# Feature guide » Operations with matrices and vectors

Introduction to essential classes of the graphics pipeline.

### Contents

Matrices and vectors are the most important part of graphics programming and one of goals of Magnum is to make their usage as intuitive as possible. They are contained in Math namespace and common variants also have aliases in root Magnum namespace. See the documentation of these namespaces for more information about usage with CMake.

## Matrix and vector classes

Magnum has three main matrix and vector classes: Math::RectangularMatrix, (square) Math::Matrix and Math::Vector. To maximize code reuse, Math::Matrix is internally square Math::RectangularMatrix and Math::RectangularMatrix is internally array of one or more Math::Vector instances. Both vectors and matrices can have arbitrary size (known at compile time) and can store any arithmetic type.

Each subclass brings some specialization to its superclass. For the most common vector and matrix sizes there are specialized classes Math::Matrix3 and Math::Matrix4, implementing various transformations in 2D and 3D and Math::Vector2, Math::Vector3 and Math::Vector4, implementing direct access to named components. Functions of each class try to return the most specialized type known to make subsequent operations more convenient — columns of Math::RectangularMatrix are returned as Math::Vector, but when accessing columns of e.g. Math::Matrix3, they are returned as Math::Vector3.

There are also even more specialized subclasses, e.g. Math::Color3 and Math::Color4 for color handling and conversion.

Commonly used types have convenience aliases in Magnum namespace, so you can write e.g. Vector3i instead of Math::Vector3<Int>. See Math type system and Magnum namespace documentation for more information.

## Constructing matrices and vectors

The default constructors of Math::RectangularMatrix and Math::Vector (and Math::Vector2, Math::Vector3, Math::Vector4, Math::Color3, Math::Color4) create zero-filled objects. Math::Matrix (and Math::Matrix3, Math::Matrix4) is by default constructed as an identity matrix.

```Matrix2x3 a;                    // zero-filled
Vector3i b;                     // zero-filled

Matrix3 identity;               // diagonal set to 1
Matrix3 zero{Math::ZeroInit};   // zero-filled```

The most common and most efficient way to create a vector is to pass all values to the constructor. A matrix is created by passing all the column vectors to the constructor. These constructors check the number of passed arguments and errors are caught at compile time.

```Vector3i vec{0, 1, 2};

Matrix3 mat{{0.0f, 1.9f, 2.2f},
{3.5f, 4.0f, 5.1f},
{6.0f, 7.3f, 8.0f}};```

You can specify all components of vectors or the diagonal of a square matrix with a single value or create a diagonal matrix from a vector:

```Matrix3 diag{Math::IdentityInit, 2.0f}; // diagonal is 2.0f, zeros elsewhere
Vector3i fill(10);                      // {10, 10, 10}
auto diag2 = Matrix3::fromDiagonal({3.0f, 2.0f, 1.0f});```

There are also shortcuts to create a vector with all but one component set to zero or one which are useful for transformations:

```auto x = Vector3::xAxis();              // {1.0f, 0.0f, 0.0f}
auto y = Vector2::yAxis(3.0f);          // {0.0f, 3.0f}
auto z = Vector3::zScale(3.0f);         // {1.0f, 1.0f, 3.0f}```

It is also possible to create matrices and vectors from a C-style array. The function performs a simple type cast without copying anything, so it's possible to conveniently operate on the array itself:

```Int mat[]{ 2, 4, 6,
1, 3, 5 };
Math::Matrix2x3<Int>::from(mat) *= 2;   // { 4, 8, 12, 2, 6, 10 }```

To make handling colors easier, their behavior is a bit different with a richer feature set. Implicit construction of Color4 from Color3 will set the alpha to the max value (thus `1.0f` for Color4 and `255` for Color4ub):

```Color4 a = Color3{0.2f, 0.7f, 0.5f};     // {0.2f, 0.7f, 0.5f, 1.0f}
Color4ub b = Color3ub{0x33, 0xb2, 0x7f}; // {0x33, 0xb2, 0x7f, 0xff}```

Similar to axes in vectors, you can create single color shades too, or create a RGB color from a HSV representation:

```auto green = Color3::green();           // {0.0f, 1.0f, 0.0f}
auto cyan = Color4::cyan(0.5f, 0.95f);  // {0.5f, 1.0f, 1.0f, 0.95f}
auto fadedRed = Color3::fromHsv({219.0_degf, 0.50f, 0.57f});```

Finally, the namespace Math::Literals provides convenient operator""_rgb() / operator""_rgbf() and operator""_rgba() / operator""_rgbaf() literals for entering colors in hex representation. These literals assume linear RGB input and don't do any gamma correction. For sRGB input, there is operator""_srgb() / operator""_srgba() and operator""_srgbf() / operator""_srgbaf(), see their documentation for more information.

```Color3ub a = 0x33b27f_rgb;      // {0x33, 0xb2, 0x7f}
Color4 b = 0x33b27fcc_rgbaf;    // {0.2f, 0.7f, 0.5f, 0.8f}
Color4 c = 0x33b27fcc_srgbaf;   // {0.0331048f, 0.445201f, 0.212231f, 0.8f}```

## Accessing matrix and vector components

Column vectors of matrices and vector components can be accessed using square brackets:

```Matrix3x2 a;
a /= 2.0f;   // third column (column major indexing, see explanation below)
a = 5.3f; // first column, second element

Vector3i b;
b = 1;       // second element```

Row vectors can be accessed too, but only for reading, and access is slower due to the way the matrix is stored (see explanation below):

`Vector3 c = a.row(1); // second row`

Fixed-size vector subclasses have functions for accessing named components and subparts:

```Vector4i a;
Int x = a.x();
a.y() += 5;

Vector3i xyz = a.xyz();
xyz.xy() *= 5;```

Color3 and Color4 name their components `rgba` instead of `xyzw`.

For more involved operations with components there is the Math::gather() and Math::scatter() functions:

```Vector4i orig{-1, 2, 3, 4};
Vector4i bgra = Math::gather<'b', 'g', 'r', 'a'>(orig); // { 3, 2, -1, 4 }
Math::Vector<6, Int> w10xyz = Math::gather<'w', '1', '0', 'x', 'y', 'z'>(orig);
// { 4, 1, 0, -1, 2, 3 }

Vector4 vec{1.5f, 3.0f, 0.1f, 1.1f};
Vector2 coords{5.0f, -2.0f};
Math::scatter<'z', 'w'>(vec, coords); // { 1.5, 3.0, 5.0, -2.0 }```

## Converting between different underlying types

All vector, matrix and other classes in Math namespace can be constructed from an instance with a different underlying type (e.g. convert between integer and floating-point or betweeen Float and Double). Unlike with plain C++ data types, the conversion is done via an explicit constructor. That might sound inconvenient, but performing the conversion explicitly avoids common issues like precision loss (or, on the other hand, expensive computation with unnecessarily high precision).

To further emphasise the intent of conversion (so it doesn't look like an accident or a typo), you are encouraged to use `auto b = Type{a}` instead of `Type b{a}`.

```Vector3 a{2.2f, 0.25f, -5.1f};
//Vector3i b = a;                   // error, implicit conversion not allowed
auto c = Vector3i{a};               // {2, 0, -5}
auto d = Vector3d{a};               // {2.2, 0.25, -5.1}```

For packing and unpacking use the Math::pack() and Math::unpack() functions:

```Color3 a{0.8f, 1.0f, 0.3f};
auto b = Math::pack<Color3ub>(a);   // {204, 255, 76}

Color3ub c{64, 127, 89};
auto d = Math::unpack<Color3>(c);   // {0.251f, 0.498f, 0.349}```

## Operations with matrices and vectors

Vectors can be added, subtracted, negated and multiplied or divided with scalars, as is common in mathematics. Magnum also adds the ability to divide a scalar with vector:

```Vector3 a{1.0f, 2.0f, 3.0f};
Vector3 b = a*5.0f - Vector3{3.0f, -0.5f, -7.5f}; // {5.0f, 9.5f, 7.5f}
Vector3 c = 1.0f/a;                             // {1.0f, 0.5f, 0.333f}```

As in GLSL, vectors can be also multiplied or divided component-wise:

```Vector3 a{1.0f, 2.0f, 3.0f};
Vector3 b = a*Vector3{-0.5f, 2.0f, -7.0f};      // {-0.5f, 4.0f, -21.0f}```

When working with integral vectors (i.e. 24bit RGB values), it is often desirable to multiply them with floating-point values but retain an integral result. In Magnum, all multiplication/division operations involving integral vectors will return integers within the result, you need to convert both arguments to the same floating-point type to have floating-point result.

```Color3ub color{80, 116, 34};
Color3ub lighter = color*1.5f;                  // {120, 174, 51}

Vector3i a{4, 18, -90};
Vector3 multiplier{2.2f, 0.25f, 0.1f};
Vector3i b = a*multiplier;                      // {8, 4, -9}
Vector3 c = Vector3(a)*multiplier;              // {8.0f, 4.5f, -9.0f}```

You can also use all bitwise operations on integral vectors:

```Vector2i size{256, 256};
Vector2i mipLevel3Size = size >> 3;             // {32, 32}```

Matrices can be added, subtracted and multiplied with matrix multiplication.

```Matrix3x2 a;
Matrix3x2 b;
Matrix3x2 c = a + (-b);

Matrix2x3 d;
Matrix2x2 e = b*d;
Matrix3x3 f = d*b;```

You can also multiply (properly sized) vectors with matrices. These operations are just convenience shortcuts for multiplying with single-column matrices:

```Matrix3x4 a;
Vector3 b;
Vector4 c = a*b;

Math::RectangularMatrix<4, 1, Float> d;
Matrix4x3 e = b*d;```

## Component-wise and inter-vector operations

As shown above, vectors can be added and multiplied component-wise using the `+` or `*` operator. You can use sum() and product() for sum or product of components in one vector:

```Float a = Vector3{1.5f, 0.3f, 8.0f}.sum();      // 8.8f
Int b = Vector3i{32, -5, 7}.product();          // 1120```

Component-wise minimum and maximum of two vectors can be done using Math::min(), Math::max() or Math::minmax(), similarly with min(), max() and minmax() for components in one vector.

```Vector3i a{-5, 7, 24};
Vector3i b{8, -2, 12};

Vector3i min = Math::min(a, b);                 // {-5, -2, 12}
Int max = a.max();                              // 24```

The vectors can be also compared component-wise, the result is returned in Math::BoolVector class:

```BoolVector3 largerOrEqual = a >= b;             // {false, true, true}
bool anySmaller = (a < b).any();                // true
bool allLarger = (a > b).all();                 // false```

There are also function for component-wise rounding, sign operations, square root, various interpolation and (de)normalization functionality:

```Vector3 a{5.5f, -0.3f, 75.0f};
Vector3 b = Math::round(a);                     // {5.0f,  0.0f, 75.0f}
Vector3 c = Math::abs(a);                       // {5.5f, -0.3f, 75.0f}
Vector3 d = Math::clamp(a, -0.2f, 55.0f);       // {5.5f, -0.2f, 55.0f}```

Component-wise functions are implemented only for vectors and not for matrices to keep the math library a sane and maintainable size. Instead, you can reinterpret the matrix as vector and do the operation on it (and vice versa):

```Matrix3x2 mat;
Math::Vector<6, Float> vec = mat.toVector();
// ...
mat = Matrix3x2::fromVector(vec);```

Note that all component-wise functions in the Math namespace also work for scalars — and on the special Deg / Rad types too.

```std::pair<Int, Int> minmax = Math::minmax(24, -5);  // -5, 24
Int a = Math::lerp(0, 360, 0.75f);                  // 270
auto b = Math::pack<UnsignedByte>(0.89f);           // 226

Deg c = Math::clamp(value, 25.0_degf, 55.0_degf);```

For types with units the only exception are power functions such as Math::pow() or Math::log() — the resulting unit of such an operation cannot be represented and thus will only work on unitless types.

## Matrices are column-major and vectors are columns

OpenGL matrices are column-major, thus in Magnum it is reasonable to use matrices also as column-major (the vectors are the columns). This naturally has some implications and it may differ from what is common in mathematics:

• Order of template arguments in specification of Math::RectangularMatrix is also column-major:

`Math::RectangularMatrix<2, 5, Int> mat; // two columns, five rows`
• Order of components in matrix constructors is also column-major, further emphasized by the requirement that you must pass column vectors directly:

```Math::Matrix3<Int> mat{{0, 1, 2},
{3, 4, 5},
{6, 7, 8}};      // first column is {0, 1, 2}```
• Element access order is also column-major, thus the bracket operator accesses columns. The returned vector also has its own bracket operator, which then indexes rows.

```mat *= 2;                            // first column
mat = 5;                          // first element of third column```
• Various algorithms which commonly operate on matrix rows (such as Gauss-Jordan elimination) have faster alternatives which operate on columns. It's then up to the user to operate with transposed matrices or use the slower non-transposed alternative of the algorithm.

Note that the Corrade::Utility::Debug utility always prints the matrices in the expected layout — rows are rows and columns are columns. You are encouraged to use it for data visualization purposes.