From b67b31b633b696f59044772368102951716c02b4 Mon Sep 17 00:00:00 2001
From: Brendan Zabarauskas <bjzaba@yahoo.com.au>
Date: Tue, 27 Aug 2013 08:59:18 +1000
Subject: [PATCH] Add matrix traits, do lots of reorganisation/re-formatting

---
 src/traits/alg/affine_space.rs                |  19 +++-
 src/traits/alg/coordinate.rs                  |  16 ++-
 src/traits/alg/euclidean_space.rs             |   9 +-
 src/traits/alg/{division_ring.rs => field.rs} |  37 ++++---
 src/traits/alg/inner_product_space.rs         |   9 +-
 src/traits/alg/matrix.rs                      |  30 ++++++
 src/traits/alg/mod.rs                         |   8 +-
 src/traits/alg/module.rs                      |  15 ++-
 src/traits/alg/ordered_ring.rs                |   7 +-
 src/traits/alg/ring.rs                        |  11 +-
 src/traits/alg/scalar_mul.rs                  |  14 ++-
 src/traits/alg/square_matrix.rs               |  33 ++++++
 src/traits/alg/vector_space.rs                |  13 ++-
 src/traits/ext/vector_ext.rs                  |  12 ++-
 src/types/matrix.rs                           |   6 ++
 src/types/point.rs                            |  40 +++----
 src/types/quaternion.rs                       |   2 +-
 src/types/vector.rs                           | 100 +++++++++---------
 18 files changed, 259 insertions(+), 122 deletions(-)
 rename src/traits/alg/{division_ring.rs => field.rs} (58%)
 create mode 100644 src/traits/alg/matrix.rs
 create mode 100644 src/traits/alg/square_matrix.rs

diff --git a/src/traits/alg/affine_space.rs b/src/traits/alg/affine_space.rs
index d79fc31..f0ce511 100644
--- a/src/traits/alg/affine_space.rs
+++ b/src/traits/alg/affine_space.rs
@@ -15,11 +15,20 @@
 
 use std::num::Zero;
 
+use traits::alg::Field;
 use traits::alg::ScalarMul;
+use traits::alg::VectorSpace;
 
 /// An affine space is a set of points closed under affine combinations.
-pub trait AffineSpace<S,V>: Eq
-                          + Zero
-                          + ScalarMul<S>
-                          + Add<V, Self>
-                          + Sub<Self, V> {}
+pub trait AffineSpace
+<
+    S: Field,
+    V: VectorSpace<S>
+>
+:   Eq
++   Zero
++   Add<V, Self>
++   Sub<Self, V>
++   ScalarMul<S>
+{
+}
diff --git a/src/traits/alg/coordinate.rs b/src/traits/alg/coordinate.rs
index f783555..f813bbb 100644
--- a/src/traits/alg/coordinate.rs
+++ b/src/traits/alg/coordinate.rs
@@ -13,8 +13,16 @@
 // See the License for the specific language governing permissions and
 // limitations under the License.
 
-pub use traits::util::Indexable;
-pub use traits::util::Swappable;
+use traits::alg::Field;
+use traits::util::Indexable;
+use traits::util::Swappable;
 
-pub trait Coordinate<S, Slice>: Indexable<S, Slice>
-                              + Swappable<S, Slice> {}
+pub trait Coordinate
+<
+    S: Field,
+    Slice
+>
+:   Indexable<S, Slice>
++   Swappable<S, Slice>
+{
+}
diff --git a/src/traits/alg/euclidean_space.rs b/src/traits/alg/euclidean_space.rs
index e24d69b..f123a7b 100644
--- a/src/traits/alg/euclidean_space.rs
+++ b/src/traits/alg/euclidean_space.rs
@@ -15,11 +15,16 @@
 
 use std::num;
 
-use traits::alg::Ring;
+use traits::alg::Field;
 use traits::alg::InnerProductSpace;
 
 /// The Euclidean space is a vector space over the Real numbers.
-pub trait EuclideanSpace<S:Real + Ring>: InnerProductSpace<S> {
+pub trait EuclideanSpace
+<
+    S:Real + Field
+>
+:   InnerProductSpace<S>
+{
     fn dot(&self, other: &Self) -> S {
         self.inner(other)
     }
diff --git a/src/traits/alg/division_ring.rs b/src/traits/alg/field.rs
similarity index 58%
rename from src/traits/alg/division_ring.rs
rename to src/traits/alg/field.rs
index 94396ce..a1fcaa7 100644
--- a/src/traits/alg/division_ring.rs
+++ b/src/traits/alg/field.rs
@@ -15,23 +15,26 @@
 
 use traits::alg::Ring;
 
-/// A ring that also requires the multiplicative inverse operation (division).
-pub trait DivisionRing: Ring
-                      + Div<Self,Self>
-                      + Rem<Self,Self> {}
+/// A commutative ring that contains a multiplicative inverse.
+pub trait Field
+:   Ring
++   Div<Self,Self>
++   Rem<Self,Self>
+{
+}
 
 // impls for concrete types
 
-impl DivisionRing for u8;
-impl DivisionRing for u16;
-impl DivisionRing for u32;
-impl DivisionRing for u64;
-impl DivisionRing for uint;
-impl DivisionRing for i8;
-impl DivisionRing for i16;
-impl DivisionRing for i32;
-impl DivisionRing for i64;
-impl DivisionRing for int;
-impl DivisionRing for f32;
-impl DivisionRing for f64;
-impl DivisionRing for float;
+impl Field for u8;
+impl Field for u16;
+impl Field for u32;
+impl Field for u64;
+impl Field for uint;
+impl Field for i8;
+impl Field for i16;
+impl Field for i32;
+impl Field for i64;
+impl Field for int;
+impl Field for f32;
+impl Field for f64;
+impl Field for float;
diff --git a/src/traits/alg/inner_product_space.rs b/src/traits/alg/inner_product_space.rs
index c20b835..9d3ed8a 100644
--- a/src/traits/alg/inner_product_space.rs
+++ b/src/traits/alg/inner_product_space.rs
@@ -13,11 +13,16 @@
 // See the License for the specific language governing permissions and
 // limitations under the License.
 
-use traits::alg::Ring;
+use traits::alg::Field;
 use traits::alg::VectorSpace;
 
 /// A vector space with the inner product operation.
-pub trait InnerProductSpace<S: Ring>: VectorSpace<S> {
+pub trait InnerProductSpace
+<
+    S: Field
+>
+:   VectorSpace<S>
+{
     fn norm(&self) -> S;
     fn inner(&self, other: &Self) -> S;
     fn is_orthogonal(&self, other: &Self) -> bool;
diff --git a/src/traits/alg/matrix.rs b/src/traits/alg/matrix.rs
new file mode 100644
index 0000000..3865536
--- /dev/null
+++ b/src/traits/alg/matrix.rs
@@ -0,0 +1,30 @@
+// Copyright 2013 The CGMath Developers. For a full listing of the authors,
+// refer to the AUTHORS file at the top-level directory of this distribution.
+//
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+//
+//     http://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+
+use traits::alg::Field;
+use traits::alg::Ring;
+use traits::alg::VectorSpace;
+
+pub trait Matrix
+<
+    S: Field,
+    RV: VectorSpace<S>,
+    CV: VectorSpace<S>,
+    MT//: Matrix<S, CV, RV, Self>
+>
+:   Ring
+{
+    fn transpose(&self) -> MT;
+}
\ No newline at end of file
diff --git a/src/traits/alg/mod.rs b/src/traits/alg/mod.rs
index 33444be..c6d891b 100644
--- a/src/traits/alg/mod.rs
+++ b/src/traits/alg/mod.rs
@@ -15,22 +15,26 @@
 
 pub use self::affine_space::AffineSpace;
 pub use self::coordinate::Coordinate;
-pub use self::division_ring::DivisionRing;
 pub use self::euclidean_space::EuclideanSpace;
+pub use self::field::Field;
 pub use self::inner_product_space::InnerProductSpace;
+pub use self::matrix::Matrix;
 pub use self::module::Module;
 pub use self::ordered_ring::OrderedRing;
 pub use self::ring::Ring;
 pub use self::scalar_mul::ScalarMul;
+pub use self::square_matrix::SquareMatrix;
 pub use self::vector_space::VectorSpace;
 
 pub mod affine_space;
 pub mod coordinate;
-pub mod division_ring;
 pub mod euclidean_space;
+pub mod field;
 pub mod inner_product_space;
+pub mod matrix;
 pub mod module;
 pub mod ordered_ring;
 pub mod ring;
 pub mod scalar_mul;
+pub mod square_matrix;
 pub mod vector_space;
diff --git a/src/traits/alg/module.rs b/src/traits/alg/module.rs
index c0df90a..5534d40 100644
--- a/src/traits/alg/module.rs
+++ b/src/traits/alg/module.rs
@@ -15,13 +15,20 @@
 
 use std::num::Zero;
 
+// use traits::alg::Field;
 use traits::alg::ScalarMul;
 
 /// An algebraic structure that generalizes the notion of a vector space.
-pub trait Module<S>: Eq
-                   + Add<Self,Self>
-                   + ScalarMul<S>
-                   + Zero {}
+pub trait Module
+<
+    S/*: Field*/
+>
+:   Eq
++   Add<Self,Self>
++   ScalarMul<S>
++   Zero
+{
+}
 
 // impls for concrete types
 
diff --git a/src/traits/alg/ordered_ring.rs b/src/traits/alg/ordered_ring.rs
index ddee8e3..4507d5a 100644
--- a/src/traits/alg/ordered_ring.rs
+++ b/src/traits/alg/ordered_ring.rs
@@ -16,8 +16,11 @@
 use traits::alg::Ring;
 
 /// A ring that can also be ordered.
-pub trait OrderedRing: Ring
-                     + Orderable {}
+pub trait OrderedRing
+:   Ring
++   Orderable
+{
+}
 
 // impls for concrete types
 
diff --git a/src/traits/alg/ring.rs b/src/traits/alg/ring.rs
index f1cedda..ba1de55 100644
--- a/src/traits/alg/ring.rs
+++ b/src/traits/alg/ring.rs
@@ -19,10 +19,13 @@ use traits::alg::Module;
 
 /// A module that also requires the additive inverse operation (subtraction)
 /// and the additive inverse.
-pub trait Ring: Module<Self>
-              + Neg<Self>
-              + Sub<Self,Self>
-              + One {}
+pub trait Ring
+:   Module<Self>
++   Neg<Self>
++   Sub<Self,Self>
++   One
+{
+}
 
 // impls for concrete types
 
diff --git a/src/traits/alg/scalar_mul.rs b/src/traits/alg/scalar_mul.rs
index 55178f0..8dd3c08 100644
--- a/src/traits/alg/scalar_mul.rs
+++ b/src/traits/alg/scalar_mul.rs
@@ -13,10 +13,18 @@
 // See the License for the specific language governing permissions and
 // limitations under the License.
 
+// use traits::alg::Field;
+
 /// Enforces the multiplication of an type by a scalar.
-pub trait ScalarMul<S>: Mul<S, Self>
-                      + Div<S, Self>
-                      + Rem<S, Self> {}
+pub trait ScalarMul
+<
+    S/*: Field*/
+>
+:   Mul<S, Self>
++   Div<S, Self>
++   Rem<S, Self>
+{
+}
 
 // impls for concrete types
 
diff --git a/src/traits/alg/square_matrix.rs b/src/traits/alg/square_matrix.rs
new file mode 100644
index 0000000..263059c
--- /dev/null
+++ b/src/traits/alg/square_matrix.rs
@@ -0,0 +1,33 @@
+// Copyright 2013 The CGMath Developers. For a full listing of the authors,
+// refer to the AUTHORS file at the top-level directory of this distribution.
+//
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+//
+//     http://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+
+use traits::alg::Field;
+use traits::alg::Matrix;
+use traits::alg::VectorSpace;
+
+pub trait SquareMatrix
+<
+    S: Field,
+    V: VectorSpace<S>
+>
+:   Matrix<S, V, V, Self>
+{
+    fn transpose_self(&mut self);
+    fn trace(&self) -> S;
+    fn det(&self) -> S;
+    fn invert(&self) -> Option<Self>;
+    fn invert_self(&mut self) -> Self;
+    fn eigenvalue(&self, v: V) -> Option<V>;
+}
\ No newline at end of file
diff --git a/src/traits/alg/vector_space.rs b/src/traits/alg/vector_space.rs
index 20a8044..dd92228 100644
--- a/src/traits/alg/vector_space.rs
+++ b/src/traits/alg/vector_space.rs
@@ -13,10 +13,17 @@
 // See the License for the specific language governing permissions and
 // limitations under the License.
 
+use traits::alg::Field;
 use traits::alg::Module;
 
 /// A vector space is a set that is closed under vector addition and
 /// scalar multiplication.
-pub trait VectorSpace<S>: Module<S>
-                        + Neg<Self>
-                        + Sub<Self,Self> {}
+pub trait VectorSpace
+<
+    S: Field
+>
+:   Module<S>
++   Neg<Self>
++   Sub<Self,Self>
+{
+}
diff --git a/src/traits/ext/vector_ext.rs b/src/traits/ext/vector_ext.rs
index 9da2bc0..cee534e 100644
--- a/src/traits/ext/vector_ext.rs
+++ b/src/traits/ext/vector_ext.rs
@@ -13,12 +13,18 @@
 // See the License for the specific language governing permissions and
 // limitations under the License.
 
-use traits::alg::Ring;
+use traits::alg::Field;
 use traits::alg::VectorSpace;
 use traits::util::Indexable;
 
-pub trait VectorExt<S: Ring, Slice>: VectorSpace<S>
-                                   + Indexable<S, Slice> {
+pub trait VectorExt
+<
+    S: Field,
+    Slice
+>
+:   VectorSpace<S>
++   Indexable<S, Slice>
+{
     #[inline] fn add_s(&self, s: S) -> Self { self.map(|x| x.add(&s)) }
     #[inline] fn sub_s(&self, s: S) -> Self { self.map(|x| x.sub(&s)) }
     #[inline] fn mul_s(&self, s: S) -> Self { self.map(|x| x.mul(&s)) }
diff --git a/src/types/matrix.rs b/src/types/matrix.rs
index 0a0d38c..e815306 100644
--- a/src/types/matrix.rs
+++ b/src/types/matrix.rs
@@ -12,3 +12,9 @@
 // WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 // See the License for the specific language governing permissions and
 // limitations under the License.
+
+use types::vector::{Vec2, Vec3, Vec4};
+
+#[deriving(Clone, Eq)] pub struct Mat2<T> { x: Vec2<T>, y: Vec2<T> }
+#[deriving(Clone, Eq)] pub struct Mat3<T> { x: Vec3<T>, y: Vec3<T>, z: Vec3<T> }
+#[deriving(Clone, Eq)] pub struct Mat4<T> { x: Vec4<T>, y: Vec4<T>, z: Vec4<T>, w: Vec4<T> }
diff --git a/src/types/point.rs b/src/types/point.rs
index f4fb37e..1c879b4 100644
--- a/src/types/point.rs
+++ b/src/types/point.rs
@@ -27,14 +27,14 @@ struct Point2<S> { x: S, y: S }
 #[deriving(Eq, Zero, Clone)]
 struct Point3<S> { x: S, y: S, z: S }
 
-impl<S> Point2<S> {
+impl<S: Field> Point2<S> {
     #[inline]
     pub fn new(x: S, y: S) -> Point2<S> {
         Point2 { x: x, y: y }
     }
 }
 
-impl<S> Point3<S> {
+impl<S: Field> Point3<S> {
     #[inline]
     pub fn new(x: S, y: S, z: S) -> Point3<S> {
         Point3 { x: x, y: y, z: z }
@@ -43,34 +43,34 @@ impl<S> Point3<S> {
 
 // Operator impls
 
-impl<S:Ring> Mul<S, Point2<S>> for Point2<S> { #[inline(always)] fn mul(&self, s: &S) -> Point2<S> { self.map(|x| x.mul(s)) }  }
-impl<S:Ring> Mul<S, Point3<S>> for Point3<S> { #[inline(always)] fn mul(&self, s: &S) -> Point3<S> { self.map(|x| x.mul(s)) }  }
+impl<S: Field> Mul<S, Point2<S>> for Point2<S> { #[inline(always)] fn mul(&self, s: &S) -> Point2<S> { self.map(|x| x.mul(s)) }  }
+impl<S: Field> Mul<S, Point3<S>> for Point3<S> { #[inline(always)] fn mul(&self, s: &S) -> Point3<S> { self.map(|x| x.mul(s)) }  }
 
-impl<S:Ring> Div<S, Point2<S>> for Point2<S> { #[inline(always)] fn div(&self, s: &S) -> Point2<S> { self.map(|x| x.div(s)) }  }
-impl<S:Ring> Div<S, Point3<S>> for Point3<S> { #[inline(always)] fn div(&self, s: &S) -> Point3<S> { self.map(|x| x.div(s)) }  }
+impl<S: Field> Div<S, Point2<S>> for Point2<S> { #[inline(always)] fn div(&self, s: &S) -> Point2<S> { self.map(|x| x.div(s)) }  }
+impl<S: Field> Div<S, Point3<S>> for Point3<S> { #[inline(always)] fn div(&self, s: &S) -> Point3<S> { self.map(|x| x.div(s)) }  }
 
-impl<S:Ring> Rem<S, Point2<S>> for Point2<S> { #[inline(always)] fn rem(&self, s: &S) -> Point2<S> { self.map(|x| x.rem(s)) }  }
-impl<S:Ring> Rem<S, Point3<S>> for Point3<S> { #[inline(always)] fn rem(&self, s: &S) -> Point3<S> { self.map(|x| x.rem(s)) }  }
+impl<S: Field> Rem<S, Point2<S>> for Point2<S> { #[inline(always)] fn rem(&self, s: &S) -> Point2<S> { self.map(|x| x.rem(s)) }  }
+impl<S: Field> Rem<S, Point3<S>> for Point3<S> { #[inline(always)] fn rem(&self, s: &S) -> Point3<S> { self.map(|x| x.rem(s)) }  }
 
-impl<S:Ring> Add<Vec2<S>, Point2<S>> for Point2<S> { #[inline(always)] fn add(&self, other: &Vec2<S>) -> Point2<S> { self.bimap(other, |a, b| a.add(b)) } }
-impl<S:Ring> Add<Vec3<S>, Point3<S>> for Point3<S> { #[inline(always)] fn add(&self, other: &Vec3<S>) -> Point3<S> { self.bimap(other, |a, b| a.add(b)) } }
+impl<S: Field> Add<Vec2<S>, Point2<S>> for Point2<S> { #[inline(always)] fn add(&self, other: &Vec2<S>) -> Point2<S> { self.bimap(other, |a, b| a.add(b)) } }
+impl<S: Field> Add<Vec3<S>, Point3<S>> for Point3<S> { #[inline(always)] fn add(&self, other: &Vec3<S>) -> Point3<S> { self.bimap(other, |a, b| a.add(b)) } }
 
-impl<S:Ring> Sub<Point2<S>, Vec2<S>> for Point2<S> { #[inline(always)] fn sub(&self, other: &Point2<S>) -> Vec2<S> { self.bimap(other, |a, b| a.sub(b)) } }
-impl<S:Ring> Sub<Point3<S>, Vec3<S>> for Point3<S> { #[inline(always)] fn sub(&self, other: &Point3<S>) -> Vec3<S> { self.bimap(other, |a, b| a.sub(b)) } }
+impl<S: Field> Sub<Point2<S>, Vec2<S>> for Point2<S> { #[inline(always)] fn sub(&self, other: &Point2<S>) -> Vec2<S> { self.bimap(other, |a, b| a.sub(b)) } }
+impl<S: Field> Sub<Point3<S>, Vec3<S>> for Point3<S> { #[inline(always)] fn sub(&self, other: &Point3<S>) -> Vec3<S> { self.bimap(other, |a, b| a.sub(b)) } }
 
 // Trait impls
 
 impl_indexable!(Point2<T>, [T, ..2])
 impl_indexable!(Point3<T>, [T, ..3])
 
-impl<S: Clone> Swappable<S, [S, ..2]> for Point2<S>;
-impl<S: Clone> Swappable<S, [S, ..3]> for Point3<S>;
+impl<S: Clone + Field> Swappable<S, [S, ..2]> for Point2<S>;
+impl<S: Clone + Field> Swappable<S, [S, ..3]> for Point3<S>;
 
-impl<S: Clone> Coordinate<S, [S, ..2]> for Point2<S>;
-impl<S: Clone> Coordinate<S, [S, ..3]> for Point3<S>;
+impl<S: Clone + Field> Coordinate<S, [S, ..2]> for Point2<S>;
+impl<S: Clone + Field> Coordinate<S, [S, ..3]> for Point3<S>;
 
-impl<S: Ring> ScalarMul<S> for Point2<S>;
-impl<S: Ring> ScalarMul<S> for Point3<S>;
+impl<S: Field> ScalarMul<S> for Point2<S>;
+impl<S: Field> ScalarMul<S> for Point3<S>;
 
-impl<S: Ring> AffineSpace<S, Vec2<S>> for Point2<S>;
-impl<S: Ring> AffineSpace<S, Vec3<S>> for Point3<S>;
+impl<S: Field> AffineSpace<S, Vec2<S>> for Point2<S>;
+impl<S: Field> AffineSpace<S, Vec3<S>> for Point3<S>;
diff --git a/src/types/quaternion.rs b/src/types/quaternion.rs
index c52e98e..8b1c044 100644
--- a/src/types/quaternion.rs
+++ b/src/types/quaternion.rs
@@ -20,7 +20,7 @@ use types::vector::Vec3;
 #[deriving(Clone, Eq)]
 pub struct Quat<T> { s: T, v: Vec3<T> }
 
-impl<T: Ring> Quat<T> {
+impl<T: Field> Quat<T> {
     /// Construct a new quaternion from one scalar component and three
     /// imaginary components
     #[inline]
diff --git a/src/types/vector.rs b/src/types/vector.rs
index 070d4db..6ff925d 100644
--- a/src/types/vector.rs
+++ b/src/types/vector.rs
@@ -30,7 +30,7 @@ use traits::util::*;
 
 macro_rules! impl_vec(
     ($Self:ident <$S:ident> { $($field:ident),+ }) => (
-        impl<$S> $Self<$S> {
+        impl<$S: Field> $Self<$S> {
             #[inline]
             pub fn new($($field: $S),+) -> $Self<$S> {
                 $Self { $($field: $field),+ }
@@ -48,7 +48,7 @@ impl_vec!(Vec6<S> { x, y, z, w, a, b })
 
 macro_rules! impl_vec_clonable(
     ($Self:ident <$S:ident>) => (
-        impl<$S:Clone> $Self<$S> {
+        impl<$S:Clone + Field> $Self<$S> {
             /// Construct a vector from a single value.
             #[inline]
             pub fn from_value(value: $S) -> $Self<$S> {
@@ -67,7 +67,7 @@ impl_vec_clonable!(Vec6<S>)
 
 macro_rules! impl_vec_ring(
     ($Self:ident <$S:ident>) => (
-        impl<$S:Ring> $Self<$S> {
+        impl<$S:Field> $Self<$S> {
             /// The additive identity of the vector.
             #[inline]
             pub fn zero() -> $Self<$S> { zero() }
@@ -90,32 +90,32 @@ macro_rules! impl_vec_ops(
             use super::*;
             use super::super::super::traits::alg::*;
 
-            impl<$S:Ring> Mul<$S, $Self<$S>> for $Self<$S> {
+            impl<$S:Field> Mul<$S, $Self<$S>> for $Self<$S> {
                 #[inline(always)]
                 fn mul(&self, s: &$S) -> $Self<$S> { self.map(|x| x.mul(s)) }
             }
 
-            impl<$S:Ring> Div<$S, $Self<$S>> for $Self<$S> {
+            impl<$S:Field> Div<$S, $Self<$S>> for $Self<$S> {
                 #[inline(always)]
                 fn div(&self, s: &$S) -> $Self<$S> { self.map(|x| x.div(s)) }
             }
 
-            impl<$S:Ring> Rem<$S, $Self<$S>> for $Self<$S> {
+            impl<$S:Field> Rem<$S, $Self<$S>> for $Self<$S> {
                 #[inline(always)]
                 fn rem(&self, s: &$S) -> $Self<$S> { self.map(|x| x.rem(s)) }
             }
 
-            impl<$S:Ring> Add<$Self<$S>, $Self<$S>> for $Self<$S> {
+            impl<$S:Field> Add<$Self<$S>, $Self<$S>> for $Self<$S> {
                 #[inline(always)]
                 fn add(&self, other: &$Self<$S>) -> $Self<$S> { self.bimap(other, |a, b| a.add(b)) }
             }
 
-            impl<$S:Ring> Sub<$Self<$S>, $Self<$S>> for $Self<$S> {
+            impl<$S:Field> Sub<$Self<$S>, $Self<$S>> for $Self<$S> {
                 #[inline(always)]
                 fn sub(&self, other: &$Self<$S>) -> $Self<$S> { self.bimap(other, |a, b| a.sub(b)) }
             }
 
-            impl<$S:Ring> Neg<$Self<$S>> for $Self<$S> {
+            impl<$S:Field> Neg<$Self<$S>> for $Self<$S> {
                 #[inline(always)]
                 fn neg(&self) -> $Self<$S> { self.map(|x| x.neg()) }
             }
@@ -131,7 +131,7 @@ impl_vec_ops!(vec5_ops, Vec5<S>)
 impl_vec_ops!(vec6_ops, Vec6<S>)
 
 /// Operations specific to two-dimensional vectors.
-impl<S: Ring> Vec2<S> {
+impl<S: Field> Vec2<S> {
     /// The perpendicular dot product of the vector and `other`.
     pub fn perp_dot(&self, other: &Vec2<S>) -> S {
         (self.x * other.y) - (self.y * other.x)
@@ -139,7 +139,7 @@ impl<S: Ring> Vec2<S> {
 }
 
 /// Operations specific to three-dimensional vectors.
-impl<S: Ring> Vec3<S> {
+impl<S: Field> Vec3<S> {
     /// Returns the cross product of the vector and `other`.
     pub fn cross(&self, other: &Vec3<S>) -> Vec3<S> {
         Vec3::new((self.y * other.z) - (self.z * other.y),
@@ -157,44 +157,44 @@ impl_indexable!(Vec4<S>, [S, ..4])
 impl_indexable!(Vec5<S>, [S, ..5])
 impl_indexable!(Vec6<S>, [S, ..6])
 
-impl<S: Clone> Swappable<S, [S, ..1]> for Vec1<S>;
-impl<S: Clone> Swappable<S, [S, ..2]> for Vec2<S>;
-impl<S: Clone> Swappable<S, [S, ..3]> for Vec3<S>;
-impl<S: Clone> Swappable<S, [S, ..4]> for Vec4<S>;
-impl<S: Clone> Swappable<S, [S, ..5]> for Vec5<S>;
-impl<S: Clone> Swappable<S, [S, ..6]> for Vec6<S>;
+impl<S: Clone + Field> Swappable<S, [S, ..1]> for Vec1<S>;
+impl<S: Clone + Field> Swappable<S, [S, ..2]> for Vec2<S>;
+impl<S: Clone + Field> Swappable<S, [S, ..3]> for Vec3<S>;
+impl<S: Clone + Field> Swappable<S, [S, ..4]> for Vec4<S>;
+impl<S: Clone + Field> Swappable<S, [S, ..5]> for Vec5<S>;
+impl<S: Clone + Field> Swappable<S, [S, ..6]> for Vec6<S>;
 
-impl<S: Clone> Coordinate<S, [S, ..1]> for Vec1<S>;
-impl<S: Clone> Coordinate<S, [S, ..2]> for Vec2<S>;
-impl<S: Clone> Coordinate<S, [S, ..3]> for Vec3<S>;
-impl<S: Clone> Coordinate<S, [S, ..4]> for Vec4<S>;
-impl<S: Clone> Coordinate<S, [S, ..5]> for Vec5<S>;
-impl<S: Clone> Coordinate<S, [S, ..6]> for Vec6<S>;
+impl<S: Clone + Field> Coordinate<S, [S, ..1]> for Vec1<S>;
+impl<S: Clone + Field> Coordinate<S, [S, ..2]> for Vec2<S>;
+impl<S: Clone + Field> Coordinate<S, [S, ..3]> for Vec3<S>;
+impl<S: Clone + Field> Coordinate<S, [S, ..4]> for Vec4<S>;
+impl<S: Clone + Field> Coordinate<S, [S, ..5]> for Vec5<S>;
+impl<S: Clone + Field> Coordinate<S, [S, ..6]> for Vec6<S>;
 
-impl<S: Ring> ScalarMul<S> for Vec1<S>;
-impl<S: Ring> ScalarMul<S> for Vec2<S>;
-impl<S: Ring> ScalarMul<S> for Vec3<S>;
-impl<S: Ring> ScalarMul<S> for Vec4<S>;
-impl<S: Ring> ScalarMul<S> for Vec5<S>;
-impl<S: Ring> ScalarMul<S> for Vec6<S>;
+impl<S: Field> ScalarMul<S> for Vec1<S>;
+impl<S: Field> ScalarMul<S> for Vec2<S>;
+impl<S: Field> ScalarMul<S> for Vec3<S>;
+impl<S: Field> ScalarMul<S> for Vec4<S>;
+impl<S: Field> ScalarMul<S> for Vec5<S>;
+impl<S: Field> ScalarMul<S> for Vec6<S>;
 
-impl<S: Ring> Module<S> for Vec1<S>;
-impl<S: Ring> Module<S> for Vec2<S>;
-impl<S: Ring> Module<S> for Vec3<S>;
-impl<S: Ring> Module<S> for Vec4<S>;
-impl<S: Ring> Module<S> for Vec5<S>;
-impl<S: Ring> Module<S> for Vec6<S>;
+impl<S: Field> Module<S> for Vec1<S>;
+impl<S: Field> Module<S> for Vec2<S>;
+impl<S: Field> Module<S> for Vec3<S>;
+impl<S: Field> Module<S> for Vec4<S>;
+impl<S: Field> Module<S> for Vec5<S>;
+impl<S: Field> Module<S> for Vec6<S>;
 
-impl<S: Ring> VectorSpace<S> for Vec1<S>;
-impl<S: Ring> VectorSpace<S> for Vec2<S>;
-impl<S: Ring> VectorSpace<S> for Vec3<S>;
-impl<S: Ring> VectorSpace<S> for Vec4<S>;
-impl<S: Ring> VectorSpace<S> for Vec5<S>;
-impl<S: Ring> VectorSpace<S> for Vec6<S>;
+impl<S: Field> VectorSpace<S> for Vec1<S>;
+impl<S: Field> VectorSpace<S> for Vec2<S>;
+impl<S: Field> VectorSpace<S> for Vec3<S>;
+impl<S: Field> VectorSpace<S> for Vec4<S>;
+impl<S: Field> VectorSpace<S> for Vec5<S>;
+impl<S: Field> VectorSpace<S> for Vec6<S>;
 
 macro_rules! impl_vec_inner_product(
     ($Self:ident <$S:ident>) => (
-        impl<$S:Real + Ring + ApproxEq<$S>> InnerProductSpace<$S> for $Self<$S> {
+        impl<$S:Real + Field + ApproxEq<$S>> InnerProductSpace<$S> for $Self<$S> {
             fn norm(&self) -> $S {
                 num::sqrt(self.inner(self))
             }
@@ -220,21 +220,21 @@ impl_vec_inner_product!(Vec6<S>)
 
 // Euclidean spaces only really make sense for 2D and 3D vector spaces
 
-impl<S:Real + Ring + ApproxEq<S>> EuclideanSpace<S> for Vec2<S> {
+impl<S:Real + Field + ApproxEq<S>> EuclideanSpace<S> for Vec2<S> {
     fn angle(&self, other: &Vec2<S>) -> S {
         atan2(self.perp_dot(other), self.dot(other))
     }
 }
 
-impl<S:Real + Ring + ApproxEq<S>> EuclideanSpace<S> for Vec3<S> {
+impl<S:Real + Field + ApproxEq<S>> EuclideanSpace<S> for Vec3<S> {
     fn angle(&self, other: &Vec3<S>) -> S {
         atan2(self.cross(other).length(), self.dot(other))
     }
 }
 
-impl<S: Ring> VectorExt<S, [S, ..1]> for Vec1<S>;
-impl<S: Ring> VectorExt<S, [S, ..2]> for Vec2<S>;
-impl<S: Ring> VectorExt<S, [S, ..3]> for Vec3<S>;
-impl<S: Ring> VectorExt<S, [S, ..4]> for Vec4<S>;
-impl<S: Ring> VectorExt<S, [S, ..5]> for Vec5<S>;
-impl<S: Ring> VectorExt<S, [S, ..6]> for Vec6<S>;
+impl<S: Clone + Field> VectorExt<S, [S, ..1]> for Vec1<S>;
+impl<S: Clone + Field> VectorExt<S, [S, ..2]> for Vec2<S>;
+impl<S: Clone + Field> VectorExt<S, [S, ..3]> for Vec3<S>;
+impl<S: Clone + Field> VectorExt<S, [S, ..4]> for Vec4<S>;
+impl<S: Clone + Field> VectorExt<S, [S, ..5]> for Vec5<S>;
+impl<S: Clone + Field> VectorExt<S, [S, ..6]> for Vec6<S>;