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<title>libgfx: Matrix Math</title>
<link rel=stylesheet href="cdoc.css" type="text/css">
<meta name="Author" content="Michael Garland">
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<body>
<h2>Matrix Math</h2>
<p>The <tt>libgfx</tt> matrix math package is the companion to the
<a href="vec.html">vector package</a> and are intended to provide a convenient
way of writing vector/matrix equations.
Currently, only square matrices
of dimension 2x2 [<a href="mat2.html"><tt>Mat2</tt></a>],
3x3 [<a href="mat3.html"><tt>Mat3</tt></a>],
and 4x4 [<a href="mat4.html"><tt>Mat4</tt></a>].
<p>A matrix consists of <i>n*n</i> double precision floating point values.
Unlike the vector package, the matrix package is not yet template-based.
Matrix elements are accessed using 0-indexed (row, column) pairs. Thus, a 2x2
identity matrix can be constructed as follows:
<pre>
Mat2 A;
A(0, 0) = 1.0;
A(0, 1) = 0.0;
A(1, 0) = 0.0;
A(1, 1) = 1.0;
</pre>
The default constructors (as used above) initialize all elements to 0. All
matrix classes also provide constructors which accept a list of row vectors to
initialize their elements. Thus, the following example is equivalent to the
code above:
<pre>
Mat2 A(Vec2(1.0, 0.0), Vec2(1.0, 0.0));
</pre>
Matrices can also be automatically case to <tt>double</tt> pointers. Since
matrices are stored in row-major order, the follow code would change element
(0, 1) of the matrix above:
<pre>
double *B = A;
B[1] = -1.0;
</pre>
<p><strong>Warning:</strong>
For efficiency reasons, <em>accessors are not range checked</em>.
Thus you can legally write
<pre>
Mat2 v;
v(-10, 37) = 1.0;
</pre>
and generate an invalid memory access.
<h3>Arithmetic Operators</h3>
<p>Like the vector package, one of the primary goals of the matrix package is
to simplify the writing of vector/matrix equations.
To accomplish this, it makes use of C++ operator overloading.
<p><strong>Assignment</strong>
Matrices can be assigned the values of other matrices or scalars.
A matrix assignment <tt>A = B</tt> copies the elements of <tt>B</tt>
into the corresponding elements of <tt>A</tt>. A scalar assignment
<tt>A = 1.0</tt> copies the given scalar, in this case <tt>1.0</tt>,
into each of the elements of <tt>A</tt>.
<p><strong>Addition/Subtraction</strong>
Matrices can be added together either with the binary addition operator (<tt>C
= A + B</tt>) or the additive assignment operator (<tt>A += B</tt>).
Subtraction operates similarly, using subtraction rather than addition
operators.
<p><strong>Scalar Multiplication/Division</strong>
Matrices can be multiplied by scalar values using either the binary
operator (<tt>A * 2.0</tt>) or the accumulation operator (<tt>A *=
2.0</tt>). Scalar division operates similarly.
<p><strong>Matrix/Vector Multiplication</strong>
Matrices can be multiplied together with a binary operator (<tt>A *
B</tt>). Matrices can also multiply vectors (<tt>A * v</tt>) which returns a
new vector.
<h3>Standard Matrix Functions</h3>
<p>All matrix classes support a standard set of functions for performing
common operations on matrices. The functions are:
<pre>
<i>// Constructs the outer product of the two vectors</i>
Mat_ outer_product(const Vec_ &u, const Vec_ &v);
<i>// Returns the determinant of the matrix</i>
double det(const Mat_ &A);
<i>// Returns the trace of the matrix (the sum of the diagonals)</i>
double trace(const Mat_ &A);
<i>// Returns the transpose of A</i>
Mat_ transpose(const Mat_ &A);
<i>// Returns the adjoint of A</i>
Mat_ adjoint(const Mat_ &A);
<i>// Places the inverse of A in A_inv and returns det(A)
// The contents of A_inv are undefined if the inverse doesn't exist.</i>
double invert(Mat_ &A_inv, const Mat_ &A);
</pre>
<p>Matrices can also be read from and written to C++ iostreams using the
standard <tt><<</tt> and <tt>>></tt> operators.
</body>
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