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use num; use std::ops::{Add, Sub, Mul, Div, MulAssign, DivAssign, Neg}; use general::{ClosedDiv, Module, Field, Real}; /// A vector space has a module structure over a field instead of a ring. pub trait VectorSpace: Module<Ring = <Self as VectorSpace>::Field> + ClosedDiv<<Self as VectorSpace>::Field> { /// The underlying scalar field. type Field: Field; } /// A normed vector space. pub trait NormedSpace: VectorSpace { /// The squared norm of this vector. fn norm_squared(&self) -> Self::Field; /// The norm of this vector. fn norm(&self) -> Self::Field; /// Returns a normalized version of this vector. fn normalize(&self) -> Self; /// Normalizes this vector in-place and returns its norm. fn normalize_mut(&mut self) -> Self::Field; /// Returns a normalized version of this vector unless its norm as smaller or equal to `eps`. fn try_normalize(&self, eps: Self::Field) -> Option<Self>; /// Normalizes this vector in-place or does nothing if its norm is smaller or equal to `eps`. /// /// If the normalization succeded, returns the old normal of this vector. fn try_normalize_mut(&mut self, eps: Self::Field) -> Option<Self::Field>; } /// A vector space aquipped with an inner product. /// /// It must be a normed space as well and the norm must agree with the inner product. /// The inner product must be symmetric, linear in its first agurment, and positive definite. pub trait InnerSpace: NormedSpace<Field = <Self as InnerSpace>::Real> { /// The result of inner product (same as the field used by this vector space). type Real: Real; /// Computes the inner product of `self` with `other`. fn inner_product(&self, other: &Self) -> Self::Real; /// Measures the angle between two vectors. #[inline] fn angle(&self, other: &Self) -> Self::Real { let prod = self.inner_product(other); let n1 = self.norm(); let n2 = self.norm(); if n1 == num::zero() || n2 == num::zero() { num::zero() } else { let cang = prod / (n1 * n2); cang.acos() } } } /// A finite-dimensional vector space. pub trait FiniteDimVectorSpace: VectorSpace { /// The vector space dimension. fn dimension() -> usize; /// Applies the given closule to each element of this vector space's canonical basis. Stops if /// `f` returns `false`. // XXX: return an iterator instead when `-> impl Iterator` will be supported by Rust. fn canonical_basis<F: FnMut(&Self) -> bool>(mut f: F) { for i in 0 .. Self::dimension() { if !f(&Self::canonical_basis_element(i)) { break; } } } /// The i-the canonical basis element. fn canonical_basis_element(i: usize) -> Self; /// Retrieves the i-th component of `Self` wrt. some basis. /// /// As usual, indexing starts with 0. The actual choice of basis is usually context-dependent /// and is not specified to this method. It is up to the user to assume the provided component /// will by wrt. the suitable basis for his application. fn component(&self, i: usize) -> Self::Field; /// The dot product between two vectors. fn dot(&self, other: &Self) -> Self::Field; /// Same as `.component(i)` but without bound-checking. unsafe fn component_unchecked(&self, i: usize) -> Self::Field; } /// A finite-dimenisonal vector space equipped with an inner product that must coincide /// with the dot product. pub trait FiniteDimInnerSpace: InnerSpace + FiniteDimVectorSpace<Field = <Self as InnerSpace>::Real> { /// Applies the given closure to each element of the orthonormal basis of the subspace /// orthogonal to free family of vectors `vs`. If `vs` is not a free family, the result is /// unspecified. // XXX: return an iterator instead when `-> impl Iterator` will be supported by Rust. fn orthonormal_subspace_basis<F: FnMut(&Self) -> bool>(vs: &[Self], f: F); // FIXME: add another method to orthogonalize a non-free family of vector? } /// A set points associated with a vector space and a transitive and free additive group action /// (the translation). pub trait AffineSpace: Sized + Clone + PartialEq + Sub<Self, Output = <Self as AffineSpace>::Translation> + Add<<Self as AffineSpace>::Translation, Output = Self> { /// The associated vector space. type Translation: VectorSpace; /// Same as `*self + *t`. Applies the additive group action of this affine space's associated /// vector space on `self`. fn translate_by(&self, t: &Self::Translation) -> Self { self.clone() + t.clone() } /// Same as `*self - *other`. Returns the unique element `v` of the associated vector space /// such that `self = right + v`. fn subtract(&self, right: &Self) -> Self::Translation { self.clone() - right.clone() } } /// The finite-dimensional affine space based on the field of reals. pub trait EuclideanSpace: AffineSpace<Translation = <Self as EuclideanSpace>::Vector> { /// The underlying finite vector space. type Vector: FiniteDimInnerSpace<Real = Self::Real> + // XXX: the following bounds should not be necessary but the compiler does not // seem to be able to find them (from the Module trait)… Also, it won't find them // even if we add ClosedMul instead of Mul and MulAssign separately… Mul<Self::Real, Output = Self::Vector> + MulAssign<Self::Real> + Div<Self::Real, Output = Self::Vector> + DivAssign<Self::Real> + Neg<Output = Self::Vector>; // XXX: we can't write the following =( : // type Vector: FiniteDimInnerSpace<Field = Self::Real> + InnerSpace<Real = Self::Real>; // The compiler won't recognize that VectorSpace::Field = Self::Real. // Though it will work if only one bound is used… looks like a compiler bug. /// The underlying reals. type Real: Real; /// The preferred origin of this euclidean space. /// /// Theoretically, an euclidean space has no clearly defined origin. Though it is almost always /// useful to have some reference point to express all the others as translations of it. fn origin() -> Self; /// Multiplies the distance of this point to `Self::origin()` by `s`. fn scale_by(&self, s: Self::Real) -> Self { Self::origin().translate_by(&(self.coordinates() * s)) } /// The coordinates of this point, i.e., the translation from the prefered origin. #[inline] fn coordinates(&self) -> Self::Vector { self.subtract(&Self::origin()) } /// The distance between two points. #[inline] fn distance_squared(&self, b: &Self) -> Self::Real { self.subtract(b).norm_squared() } /// The distance between two points. #[inline] fn distance(&self, b: &Self) -> Self::Real { self.subtract(b).norm() } }