magnum/doc/matrix-vector.dox

namespace Magnum { namespace Math {
/** @page matrix-vector Operations with matrices and vectors

@brief Introduction to essential classes of the graphics pipeline.

@tableofcontents

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. This
page will overview their usage and introduce some tricks to make your life
easier.

@section matrix-vector-hierarchy Matrix and vector classes

%Magnum has three main matrix and vector classes: RectangularMatrix, (square)
Matrix and Vector. To achieve greatest code reuse, %Matrix is internally
square %RectangularMatrix and %RectangularMatrix is internally array of one or
more %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 and for most common
vector and matrix sizes there are specialized classes Matrix3 and Matrix4,
implementing various transformation in 2D and 3D, Vector2, Vector3 and 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 %RectangularMatrix are returned as %Vector, but when
accessing columns of e.g. %Matrix3, they are returned as %Vector3.

There are also even more specialized subclasses - Point2D, Point3D for
creating points with homogeneous coordinates and Color3, Color4 for color
handling and conversion.

@section matrix-vector-construction Constructing matrices and vectors

Default constructors of RectangularMatrix and Vector (and Vector2, Vector3,
Vector4, Color3) create zero-filled objects. Matrix (and Matrix3, Matrix4) is
by default constructed as identity matrix. Point2D and Point3D have
homogeneous component set to one, Color4 has alpha value set to opaque.
@code
RectangularMatrix<2, 3, int> a; // zero-filled
Vector<3, int> b;               // zero-filled

Matrix<3, int> identity;                    // diagonal set to 1
Matrix<3, int> zero(Matrix<3, int>::Zero);  // zero-filled

Point2D<int> c; // {0, 0, 1}
Point3D<int> d; // {0, 0, 0, 1}

Color4<float> black1;           // {0.0f, 0.0f, 0.0f, 1.0f}
Color4<unsigned char> black2;   // {0, 0, 0, 255}
@endcode

Most common and most efficient way to create vector is to pass all values to
constructor, matrix is created by passing all column vectors to the
constructor.
@code
Vector3<int> vec(0, 1, 2);

Matrix3<int> mat({0, 1, 2},
                 {3, 4, 5},
                 {6, 7, 8});
@endcode
All constructors check number of passed arguments and the errors are catched
at compile time.

You can specify all components of vector or whole diagonal of square matrix at
once:
@code
Matrix3<int> diag(Matrix3<int>::Identity, 2); // diagonal set to 2, zeros elsewhere
Vector3<int> fill(10);    // {10, 10, 10}
@endcode

Vectors are commonly used to specify various axes and scaling coefficients in
transformations, you can use convenience functions instead of typing out all
other elements:
@code
Matrix4::rotation(5.0_degf, Vector3::xAxis());  // {1.0f, 0.0f, 0.0f}
Matrix3::translation(Vector2::yAxis(2.0f));     // {0.0f, 2.0f}
Matrix4::scaling(Vector3::zScale(-10.0f));      // {1.0f, 1.0f, -10.0f}
@endcode

It is possible to create matrices from other matrices and vectors with the
same row count; vectors from vector and scalar:
@code
RectangularMatrix<2, 3, int> a;
Vector3<int> b, c;
Matrix3<int> mat(a, b);
Vector<8, int> vec(1, b, 2, c);
@endcode

It is also possible to create them from an C-style array. The function does
simple type cast without any copying, so it's possible to conveniently operate
on the array itself:
@code
int[] mat = { 2, 4, 6,
              1, 3, 5 };
RectangularMatrix<2, 3, int>::from(mat) *= 2; // mat == { 4, 8, 12, 2, 6, 10 }
@endcode
Note that unlike constructors, this function has no way to check whether the
array is long enough to contain all elements, so use with caution.

You can also *explicitly* convert between data types:
@code
Vector4<float> floating(1.3f, 2.7f, -15.0f, 7.0f);
Vector4<int> integral(floating); // {1, 2, -15, 7}
@endcode

@section matrix-vector-component-access Accessing matrix and vector components

Column vectors of matrices and vector components can be accessed using square
brackets, there is also round bracket operator for accessing matrix components
directly:
@code
RectangularMatrix<3, 2, int> a;
a[2] /= 2;      // third column (column major indexing, see explanation below)
a[0][1] = 5;    // first column, second element

Vector<3, int> b;
b[1] = 1;       // second element
@endcode

Fixed-size vector subclasses have functions for accessing named components
and subparts:
@code
Vector4<int> a;
int x = a.x();
a.y() += 5;

Vector3<int> xyz = a.xyz();
xyz.xy() *= 5;
@endcode
Color3 and Color4 name their components `rgba` instead of `xyzw`.

For more involved operations with components there is the swizzle() function:
@code
Vector4<int> original(-1, 2, 3, 4);
Vector4<int> bgra = swizzle<'b', 'g', 'r', 'a'>(original); // { 3, 2, -1, 4 }
Vector<6, int> w10xyz = swizzle<'w', '1', '0', 'x', 'y', 'z'>(original); // { 4, 1, 0, -1, 2, 3 }
@endcode

Matrix3 and Matrix4 have functions for accessing properties of given 2D/3D
transformation:
@code
Matrix4<float> a = Matrix4<float>::translation(Vector3::yAxis(4.0f));
Vector3<float> translation = a.translation();

Matrix3<float> b = Matrix3<float>::rotation(15.0_degf));
Matrix<2, float> rotationScaling = b.rotationScaling();
Vector2<float> up = b.up();
@endcode

@section matrix-vector-column-major Matrices are column-major and vectors are columns

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

- Order of template arguments in specification of RectangularMatrix is also
  column-major:
@code
RectangularMatrix<2, 3, int> mat; // two columns, three rows
@endcode
- Order of components in matrix constructors is also column-major, further
  emphasized by requirement that you have to pass directly column vectors:
@code
Matrix3<int> mat({0, 1, 2},
                 {3, 4, 5},
                 {6, 7, 8}); // first column is {0, 1, 2}
@endcode
- Element accessing order is also column-major, thus the bracket operator is
  accessing columns. Returned vector has also its own bracket operator, which
  is then indexing rows.
@code
mat[0] *= 2;    // first column
mat[2][0] = 5;  // first element of first column
@endcode
- Various algorithms which commonly operate on matrix rows (such as
  @ref Algorithms::gaussJordanInPlace() "Gauss-Jordan elimination") have faster
  alternatives which operate on columns. It's then up to user decision to
  operate with transposed matrices or use the slower non-transposed
  alternative of the algorithm.
*/
}}