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    use crate::general::{
        ClosedDiv, ClosedMul, ClosedNeg, ComplexField, Id, MultiplicativeGroup, MultiplicativeMonoid,
        RealField, SubsetOf, TwoSidedInverse,
    };
    use crate::linear::{EuclideanSpace, NormedSpace};

    // NOTE: A subgroup trait inherit from its parent groups.

    /// A general transformation acting on an euclidean space. It may not be inversible.
    pub trait Transformation<E: EuclideanSpace>: MultiplicativeMonoid {
        /// Applies this group's action on a point from the euclidean space.
        fn transform_point(&self, pt: &E) -> E;

        /// Applies this group's action on a vector from the euclidean space.
        ///
        /// If `v` is a vector and `a, b` two point such that `v = a - b`, the action `∘` on a vector
        /// is defined as `self ∘ v = (self × a) - (self × b)`.
        fn transform_vector(&self, v: &E::Coordinates) -> E::Coordinates;
    }

    /// The most general form of invertible transformations on an euclidean space.
    pub trait ProjectiveTransformation<E: EuclideanSpace>:
        MultiplicativeGroup + Transformation<E>
    {
        /// Applies this group's two_sided_inverse action on a point from the euclidean space.
        fn inverse_transform_point(&self, pt: &E) -> E;

        /// Applies this group's two_sided_inverse action on a vector from the euclidean space.
        ///
        /// If `v` is a vector and `a, b` two point such that `v = a - b`, the action `∘` on a vector
        /// is defined as `self ∘ v = (self × a) - (self × b)`.
        fn inverse_transform_vector(&self, v: &E::Coordinates) -> E::Coordinates;
    }

    /// The group of affine transformations. They are decomposable into a rotation, a non-uniform
    /// scaling, a second rotation, and a translation (applied in that order).
    pub trait AffineTransformation<E: EuclideanSpace>: ProjectiveTransformation<E> {
        /// Type of the first rotation to be applied.
        type Rotation: Rotation<E>;
        /// Type of the non-uniform scaling to be applied.
        type NonUniformScaling: AffineTransformation<E>;
        /// The type of the pure translation part of this affine transformation.
        type Translation: Translation<E>;

        /// Decomposes this affine transformation into a rotation followed by a non-uniform scaling,
        /// followed by a rotation, followed by a translation.
        fn decompose(
            &self,
        ) -> (
            Self::Translation,
            Self::Rotation,
            Self::NonUniformScaling,
            Self::Rotation,
        );
        // FIXME: add a `recompose` method?

        /*
         * Composition with components.
         */
        /// Appends a translation to this similarity.
        fn append_translation(&self, t: &Self::Translation) -> Self;

        /// Prepends a translation to this similarity.
        fn prepend_translation(&self, t: &Self::Translation) -> Self;

        /// Appends a rotation to this similarity.
        fn append_rotation(&self, r: &Self::Rotation) -> Self;

        /// Prepends a rotation to this similarity.
        fn prepend_rotation(&self, r: &Self::Rotation) -> Self;

        /// Appends a scaling factor to this similarity.
        fn append_scaling(&self, s: &Self::NonUniformScaling) -> Self;

        /// Prepends a scaling factor to this similarity.
        fn prepend_scaling(&self, s: &Self::NonUniformScaling) -> Self;

        /// Appends to this similarity a rotation centered at the point `p`, i.e., this point is left
        /// invariant.
        ///
        /// May return `None` if `Self` does not have enough translational degree of liberty to perform
        /// this computation.
        #[inline]
        fn append_rotation_wrt_point(&self, r: &Self::Rotation, p: &E) -> Option<Self> {
            if let Some(t) = Self::Translation::from_vector(p.coordinates()) {
                let it = t.two_sided_inverse();
                Some(
                    self.append_translation(&it)
                        .append_rotation(&r)
                        .append_translation(&t),
                )
            } else {
                None
            }
        }
    }

    /// Subgroups of the similarity group `S(n)`, i.e., rotations, translations, and (signed) uniform scaling.
    ///
    /// Similarities map lines to lines and preserve angles.
    pub trait Similarity<E: EuclideanSpace>:
        AffineTransformation<E, NonUniformScaling = <Self as Similarity<E>>::Scaling>
    {
        /// The type of the pure (uniform) scaling part of this similarity transformation.
        type Scaling: Scaling<E>;

        /*
         * Components retrieval.
         */
        /// The pure translational component of this similarity transformation.
        fn translation(&self) -> Self::Translation;

        /// The pure rotational component of this similarity transformation.
        fn rotation(&self) -> Self::Rotation;

        /// The pure scaling component of this similarity transformation.
        fn scaling(&self) -> Self::Scaling;

        /*
         * Transformations.
         */
        /// Applies this transformation's pure translational part to a point.
        #[inline]
        fn translate_point(&self, pt: &E) -> E {
            self.translation().transform_point(pt)
        }

        /// Applies this transformation's pure rotational part to a point.
        #[inline]
        fn rotate_point(&self, pt: &E) -> E {
            self.rotation().transform_point(pt)
        }

        /// Applies this transformation's pure scaling part to a point.
        #[inline]
        fn scale_point(&self, pt: &E) -> E {
            self.scaling().transform_point(pt)
        }

        /// Applies this transformation's pure rotational part to a vector.
        #[inline]
        fn rotate_vector(&self, pt: &E::Coordinates) -> E::Coordinates {
            self.rotation().transform_vector(pt)
        }

        /// Applies this transformation's pure scaling part to a vector.
        #[inline]
        fn scale_vector(&self, pt: &E::Coordinates) -> E::Coordinates {
            self.scaling().transform_vector(pt)
        }

        /*
         * Inverse transformations.
         */
        /// Applies this transformation inverse's pure translational part to a point.
        #[inline]
        fn inverse_translate_point(&self, pt: &E) -> E {
            self.translation().inverse_transform_point(pt)
        }

        /// Applies this transformation inverse's pure rotational part to a point.
        #[inline]
        fn inverse_rotate_point(&self, pt: &E) -> E {
            self.rotation().inverse_transform_point(pt)
        }

        /// Applies this transformation inverse's pure scaling part to a point.
        #[inline]
        fn inverse_scale_point(&self, pt: &E) -> E {
            self.scaling().inverse_transform_point(pt)
        }

        /// Applies this transformation inverse's pure rotational part to a vector.
        #[inline]
        fn inverse_rotate_vector(&self, pt: &E::Coordinates) -> E::Coordinates {
            self.rotation().inverse_transform_vector(pt)
        }

        /// Applies this transformation inverse's pure scaling part to a vector.
        #[inline]
        fn inverse_scale_vector(&self, pt: &E::Coordinates) -> E::Coordinates {
            self.scaling().inverse_transform_vector(pt)
        }
    }

    /// Subgroups of the isometry group `E(n)`, i.e., rotations, reflexions, and translations.
    pub trait Isometry<E: EuclideanSpace>: Similarity<E, Scaling = Id> {}

    /// Subgroups of the orientation-preserving isometry group `SE(n)`, i.e., rotations and translations.
    pub trait DirectIsometry<E: EuclideanSpace>: Isometry<E> {}

    /// Subgroups of the n-dimensional rotations and scaling `O(n)`.
    pub trait OrthogonalTransformation<E: EuclideanSpace>: Isometry<E, Translation = Id> {}

    /// Subgroups of the (signed) uniform scaling group.
    pub trait Scaling<E: EuclideanSpace>:
        AffineTransformation<E, NonUniformScaling = Self, Translation = Id, Rotation = Id>
        + SubsetOf<E::RealField>
    {
        /// Converts this scaling factor to a real. Same as `self.to_superset()`.
        #[inline]
        fn to_real(&self) -> E::RealField {
            self.to_superset()
        }

        /// Attempts to convert a real to an element of this scaling subgroup. Same as
        /// `Self::from_superset()`. Returns `None` if no such scaling is possible for this subgroup.
        #[inline]
        fn from_real(r: E::RealField) -> Option<Self> {
            Self::from_superset(&r)
        }

        /// Raises the scaling to a power. The result must be equivalent to
        /// `self.to_superset().powf(n)`. Returns `None` if the result is not representable by `Self`.
        #[inline]
        fn powf(&self, n: E::RealField) -> Option<Self> {
            Self::from_superset(&self.to_superset().powf(n))
        }

        /// The scaling required to make `a` have the same norm as `b`, i.e., `|b| = |a| * norm_ratio(a,
        /// b)`.
        #[inline]
        fn scale_between(a: &E::Coordinates, b: &E::Coordinates) -> Option<Self> {
            Self::from_superset(&(b.norm() / a.norm()))
        }
    }

    /// Subgroups of the n-dimensional translation group `T(n)`.
    pub trait Translation<E: EuclideanSpace>:
        DirectIsometry<E, Translation = Self, Rotation = Id> /* + SubsetOf<E::Coordinates> */
    {
        // NOTE: we must define those two conversions here (instead of just using SubsetOf) because the
        // structure of Self uses the multiplication for composition, while E::Coordinates uses addition.
        // Having a trait that says "remap this operator to this other one" does not seem to be
        // possible without higher kinded traits.
        /// Converts this translation to a vector.
        fn to_vector(&self) -> E::Coordinates;

        /// Attempts to convert a vector to this translation. Returns `None` if the translation
        /// represented by `v` is not part of the translation subgroup represented by `Self`.
        fn from_vector(v: E::Coordinates) -> Option<Self>;

        /// Raises the translation to a power. The result must be equivalent to
        /// `self.to_superset() * n`.  Returns `None` if the result is not representable by `Self`.
        #[inline]
        fn powf(&self, n: E::RealField) -> Option<Self> {
            Self::from_vector(self.to_vector() * n)
        }

        /// The translation needed to make `a` coincide with `b`, i.e., `b = a * translation_to(a, b)`.
        #[inline]
        fn translation_between(a: &E, b: &E) -> Option<Self> {
            Self::from_vector(b.clone() - a.clone())
        }
    }

    /// Subgroups of the n-dimensional rotation group `SO(n)`.
    pub trait Rotation<E: EuclideanSpace>:
        OrthogonalTransformation<E, Rotation = Self> + DirectIsometry<E, Rotation = Self>
    {
        /// Raises this rotation to a power. If this is a simple rotation, the result must be
        /// equivalent to multiplying the rotation angle by `n`.
        fn powf(&self, n: E::RealField) -> Option<Self>;

        /// Computes a simple rotation that makes the angle between `a` and `b` equal to zero, i.e.,
        /// `b.angle(a * delta_rotation(a, b)) = 0`. If `a` and `b` are collinear, the computed
        /// rotation may not be unique. Returns `None` if no such simple rotation exists in the
        /// subgroup represented by `Self`.
        fn rotation_between(a: &E::Coordinates, b: &E::Coordinates) -> Option<Self>;

        /// Computes the rotation between `a` and `b` and raises it to the power `n`.
        ///
        /// This is equivalent to calling `self.rotation_between(a, b)` followed by `.powf(n)` but will
        /// usually be much more efficient.
        #[inline]
        fn scaled_rotation_between(
            a: &E::Coordinates,
            b: &E::Coordinates,
            s: E::RealField,
        ) -> Option<Self>;

        // FIXME: add a function that computes the rotation with the axis orthogonal to Span(a, b) and
        // with angle equal to `n`?
    }

    /*
     *
     * Implementation for floats.
     *
     */

    impl<R, E> Transformation<E> for R
    where
        R: RealField,
        E: EuclideanSpace<RealField = R>,
        E::Coordinates: ClosedMul<R> + ClosedDiv<R> + ClosedNeg,
    {
        #[inline]
        fn transform_point(&self, pt: &E) -> E {
            pt.scale_by(*self)
        }

        #[inline]
        fn transform_vector(&self, v: &E::Coordinates) -> E::Coordinates {
            v.clone() * *self
        }
    }

    impl<R, E> ProjectiveTransformation<E> for R
    where
        R: RealField,
        E: EuclideanSpace<RealField = R>,
        E::Coordinates: ClosedMul<R> + ClosedDiv<R> + ClosedNeg,
    {
        #[inline]
        fn inverse_transform_point(&self, pt: &E) -> E {
            assert!(*self != R::zero());
            pt.scale_by(R::one() / *self)
        }

        #[inline]
        fn inverse_transform_vector(&self, v: &E::Coordinates) -> E::Coordinates {
            assert!(*self != R::zero());
            v.clone() * (R::one() / *self)
        }
    }

    impl<R, E> AffineTransformation<E> for R
    where
        R: RealField,
        E: EuclideanSpace<RealField = R>,
        E::Coordinates: ClosedMul<R> + ClosedDiv<R> + ClosedNeg,
    {
        type Rotation = Id;
        type NonUniformScaling = R;
        type Translation = Id;

        #[inline]
        fn decompose(&self) -> (Id, Id, R, Id) {
            (Id::new(), Id::new(), *self, Id::new())
        }

        #[inline]
        fn append_translation(&self, _: &Self::Translation) -> Self {
            *self
        }

        #[inline]
        fn prepend_translation(&self, _: &Self::Translation) -> Self {
            *self
        }

        #[inline]
        fn append_rotation(&self, _: &Self::Rotation) -> Self {
            *self
        }

        #[inline]
        fn prepend_rotation(&self, _: &Self::Rotation) -> Self {
            *self
        }

        #[inline]
        fn append_scaling(&self, s: &Self::NonUniformScaling) -> Self {
            *s * *self
        }

        #[inline]
        fn prepend_scaling(&self, s: &Self::NonUniformScaling) -> Self {
            *self * *s
        }
    }

    impl<R, E> Scaling<E> for R
    where
        R: RealField + SubsetOf<R>,
        E: EuclideanSpace<RealField = R>,
        E::Coordinates: ClosedMul<R> + ClosedDiv<R> + ClosedNeg,
    {
        #[inline]
        fn to_real(&self) -> E::RealField {
            *self
        }

        #[inline]
        fn from_real(r: E::RealField) -> Option<Self> {
            Some(r)
        }

        #[inline]
        fn powf(&self, n: E::RealField) -> Option<Self> {
            Some(n.powf(n))
        }

        #[inline]
        fn scale_between(a: &E::Coordinates, b: &E::Coordinates) -> Option<Self> {
            Some(b.norm() / a.norm())
        }
    }

    impl<R, E> Similarity<E> for R
    where
        R: RealField + SubsetOf<R>,
        E: EuclideanSpace<RealField = R>,
        E::Coordinates: ClosedMul<R> + ClosedDiv<R> + ClosedNeg,
    {
        type Scaling = R;

        fn translation(&self) -> Self::Translation {
            Id::new()
        }

        fn rotation(&self) -> Self::Rotation {
            Id::new()
        }

        fn scaling(&self) -> Self::Scaling {
            *self
        }
    }

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