionComposition<N> for M {}

/// The result of prepending a translation to a rigid motion.
pub struct PrependTranslation<'a, N: RealField, M: ?Sized> {
motion: &'a M,
translation: Vector<N>,
}

impl<'a, N: RealField, M: ?Sized + RigidMotion<N>> RigidMotion<N> for PrependTranslation<'a, N, M> {
fn position_at_time(&self, t: N) -> Isometry<N> {
let m = self.motion.position_at_time(t);
m * Translation::from(self.translation)
}
}

/// The result of prepending an isometric transformation to a rigid motion.
pub struct PrependTransformation<'a, N: RealField, M: ?Sized> {
motion: &'a M,
transformation: Isometry<N>,
}

impl<'a, N: RealField, M: ?Sized + RigidMotion<N>> RigidMotion<N>
for PrependTransformation<'a, N, M>
{
fn position_at_time(&self, t: N) -> Isometry<N> {
let m = self.motion.position_at_time(t);
m * self.transformation
}
}
```