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    use alga::general::{
        AbstractGroup, AbstractLoop, AbstractMagma, AbstractMonoid, AbstractQuasigroup,
        AbstractSemigroup, Identity, TwoSidedInverse, Multiplicative, RealField,
    };
    use alga::linear::{ProjectiveTransformation, Transformation};

    use crate::base::allocator::Allocator;
    use crate::base::dimension::{DimNameAdd, DimNameSum, U1};
    use crate::base::{DefaultAllocator, VectorN};

    use crate::geometry::{Point, SubTCategoryOf, TCategory, TProjective, Transform};

    /*
     *
     * Algebraic structures.
     *
     */
    impl<N: RealField, D: DimNameAdd<U1>, C> Identity<Multiplicative> for Transform<N, D, C>
    where
        C: TCategory,
        DefaultAllocator: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>>,
    {
        #[inline]
        fn identity() -> Self {
            Self::identity()
        }
    }

    impl<N: RealField, D: DimNameAdd<U1>, C> TwoSidedInverse<Multiplicative> for Transform<N, D, C>
    where
        C: SubTCategoryOf<TProjective>,
        DefaultAllocator: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>>,
    {
        #[inline]
        fn two_sided_inverse(&self) -> Self {
            self.clone().inverse()
        }

        #[inline]
        fn two_sided_inverse_mut(&mut self) {
            self.inverse_mut()
        }
    }

    impl<N: RealField, D: DimNameAdd<U1>, C> AbstractMagma<Multiplicative> for Transform<N, D, C>
    where
        C: TCategory,
        DefaultAllocator: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>>,
    {
        #[inline]
        fn operate(&self, rhs: &Self) -> Self {
            self * rhs
        }
    }

    macro_rules! impl_multiplicative_structures(
        ($($marker: ident<$operator: ident>),* $(,)*) => {$(
            impl<N: RealField, D: DimNameAdd<U1>, C> $marker<$operator> for Transform<N, D, C>
                where C: TCategory,
                      DefaultAllocator: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>> { }
        )*}
    );

    macro_rules! impl_inversible_multiplicative_structures(
        ($($marker: ident<$operator: ident>),* $(,)*) => {$(
            impl<N: RealField, D: DimNameAdd<U1>, C> $marker<$operator> for Transform<N, D, C>
                where C: SubTCategoryOf<TProjective>,
                      DefaultAllocator: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>> { }
        )*}
    );

    impl_multiplicative_structures!(
        AbstractSemigroup<Multiplicative>,
        AbstractMonoid<Multiplicative>,
    );

    impl_inversible_multiplicative_structures!(
        AbstractQuasigroup<Multiplicative>,
        AbstractLoop<Multiplicative>,
        AbstractGroup<Multiplicative>
    );

    /*
     *
     * Transformation groups.
     *
     */
    impl<N, D: DimNameAdd<U1>, C> Transformation<Point<N, D>> for Transform<N, D, C>
    where
        N: RealField,
        C: TCategory,
        DefaultAllocator: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>>
            + Allocator<N, DimNameSum<D, U1>>
            + Allocator<N, D, D>
            + Allocator<N, D>,
    {
        #[inline]
        fn transform_point(&self, pt: &Point<N, D>) -> Point<N, D> {
            self.transform_point(pt)
        }

        #[inline]
        fn transform_vector(&self, v: &VectorN<N, D>) -> VectorN<N, D> {
            self.transform_vector(v)
        }
    }

    impl<N, D: DimNameAdd<U1>, C> ProjectiveTransformation<Point<N, D>> for Transform<N, D, C>
    where
        N: RealField,
        C: SubTCategoryOf<TProjective>,
        DefaultAllocator: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>>
            + Allocator<N, DimNameSum<D, U1>>
            + Allocator<N, D, D>
            + Allocator<N, D>,
    {
        #[inline]
        fn inverse_transform_point(&self, pt: &Point<N, D>) -> Point<N, D> {
            self.inverse_transform_point(pt)
        }

        #[inline]
        fn inverse_transform_vector(&self, v: &VectorN<N, D>) -> VectorN<N, D> {
            self.inverse_transform_vector(v)
        }
    }

    // FIXME: we need to implement an SVD for this.
    //
    // impl<N, D: DimNameAdd<U1>, C> AffineTransformation<Point<N, D>> for Transform<N, D, C>
    //     where N:  RealField,
    //           C: SubTCategoryOf<TAffine>,
    //           DefaultAllocator: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>> +
    //                             Allocator<N, D, D> +
    //                             Allocator<N, D> {
    //     type PreRotation       = Rotation<N, D>;
    //     type NonUniformScaling = VectorN<N, D>;
    //     type PostRotation      = Rotation<N, D>;
    //     type Translation       = Translation<N, D>;
    //
    //     #[inline]
    //     fn decompose(&self) -> (Self::Translation, Self::PostRotation, Self::NonUniformScaling, Self::PreRotation) {
    //         unimplemented!()
    //     }
    // }

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