/linalg

linalg.h is a single header, public domain, short vector math library for C++

Primary LanguageC++The UnlicenseUnlicense

NOTE: This is the development branch for linalg.h v3.0. Breaking changes may occur until the v3.0 tag is published.

Release is 3.0-beta License is Unlicense Travis CI build status Appveyor build status

linalg.h is a single header, public domain linear algebra library for C++11. It is inspired by the syntax of popular shading and compute languages and is intended to serve as a lightweight alternative to projects such as GLM, Boost.QVM or Eigen in the domains such as computer graphics, computational geometry, and physical simulation. It allows you to easily write programs like the following:

#include <linalg.h>
using namespace linalg::aliases;

// Compute the equation of a plane containing points a, b, and c
float4 compute_plane(float3 a, float3 b, float3 c)
{
    float3 n = cross(b-a, c-a);
    return {n, -dot(n,a)};
}

Data structures

Vectors

linalg::vec<T,M> defines a fixed-length vector containing exactly M elements of type T. Convenience aliases such as float3, float4, or int2 are provided in the linalg::aliases namespace. This data structure can be used to store a wide variety of types of data, including geometric vectors, points, homogeneous coordinates, plane equations, colors, texture coordinates, or any other situation where you need to manipulate a small sequence of numbers. As such, vec<T,M> is supported by a set of algebraic and component-wise functions, as well as a set of standard reductions.

vec<T,M>:

  • is DefaultConstructible: 
    float3 v; // v contains 0,0,0
  • is constructible from M elements of type T: 
    float3 v {1,2,3}; // v contains 1,2,3
  • is CopyConstructible and CopyAssignable: 
    float3 v {1,2,3}; // v contains 1,2,3
float3 u {v};     // u contains 1,2,3
float3 w;         // w contains 0,0,0 
w = u;            // w contains 1,2,3
  • is EqualityComparable and LessThanComparable: 
    if(v == y) cout << "v and u contain equal elements in the same positions" << endl;
if(v < u) cout << "v precedes u lexicographically" << endl;
  • is explicitly constructible from a single element of type T: 
    float3 v = float3{4}; // v contains 4,4,4
  • is explicitly constructible from a vec<U,M> of some other type U: 
    float3 v {1.1f,2.3f,3.5f}; // v contains 1.1,2.3,3.5
int3 u = int3{v};          // u contains 1,2,3
  • supports indexing: 
    float x = v[0]; // x contains first element of v
v[2] = 5;       // third element of v set to 5
  • supports named accessors x,y,z,w, r,g,b,a, or s,t,p,q and swizzles: 
    float y = point.y;    // y contains second element of point
pixel.a = 0.5;        // fourth element of pixel set to 0.5
float s = tc.s;       // s contains first element of tc
float3 c = pixel.bgr; // c contains pixel[2],pixel[1],pixel[0]
  • supports unary operators +, -, ! and ~ in component-wise fashion: 
    auto v = -float{2,3}; // v is float2{-2,-3}
  • supports binary operators +, -, *, /, %, |, &, ^, << and >> in component-wise fashion: 
    auto v = float2{1,1} + float2{2,3}; // v is float2{3,4}
  • supports binary operators with a scalar on the left or the right: 
    auto v = 2 * float3{1,2,3}; // v is float3{2,4,6}
auto u = float3{1,2,3} + 1; // u is float3{2,3,4}
  • supports operators +=, -=, *=, /=, %=, |=, &=, ^=, <<= and >>= with vectors or scalars on the right: 
    float2 v {1,2}; v *= 3; // v is float2{3,6}
  • supports operations on mixed element types: 
    auto v = float3{1,2,3} + int3{4,5,6}; // v is float3{5,7,9}
  • supports range-based for: 
    for(auto elem : float3{1,2,3}) cout << elem << ' '; // prints "1 2 3 "
  • has a flat memory layout: 
    float3 v {1,2,3}; 
float * p = v.data(); // &v[i] == p+i
p[1] = 4; // v contains 1,4,3

Matrices

linalg::mat<T,M,N> defines a fixed-size matrix containing exactly M rows and N columns of type T, in column-major order. Convenience aliases such as float4x4 or double3x3 are provided in the linalg::aliases namespace. Unlike vec<T,M>, this data structure is not intended for general storage of two dimensional arrays of data, and is supported only by a set of algebraic functions. However, component-wise and reduction operations can be invoked explicitly via higher-order functions.

mat<T,M,N>:

  • is DefaultConstructible: 
    float2x2 m; // m contains columns 0,0; 0,0
  • is constructible from N columns of type vec<T,M>: 
    float2x2 m {{1,2},{3,4}}; // m contains columns 1,2; 3,4
  • is constructible from linalg::identity: 
    float3x3 m = linalg::identity; // m contains columns 1,0,0; 0,1,0; 0,0,1
  • is CopyConstructible and CopyAssignable: 
    float2x2 m {{1,2},{3,4}}; // m contains columns 1,2; 3,4
float2x2 n {m};           // n contains columns 1,2; 3,4
float2x2 p;               // p contains columns 0,0; 0,0
p = n;                    // p contains columns 1,2; 3,4
  • is EqualityComparable and LessThanComparable: 
    if(m == n) cout << "m and n contain equal elements in the same positions" << endl;
if(m < n) cout << "m precedes n lexicographically when compared in column-major order" << endl;
  • is explicitly constructible from a single element of type T: 
    float2x2 m {5}; // m contains columns 5,5; 5,5
  • is explicitly constructible from a mat<U,M,N> of some other type U: 
    float2x2 m {int2x2{{5,6},{7,8}}}; // m contains columns 5,6; 7,8
  • supports indexing into columns: 
    float2x3 m {{1,2},{3,4},{5,6}}; // m contains columns 1,2; 3,4; 5,6
float2 c = m[0];                // c contains 1,2
m[1]     = {7,8};               // m contains columns 1,2; 7,8; 5,6
  • supports retrieval of rows: 
    float2x3 m {{1,2},{3,4},{5,6}}; // m contains columns 1,2; 3,4; 5,6
float3 r = m.row(1);            // r contains 2,4,6
  • supports unary operators +, -: 
    float2x2 m {{1,2},{3,4}}; // m contains columns 1,2; 3,4
float2x2 n = -m;          // n contains columns -1,-2; -3,-4
  • supports binary operators +, - between matrices of the same size: 
    float2x2 a {{0,0},{2,2}}; // a contains columns 0,0; 2,2
float2x2 b {{1,2},{1,2}}; // b contains columns 1,2; 1,2
float2x2 c = a + b;       // c contains columns 1,2; 3,4
  • supports operator * with a scalar on the left or on the right
  • supports operator / with a scalar on the right
  • supports operator * with a vec<T,N> on the right (matrix product)
  • supports operator * with a mat<T,N,P> on the right (matrix product)
  • supports operators +=, -=, *=, /= with appropriate types on the right
  • supports operations on mixed element types
  • supports range-based for over columns
  • has a flat memory layout

TODO: Finish explaining linalg::mat<T,M,N>

Quaternions

linalg::quat<T> defines a quaternion using four elements of type T, representing the quantity xi + yi + zj + w in x,y,z,w order. Convenience aliases such as quatf and quatd are provided in the linalg::aliases namespace. Unlike vec<T,4>, this data structure is not intended for general storage of four dimensional quantities, and is supported only by a set of algebraic functions. However, component-wise and reduction operations can be invoked explicitly via higher-order functions.

quat<T>:

  • is DefaultConstructible: to zero: 
    quatf q; // q contains 0,0,0,0
  • is constructible from four elements of type T: 
    quatf q {1,0,0,0}; // q contains 1,0,0,0
  • is constructible from linalg::identity: 
    quatf q = linalg::identity; // q contains 0,0,0,1
  • is constructible from a vec<T,3> and a T: 
    float3 axis {0,1,0};
float angle = 2;
quatf q {axis*sinf(angle/2), cosf(angle/2)}; // q represents rotation of angle about axis
  • is CopyConstructible and CopyAssignable: 
    quatf q {0,1,0,0}; // q contains 0,1,0,0
quatf r {q};       // r contains 0,1,0,0
quatf s;           // s contains 0,0,0,0
s = r;             // s contains 0,1,0,0
  • is EqualityComparable and LessThanComparable: 
    if(q == r) cout << "q and r are equal quaternions" << endl;
if(q < r) cout << "q precedes r lexicographically in x,y,z,w order" << endl;
  • is explicitly constructible from a vec<T,4>: 
    float4 v {1,2,3,4}; // v contains 1,2,3,4
quatf q {v};        // q contains 1,2,3,4
  • is explicitly constructible from a quat<U> of some other type U: 
    quatf q {quatd{1,0,0,0}}; // q contains 1,0,0,0`
  • has named fields x,y,z,w: 
    float real = q.w; // real contains the real-valued part of q
  • supports unary operators +, -
  • supports binary operators +, - between quaternions
  • supports operator * with quaternions or scalars
  • supports operator / with a scalar on the right
  • supports operators +=, -=, *=, /= with appropriate types on the right
  • supports operations on mixed element types
  • exposes explicit cast to vec<T,4> and member function xyz()

TODO: Finish explaining linalg::quat<T>

Function listing

Vector algebra

TODO: Explain cross, dot, length, length2, normalize, distance, distance2, angle, uangle, nlerp, slerp

Matrix algebra

TODO: Explain diagonal, trace, outerprod, transpose, adjugate, comatrix, determinant, inverse

Quaternion algebra

TODO: Explain exp, log, pow, conjugate, dot, length, length2, inverse, normalize, uangle, lerp, nlerp, slerp TODO: Explain qxdir, qydir, qzdir, qmat, qrot, qangle, qaxis

Component-wise operations

The unary functions abs, floor, ceil, exp, log, log10, sqrt, sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, round accept a vector-valued argument and produce a vector-valued result by passing individual elements to the function of the same name in the std:: namespace, as defined by <cmath> or <cstdlib>.

float4 a {1,-4,9,-16}; // a contains 1,-4,9,-16
float4 b = abs(a);     // b contains 1,4,9,16
float4 c = sqrt(b);    // c contains 1,2,3,4

The binary functions fmod, pow, atan2, and copysign function similarly, except that either argument can be a vector or a scalar.

float2 a {5,4}, b {2,3};
float2 c = pow(a, 2);    // c contains 25,16
float2 d = pow(2, b);    // d contains 4,8
float2 e = pow(a, b);    // e contains 25,64

The binary functions equal, nequal, less, greater, lequal, and gequal apply operators ==, !=, <, >, <= and >= respectively in a component-wise fashion, returning a vec<bool,M>. As before, either argument can be a vector or a scalar.

int2 a {2,5}, b {3,4};
bool2 c = less(a,3);    // c contains true, false
bool2 d = equal(4,b);   // d contains false, true
bool2 e = greater(a,b); // e contains false, true

TODO: Explain min, max, clamp, select, lerp

Reductions

  • any :: vec<bool,M> => bool returns true if any element of the vector is true
  • all :: vec<bool,M> => bool returns true if all elements of the vector are true
  • sum :: vec<T,M> => T returns the sum of all elements in the vector
  • product :: vec<T,M> => T returns the product of all elements in the vector
  • minelem :: vec<T,M> => T returns the value of the least element in the vector
  • maxelem :: vec<T,M> => T returns the value of the greatest element in the vector
  • argmin :: vec<T,M> => int returns the index of the least element in the vector
  • argmax :: vec<T,M> => int returns the index of the greatest element in the vector

Comparisons

TODO: Explain compare

Optional features

Type aliases

By default, linalg.h does not define any symbols in the global namespace, and a three-element vector of single-precision floating point values must be spelled linalg::vec<float,3>. In various libraries and shading languages, such a type might be spelled float3, vec3, vec3f, point3f, simd_float3, or any one of a hundred other possibilities. linalg.h provides a collection of useful aliases in the linalg::aliases namespace. If the names specified in this namespace are suitable for a user's purposes, they can quickly be brought into scope as follows:

#include <linalg.h>
using namespace linalg::aliases;

float3 a_vector;
float4x4 a_matrix;

Note that this only brings the type aliases into global scope. The core types and all functions and operator overloads defined by the library remain in namespace linalg.

If the spellings in namespace linalg::aliases conflict with other types that have been defined in the global namespace or in other namespaces of interest, the user can choose to omit the using namespace directive and instead define their own aliases as desired.

#include <linalg.h>
using v3f = linalg::vec<float,3>;
using m44f = linalg::mat<float,4,4>;

v3f a_vector;
m44f a_matrix;

It is, of course, always possible to use the core linalg.h types directly if operating in an environment where no additional symbols should be defined.

#include <linalg.h>

linalg::vec<float,3> a_vector;
linalg::mat<float,4,4> a_matrix;

The set of type aliases defined in namespace linalg::aliases is as follows:

  • vec<float,M> aliased to floatM, as in: float1, float2, float3, float4
  • vec<double,M> aliased to doubleM, as in: double1, double2, double3, double4
  • vec<int,M> aliased to intM as in: int1, int2, int3, int4
  • vec<bool,M> aliased to boolM as in: bool1, bool2, bool3, bool4
  • mat<float,M,N> aliased to floatMxN as in: float1x3, float3x2, float4x4, etc.
  • mat<double,M,N> aliased to doubleMxN as in: double1x3, double3x2, double4x4, etc.
  • mat<int,M,N> aliased to intMxN as in: int1x3, int3x2, int4x4, etc.
  • quat<float> aliased to quatf
  • quat<double> aliased to quatd
  • quat<int> aliased to quati

ostream overloads

TODO: Explain namespace linalg::ostream_overloads

User-defined conversions

TODO: Explain converter<T,U>

Extensions in linalg-ext.h

A second header linalg-ext.h is under development, providing functionality that is not essential to the use of linalg.h. This header will be considered unstable even after the publication of v3.0, but should stabilize shortly thereafter.

Higher order functions

linalg::fold(f, a, b)

fold(f, a, b) is a higher order function which accepts a function of the form A,B => A and repeatedly invokes a = f(a, element_of_b) until all elements have been consumed, before returning a. It is approximately equal to a left fold with an initial value. When b is a vec<T,M>, elements are folded from least to greatest index. When b is a mat<T,M,N>, elements are folded in column-major order. When b is a quat<T>, elements are folded in x,y,z,w order.

See also: Reductions

linalg::apply(f, a...)

apply(f, a...) is a higher order function which accepts a function of the form A... => T and applies it to component-wise sets of elements from data structures of compatible shape and dimensions. It is approximately equal to a convolution followed by a map. The shape of the result (that is, whether it is a scalar, vector, matrix, or quaternion, and the dimensions thereof) is determined by the arguments. If more than one argument is a non-scalar, the shape of those arguments must agree. Scalars can be freely intermixed with non-scalars, and element types can also be freely mixed. The element type of the returned value is determined by the return type of the provided mapping function f. The supported call signatures are enumerated in the following table:

call type of a type of b type of c result type result elements
apply(f,a) A T f(a)
apply(f,a) vec<A,M> vec<T,M> f(a[i])...
apply(f,a) mat<A,M,N> mat<T,M,N> f(a[j][i])...
apply(f,a) quat<A> quat<T> f(a.x)...
apply(f,a,b) A B T f(a, b)...
apply(f,a,b) A vec<B,M> vec<T,M> f(a, b[i])...
apply(f,a,b) vec<A,M> B vec<T,M> f(a[i], b)...
apply(f,a,b) vec<A,M> vec<B,M> vec<T,M> f(a[i], b[i])...
apply(f,a,b) A mat<B,M,N> mat<T,M,N> f(a, b[j][i])...
apply(f,a,b) mat<A,M,N> B mat<T,M,N> f(a[j][i], b)...
apply(f,a,b) mat<A,M,N> mat<B,M,N> mat<T,M,N> f(a[j][i], b[j][i])...
apply(f,a,b) A quat<B> quat<T> f(a, b.x)...
apply(f,a,b) quat<A> B quat<T> f(a.x, b)...
apply(f,a,b) quat<A> quat<B> quat<T> f(a.x, b.x)...
apply(f,a,b,c) A B C T f(a, b, c)...
apply(f,a,b,c) A B vec<C,M> vec<T,M> f(a, b, c[i])...
apply(f,a,b,c) A vec<B,M> C vec<T,M> f(a, b[i], c)...
apply(f,a,b,c) A vec<B,M> vec<C,M> vec<T,M> f(a, b[i], c[i])...
apply(f,a,b,c) vec<A,M> B C vec<T,M> f(a[i], b, c)...
apply(f,a,b,c) vec<A,M> B vec<C,M> vec<T,M> f(a[i], b, c[i])...
apply(f,a,b,c) vec<A,M> vec<B,M> C vec<T,M> f(a[i], b[i], c)...
apply(f,a,b,c) vec<A,M> vec<B,M> vec<C,M> vec<T,M> f(a[i], b[i], c[i])...

TODO: Explain apply_t<F, A...> and SFINAE helpers.

See also: Component-wise operations

Design rationale

This space exists for me to explain the design decisions that went into linalg.h.

Stability guarantees

As of the v3.0 tag, the functionality provided by linalg.h will be considered stable, except for:

  • The contents of the linalg::detail namespace
  • The type or implementation details of any types which begin with an underscore, such as _scalar<...>, _lswizzle<...>, _rswizzle<...>
  • Direct access to any member variable whose name begins with an underscore, such as vec<T,M>::_ or mat<T,M,N>::_

The linalg-ext.h header will continue to experience breaking changes. As functionality in linalg-ext.h stabilizes, it may be migrated into linalg.h, in which case it will be considered stable.

Changes from v2.1

TODO: Rework this to include a migration guide for users coming from previous versions of linalg.h.

Improvements in v3.0

  • mat<T,M,N> now defines operator * as the matrix product
  • New type quat<T> models quaternions, and defines operator * as the quaternion product
  • map(a,f) and zip(a,b,f) subsumed by new apply(f,a...)
    • apply(...) supports unary, binary, and ternary operations for vec
    • apply(...) supports unary and binary operations for mat and quat
    • apply(...) can also be invoked exclusively with scalars, and supports arbitrary numbers of arguments
    • apply(...) supports mixed element types
    • Template type alias apply_t<F,A...> provides the return type of apply(f,a...)
  • vec<T,1> and mat<T,M,1> specializations are now provided
  • compare(a,b) provide three-way comparison between compatible types
  • clamp(a,b,c) can be invoked with three distinct (but compatible) types
  • select(a,b,c) provides the a component-wise equivalent to a ? b : c
  • lerp(a,b,t) has been generalized to a component-wise operation where any of a, b, and t can be vectors or scalars
  • vec<T,M> elements can be referred to via x,y,z,w or r,g,b,a or s,t,p,q
  • vec<T,M> and mat<T,M,N> contain a member function .data() which returns a pointer to the elements
  • Groups of vec<T,M> elements can be accessed via named swizzles, such as xyz, bgr, or pq
    • All swizzles can be used as rvalues
    • Only permutation swizzles (where no accessor is repeated) can be used as lvalues
  • User can specialize converter<T,U> to enable implicit conversions from U to T, if either type is a vec, mat, or quat
    • identity is implemented using this facility to serve as an in-library example
  • No undefined behavior according to the C++11 standard
  • Almost all operations which do not internally call <cmath> functions are constexpr, except for argmin and argmax
  • No lambdas are used in linalg.h, avoidng potential ODR violations

Breaking changes in v3.0

  • linalg.h no longer supports Visual Studio 2013. However, it is known to work on GCC 4.9+, Clang 3.5+ in C++11 mode and Visual Studio 2015+.
  • vec<T,M> and mat<T,M,N> may only be used with a T which is an arithmetic type
    • This requirement will likely be relaxed by the official release of v3.0 but will require specializing some trait type to indicate additional scalar types
  • As stated above, the semantics of operator * for mat<T,M,N> has changed
    • Additional, the set of operator overloads on matrices has been reduced to those with an obvious arithmetic meaning.
    • mat+mat, mat-mat, mat*scalar, scalar*mat, and mat/scalar are implemented as elementwise operations
    • mat*mat and mat*vec computes the matrix product
    • All other binary matrix overloads have been deleted
    • Most componentwise and reduction functions no longer apply to matrices
    • However, componentwise operations can still be invoked manually with the apply(...) function
  • vec<T,M> and mat<T,M,N> are now defined in terms of arrays, instead of explicit fields x, y, z, w
    • However, vec<T,M> can still be accessed with x,y,z,w due to special scalar accessor types which should silently convert to T in most scenarios
  • Some constructors have been removed from vec and mat
    • vec<T,M> no longer has an implicit constructor from std::array<T,M>
    • vec<T,M> and mat<T,M,N> no longer have an explicit constructor from const T *
    • These capabilities can be added by specializing converter<T,U>, as shown in test-user-defined-conversions.cpp
  • The functions vec::xy() and vec::xyz() have been replaced by the swizzles vec::xy and vec::xyz
  • Some functionality has been moved from linalg.h to optional linalg-ext.h header
    • quat/matrix factory functions for 3D transformations
    • std::hash<...> specializations