magnum/doc/matrix-vector.dox

/*
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              Vladimír Vondruš <mosra@centrum.cz>

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namespace Magnum {
/** @page matrix-vector Operations with matrices and vectors
@brief Introduction to essential classes of the graphics pipeline.

@tableofcontents
@m_footernavigation

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 @ref Math namespace and common variants also have aliases in
root @ref Magnum namespace. See documentation of these namespaces for more
information about usage with CMake.

@section matrix-vector-hierarchy Matrix and vector classes

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

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

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

@section matrix-vector-construction Constructing matrices and vectors

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

@code{.cpp}
Matrix2x3 a;                    // zero-filled
Vector3i b;                     // zero-filled

Matrix3 identity;               // diagonal set to 1
Matrix3 zero{Math::ZeroInit};   // zero-filled
@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. All constructors check number of passed arguments and the errors
are catched at compile time.

@code{.cpp}
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});
@endcode

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

@code{.cpp}
Matrix3 diag(Matrix3::Identity, 2.0f);  // diagonal is 2.0f, zeros elsewhere
Vector3i fill(10);                      // {10, 10, 10}
auto diag2 = Matrix3::fromDiagonal({3.0f, 2.0f, 1.0f});
@endcode

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

@code{.cpp}
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}
@endcode

It is also possible to create matrices and vectors 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{.cpp}
Int[] mat = { 2, 4, 6,
              1, 3, 5 };
Math::Matrix2x3<Int>::from(mat) *= 2;   // { 4, 8, 12, 2, 6, 10 }
@endcode

@attention 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.

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

@code{.cpp}
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}
@endcode

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

@code{.cpp}
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)
@endcode

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

@code{.cpp}
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}
@endcode

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

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

@code{.cpp}
Matrix3x2 a;
a[2] /= 2.0f;   // third column (column major indexing, see explanation below)
a[0][1] = 5.3f; // first column, second element

Vector3i b;
b[1] = 1;       // second element
@endcode

Row vectors can be accessed too, but only for reading, and the access is slower
due to the way the matrix is stored (see @ref matrix-vector-column-major "explanation below"):

@code{.cpp}
Vector2i c = a.row(2); // third row
@endcode

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

@code{.cpp}
Vector4i a;
Int x = a.x();
a.y() += 5;

Vector3i xyz = a.xyz();
xyz.xy() *= 5;
@endcode

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

For more involved operations with components there is the @ref Math::swizzle()
function:

@code{.cpp}
Vector4i original(-1, 2, 3, 4);
Vector4i bgra = Math::swizzle<'b', 'g', 'r', 'a'>(original); // { 3, 2, -1, 4 }
Math::Vector<6, Int> w10xyz = Math::swizzle<'w', '1', '0', 'x', 'y', 'z'>(original); // { 4, 1, 0, -1, 2, 3 }
@endcode

@section matrix-vector-conversion Converting between different underlying types

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

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

@code{.cpp}
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}
@endcode

For packing and unpacking there are @ref Math::pack() and @ref Math::unpack()
functions:

@code{.cpp}
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}
@endcode

See @ref matrix-vector-componentwise "below" for more information about other
available component-wise operations.

@section matrix-vector-operations 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
scalar with vector:

@code{.cpp}
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}
@endcode

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

@code{.cpp}
Vector3 a{1.0f, 2.0f, 3.0f};
Vector3 b = a*Vector3{-0.5f, 2.0f, -7.0f};      // {-0.5f, 4.0f, -21.0f}
@endcode

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

@code{.cpp}
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}
@endcode

You can also use all bitwise operations on integral vectors:

@code{.cpp}
Vector2i size{256, 256};
Vector2i mipLevel3Size = size >> 3;             // {32, 32}
@endcode

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

@code{.cpp}
Matrix3x2 a;
Matrix3x2 b;
Matrix3x2 c = a + (-b);

Matrix2x3 d;
Matrix2x2 e = d*b;
Matrix3x3 f = b*d;
@endcode

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

@code{.cpp}
Matrix3x4 a;
Vector3 b;
Vector4 c = a*b;

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

@section matrix-vector-componentwise Component-wise and inter-vector operations

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

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

Component-wise minimum and maximum of two vectors can be done using
@ref Math::min(), @ref Math::max() or @ref Math::minmax(), similarly with
@ref Vector::min() "min()", @ref Vector::max() "max()" and
@ref Vector2::minmax() "minmax()" for components in one vector.

@code{.cpp}
Vector3i a{-5, 7, 24};
Vector3i b{8, -2, 12};

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

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

@code{.cpp}
BoolVector<3> largerOrEqual = a >= b;           // {false, true, true}
bool anySmaller = (a < b).any();                // true
bool allLarger = (a > b).all();                 // false
@endcode

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

@code{.cpp}
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}
@endcode

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

@code{.cpp}
Matrix3x2 mat;
Math::Vector<6, Float> vec = mat.toVector();
// ...
mat = Matrix3x2::fromVector(vec);
@endcode

Note that all component-wise functions in @ref Math namespace work also for
scalars:

@code{.cpp}
std::pair<Int, Int> minmax = Math::minmax(24, -5);  // -5, 24
Int a = Math::lerp(0, 360, 0.75f);                  // 270
auto b = Math::denormalize<UnsignedByte>(0.89f);    // 226
@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:

<ul><li>
    Order of template arguments in specification of @ref Math::RectangularMatrix
    is also column-major:

    @code{.cpp}
    Math::RectangularMatrix<2, 5, Int> mat; // two columns, five rows
    @endcode
</li><li>
    Order of components in matrix constructors is also column-major, further
    emphasized by requirement that you have to pass directly column vectors:

    @code{.cpp}
    Math::Matrix3<Int> mat({0, 1, 2},
                           {3, 4, 5},
                           {6, 7, 8});      // first column is {0, 1, 2}
    @endcode
</li><li>
    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{.cpp}
    mat[0] *= 2;                            // first column
    mat[2][0] = 5;                          // first element of third column
    @endcode
</li><li>
    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.
</li></ul>
*/
}