Most call sites still invoke UB through `assume_init`. Said call sites instead invoke `unimplemented!()` if the `no_unsound_assume_init` feature is enabled, to make it easier to gradually fix them.
Progress towards #556.
This implements `UnitDualQuaternion` as an alternative to `Isometry3`
for representing 3D isometries, which also provides the `sclerp`
operation which can be used to perform screw-linear interpolation
between two unit dual quaternions.
The various nalgebra-lapack FooScalars are still Copy because they make use of uninitialized memory.
nalgebgra-glm Number still uses Copy because upstream `approx` requires it.
`./ci/test.sh` now passes locally.
Refactoring done via the following sed commands:
```bash
export RELEVANT_SOURCEFILES="$(find src -name '*.rs') $(find examples -name '*.rs')"
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N\([0-9]\?\): *Scalar + \(Arbitrary\)/N\1: Scalar + Copy + \2/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N\([0-9]\?\): *Scalar + \(Serialize\)/N\1: Scalar + Copy + \2/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N\([0-9]\?\): *Scalar + \(Deserialize\)/N\1: Scalar + Copy + \2/' $f; do
export RELEVANT_SOURCEFILES="$(find nalgebra-glm -name '*.rs')"
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N\([0-9]\?\): *Scalar,/N\1: Scalar + Copy,/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N\([0-9]\?\): *Scalar>/N\1: Scalar + Copy>/' $f; done
for f in algebra-glm/src/traits.rs; do sed -i 's/Scalar + Ring/Scalar + Copy + Ring>/' $f; done # Number trait definition
```
This should semantically be a no-op, but enables refactorings to use non-Copy scalars on a case-by-case basis.
Also, the only instance of a `One + Zero` trait bound was changed into a `Zero + One` bound to match the others.
The following sed scripts were used in the refactoring (with each clause added to reduce the error count of `cargo check`):
```bash
export RELEVANT_SOURCEFILES="$(find src -name '*.rs') $(find examples -name '*.rs')"
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N: Scalar,/N: Scalar+Copy,/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N: Scalar + Field/N: Scalar + Copy + Field/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N: Scalar + Zero/N: Scalar + Copy + Zero/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N: Scalar + Closed/N: Scalar + Copy + Closed/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N: Scalar + Eq/N: Scalar + Copy + Eq/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N: Scalar + PartialOrd/N: Scalar + Copy + PartialOrd/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N: *Scalar + Zero/N: Scalar + Copy + Zero/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N: Scalar + PartialEq/N: Scalar + Copy + PartialEq/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N: Scalar>/N: Scalar+Copy>/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N: Scalar + $bound/N: Scalar + Copy + $bound/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N: *Scalar + $bound/N: Scalar + Copy + $bound/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N\([0-9]\): *Scalar,/N\1: Scalar+Copy,/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N: *Scalar + $trait/N: Scalar + Copy + $trait/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N\([0-9]\): *Scalar + Superset/N\1: Scalar + Copy + Superset/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N\([0-9]\): *Scalar + \([a-zA-Z]*Eq\)/N\1: Scalar + Copy + \2/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N\([0-9]\?\): *Scalar + \([a-zA-Z]*Eq\)/N\1: Scalar + Copy + \2/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N\([0-9]\?\): *Scalar + \(hash::\)/N\1: Scalar + Copy + \2/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N\([0-9]\?\): *Scalar {/N\1: Scalar + Copy {/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N\([0-9]\?\): *Scalar + \(Zero\)/N\1: Scalar + Copy + \2/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N\([0-9]\?\): *Scalar + \(Bounded\)/N\1: Scalar + Copy + \2/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N\([0-9]\?\): *Scalar + \(Lattice\)/N\1: Scalar + Copy + \2/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N\([0-9]\?\): *Scalar + \(Meet\|Join\)/N\1: Scalar + Copy + \2/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N\([0-9]\?\): *Scalar + \(fmt::\)/N\1: Scalar + Copy + \2/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N\([0-9]\?\): *Scalar + \(Ring\)/N\1: Scalar + Copy + \2/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N\([0-9]\?\): *Scalar + \(Hash\)/N\1: Scalar + Copy + \2/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N\([0-9]\?\): *Scalar + \(Send\|Sync\)/N\1: Scalar + Copy + \2/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/One + Zero/Zero + One/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N\([0-9]\?\): *Scalar + \(Zero\)/N\1: Scalar + Copy + \2/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N\([0-9]\?\): *Scalar + \($marker\)/N\1: Scalar + Copy + \2/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/N\([0-9]\?\): *Scalar>/N\1: Scalar + Copy>/' $f; done
for f in $RELEVANT_SOURCEFILES; do sed -i 's/Scalar+Copy/Scalar + Copy/' $f; done
```
The previous implementation had stability problems for small angles due
to the behaviour of the arccosine it used. In particular, it needs a
hack to handle "cosines" greater than 1 and the smallest obtainable
nonzero angle for e.g. f32 is acos(1-2^-22) = 0.00069...
These problems can be fixed by using an arctangent-based formula.
This adds `transform_point`, `transform_vector`,
`inverse_transform_point` and `inverse_transform_vector` as inherent
methods on the applicable geometric transformation structures, such
that they can be used without the need to import the `Transformation`
and `ProjectiveTransformation` traits from `alga`.
