magnum/doc/matrix-vector.dox

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

@m_keyword{Matrices and vectors,,}

@tableofcontents
@m_footernavigation

Matrices and vectors are the most important part of graphics programming and
one of Magnum goals is to make their usage as intuitive as possible.

@section matrix-vector-hierarchy Matrix and vector classes

Magnum has three main matrix and vector classes --- a
@ref Math::RectangularMatrix, a (square) @ref Math::Matrix and a
@ref Math::Vector. To maximize code reuse, the @ref Math::Matrix is internally
a square @ref Math::RectangularMatrix and a @ref Math::RectangularMatrix is
internally an 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, including a @ref Half or e.g. a @ref Rad.

Each subclass brings some specialization to its superclass. The
@ref Math::Vector2, @ref Math::Vector3 and @ref Math::Vector4 classes implement
direct access to named components, @ref Math::Matrix adds determinant
calculation or matrix inversion and @ref Math::Matrix3 / @ref Math::Matrix4
implement 2D and 3D transformations. Functions of each class return the most
specialized type known to make subsequent operations more convenient ---
columns of @ref Math::RectangularMatrix are returned as a @ref Math::Vector,
but when accessing columns of e.g. @ref Math::Matrix3, they are returned as a
@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 the @ref Magnum namespace, so
you can write e.g. @ref Vector3i instead of
@ref Math::Vector3 "Math::Vector3<Int>". See @ref types and the @ref Magnum
namespace documentation for more information.

@section matrix-vector-construction Constructing matrices and vectors

The 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, equivalent
to using the @ref Math::ZeroInit tag. @ref Math::Matrix (and
@ref Math::Matrix3, @ref Math::Matrix4) is by default constructed as an
identity matrix, equivalent to using the @ref Math::IdentityInit tag.

@snippet Math.cpp matrix-vector-construct

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 *column* vectors to the
constructor. The constructors check correct number of passed arguments at
compile time.

@snippet Math.cpp matrix-vector-construct-value

You can specify all components of a vector or a matrix with a single value, or
specify just values on the matrix diagonal:

@snippet Math.cpp matrix-vector-construct-diagonal

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

@snippet Math.cpp matrix-vector-construct-axis

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:

@snippet Math.cpp matrix-vector-construct-from

@attention Note that, unlike a constructor, this function has no way to check
    whether the array is long enough to contain all the elements, so use it
    with caution.

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

@snippet Math.cpp matrix-vector-construct-color

Similar to axes and scales in vectors, you can create single color shades too:

@snippet Math.cpp matrix-vector-construct-color-axis

There are also builtin colorspace conversion functions --- it's possible to
create a RGB color from a HSV value, a linear color value from a sRGB
representation, or convert from CIE XYZ / xyY. And the other way as well:

@snippet Math.cpp matrix-vector-construct-color-colorspace

Finally, the namespace @ref Math::Literals provides convenient
@link Math::Literals::ColorLiterals::operator""_rgb() operator""_rgb() @endlink
/ @link Math::Literals::ColorLiterals::operator""_rgbf() operator""_rgbf() @endlink
and @link Math::Literals::ColorLiterals::operator""_rgba() operator""_rgba() @endlink
/ @link Math::Literals::ColorLiterals::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. For sRGB input, there is
@link Math::Literals::ColorLiterals::operator""_srgb() operator""_srgb() @endlink
/ @link Math::Literals::ColorLiterals::operator""_srgba() operator""_srgba() @endlink
and @link Math::Literals::ColorLiterals::operator""_srgbf() operator""_srgbf() @endlink
/ @link Math::Literals::ColorLiterals::operator""_srgbaf() operator""_srgbaf() @endlink,
see their documentation for more information.

@snippet Math.cpp matrix-vector-construct-color-literal

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

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

@snippet Math.cpp matrix-vector-access

Row vectors can be accessed too, but only for reading, and access is slower
due to the matrix being stored @ref matrix-vector-column-major "in column-major order":

@snippet Math.cpp matrix-vector-access-row

Fixed-size vector subclasses have functions for accessing named components
and subparts using either `xyzw` or `rgba`:

@snippet Math.cpp matrix-vector-access-named

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

@snippet Math.cpp matrix-vector-access-swizzle

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

All classes in @ref Math namespace can be constructed from an instance with a
different underlying type (e.g. convert between integer and floating-point type
or betweeen @ref Float, @ref Double and @ref Half). Unlike with plain C++ data
types, the conversion is *explicit*. That might sound inconvenient at first,
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 @cpp auto b = Type{a} @ce
instead of @cpp Type b{a} @ce.

@snippet Math.cpp matrix-vector-convert

For packing floats into integers and unpacking them back use the
@ref Math::pack() and @ref Math::unpack() functions:

@snippet Math.cpp matrix-vector-convert-pack

See below for more information about other available
@ref matrix-vector-componentwise "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
a scalar with vector:

@snippet Math.cpp matrix-vector-operations-vector

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

@snippet Math.cpp matrix-vector-operations-multiply

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 an integer result, you need to convert both arguments to
the same floating-point type to have a floating-point result.

@snippet Math.cpp matrix-vector-operations-integer

You can also use all bitwise operations on integral vectors:

@snippet Math.cpp matrix-vector-operations-bitwise

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

@snippet Math.cpp matrix-vector-operations-matrix

You can also multiply (properly sized) vectors with matrices. These operations
are equivalent to multiplying with single-column matrices:

@snippet Math.cpp matrix-vector-operations-multiply-matrix

@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. For a sum or product of components *inside*
a vector you can use @ref Math::Vector::sum() "sum()" and
@ref Math::Vector::product() "product()" instead:

@snippet Math.cpp matrix-vector-operations-componentwise

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 Math::Vector::min() "min()", @ref Math::Vector::max() "max()" and
@ref Vector2::minmax() "minmax()" for components in one vector.

@snippet Math.cpp matrix-vector-operations-minmax

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

@snippet Math.cpp matrix-vector-operations-compare

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

@snippet Math.cpp matrix-vector-operations-functions

Component-wise functions are implemented only for vectors and not for matrices
to keep the math library in a sane and maintainable size. Instead, you can
reinterpret the matrix as a vector and do the operation on it (and vice versa)
--- because you get a reference that way, the operation will affect the
original data:

@snippet Math.cpp matrix-vector-operations-functions-componentwise

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

@snippet Math.cpp matrix-vector-operations-functions-scalar

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

@section matrix-vector-linear-algebra Linear algebra operations

Also available are common linear algebra operations --- the dot or cross
product, vector, reflection or angle calculation. These are mostly available
as free functions in the @ref Math namespace, with more advanced functionality
such as QR or SVD decomposition in @ref Math::Algorithms.

@snippet Math.cpp matrix-vector-linear-algebra

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

For consistency with GLSL, Magnum stores matrices as column-major (the vectors
are columns). This naturally has some implications and it may differ from what
you're used to from linear algebra or other graphics toolkits:

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

    @snippet Math.cpp matrix-vector-column-major-template
</li><li>
    Order of components in matrix constructors is also column-major, further
    emphasized by the requirement that you must pass column vectors directly:

    @snippet Math.cpp matrix-vector-column-major-construct
</li><li>
    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.

    @snippet Math.cpp matrix-vector-column-major-access
</li><li>
    Various algorithms which commonly operate on matrix rows (such as
    @ref Math::Algorithms::gaussJordanInPlace() "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.
</li></ul>

Note that the @ref 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.
*/
}