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 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 the 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 maximize 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

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. @ref Math::Matrix (and
@ref Math::Matrix3, @ref Math::Matrix4) is by default constructed as an identity
matrix.

@snippet MagnumMath.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 the column vectors to the
constructor. These constructors check the number of passed arguments and errors
are caught at compile time.

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

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:

@snippet MagnumMath.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 MagnumMath.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 MagnumMath.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 Color4 from @ref Color3 will
set the alpha to the max value (thus @cpp 1.0f @ce for @ref Color4 and @cpp 255 @ce
for @ref Color4ub):

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

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

@snippet MagnumMath.cpp matrix-vector-construct-color-hue

Finally, the 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. 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.

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

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

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

@snippet MagnumMath.cpp matrix-vector-access

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

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

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

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

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

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

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

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

All vector, matrix and other classes in @ref Math namespace can be
constructed from an instance with a 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 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 @cpp auto b = Type{a} @ce instead of
@cpp Type b{a} @ce.

@snippet MagnumMath.cpp matrix-vector-convert

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

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

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
a scalar with vector:

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

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

@snippet MagnumMath.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 integers within the result, you need to convert both arguments to the same
floating-point type to have floating-point result.

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

You can also use all bitwise operations on integral vectors:

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

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

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

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

@snippet MagnumMath.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. You can use @ref Math::Vector::sum() "sum()"
and @ref Math::Vector::product() "product()" for sum or product of components
in one vector:

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

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

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

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

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

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

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):

@snippet MagnumMath.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 MagnumMath.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-column-major 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:

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

    @snippet MagnumMath.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 MagnumMath.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 MagnumMath.cpp matrix-vector-column-major-access
</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 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.
*/
}