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<center><h1>The <a href="http://www.complang.tuwien.ac.at/schani/mathmap/">MathMap</a> Language Reference Manual</h1></center>
<h2>Filter Syntax</h2>
<blockquote>
<p><em>options</em> <tt>filter</tt> <em>name</em> <tt>(</tt><em>user values</em></tt>)</tt><br>
<em>expression</em><br>
<tt>end</tt>
<p><em>options</em> can be zero or more options for the filter.
They are discussed in detail <a href="#options">below</a>.
<p>The <em>name</em> of the filter can be an arbitrary
identifier (identifiers can contain letters, digits and the
underscore and must not begin with a digit).
<p>The <em>user values</em> must be separated by commas. Any
number of user values can be specified. See <a href="#userValues">below</a> for a detailed
discussion of user values.
</blockquote>
<h2><a name="userValues">User Values</a></h2>
<blockquote>
<dl>
<dt><tt>float</tt> <em>name</em><tt>:</tt> <em>min</em><tt>-</tt><em>max</em> <tt>(</tt><em>default</em><tt>)</tt>
<dd>A tuple of type <tt>nil:1</tt> with the value of a floating point number
chosen in the range from <em>min</em> to <em>max</em>. The default
value is <em>default</em>.
<dt><tt>int</tt> <em>name</em><tt>:</tt> <em>min</em><tt>-</tt><em>max</em> <tt>(</tt><em>default</em><tt>)</tt>
<dd>A tuple of type <tt>nil:1</tt> with the value of an integer
chosen in the range from <em>min</em> to <em>max</em>. The default
value is <em>default</em>.
<dt><tt>bool</tt> <em>name</em>
<dd>A tuple of type <tt>nil:1</tt> with the value <tt>0</tt> or <tt>1</tt>.
<dt><tt>color</tt> <em>name</em>
<dd>A tuple of type <tt>rgba:4</tt>, representing a color.
<dt><tt>curve</tt> <em>name</em>
<dd>A function taking a single argument between <tt>0</tt> and <tt>1</tt> and returning
the value of the corresponding point on the curve, also between <tt>0</tt> and <tt>1</tt>.
<dt><tt>gradient</tt> <em>name</em>
<dd>A function taking a single argument between <tt>0</tt> and <tt>1</tt> and returning
the color at that point as a tuple of type <tt>rgba:4</tt>.
<dt><em>options</em> <tt>image</tt> <em>name</em>
<dd>A function taking a single argument of type <tt>xy:2</tt> or <tt>ra:2</tt> and returning
the color of the point with those coordinates in the image as a tuple of type <tt>rgba:4</tt>.
<em>options</em> can be left out or it can be the <tt>unit</tt> option, discussed <a href="#options">below</a>.
</dl>
</blockquote>
<h2><a name="options">Options</a></h2>
<blockquote>
<h3><tt>unit</tt></h3>
<blockquote>
<p>Filters and image user values can be given the
<tt>unit</tt> option to use a differently scaled coordinate
system. Instead of pixel coordinates, which are the
default, a filter or image with the <tt>unit</tt> option
will be scaled such that the value along its shorter axis
goes from <tt>-1</tt> to <tt>1</tt>. The longer axis will
be scaled to preserve the aspect ratio.
<p>If the <tt>unit</tt> option is given the <tt>stretched</tt>
sub-option in addition, the values along both axes will
go from <tt>-1</tt> to <tt>1</tt>.
</blockquote>
</blockquote>
<h2>The Type System</h2>
<blockquote>
<p>Every value in MathMap is a tagged tuple. This means it
consists of one or more numbers and a symbolic tag. RGBA colors, for example are
represented by tuples of length 4 with the tag <tt>rgba</tt>
(standing for "red, green, blue, alpha"). The value
<tt>rgba:[1,0,0,1]</tt> for example, represents the fully
opaque color red.
<p>Single numbers are represented by tuples of length 1 with
the tag <tt>nil</tt> (which is the default tag). For
simplicity's sake, single numbers need not be written in tuple
syntax, i.e. <tt>3.1415</tt> is equivalent to
<tt>nil:[3.1415]</tt>.
<p>There are several other tuple tags which are used by
MathMap:
<table>
<tr><td><b>Tag</b> <td><b>Purpose</b></tr>
<tr><td><tt>nil</tt> <td>Default tag</tr>
<tr><td><tt>rgba</tt> <td>RGBA Color</tr>
<tr><td><tt>hsva</tt> <td>HSVA Color</tr>
<tr><td><tt>ri</tt> <td>Complex number</tr>
<tr><td><tt>xy</tt> <td>Cartesian coordinate</tr>
<tr><td><tt>ra</tt> <td>Polar coordinate</tr>
<tr><td><tt>v2</tt> <td>2d vector</tr>
<tr><td><tt>v3</tt> <td>3d vector</tr>
<tr><td><tt>m2x2</tt> <td>2x2 matrix</tr>
<tr><td><tt>m3x3</tt> <td>3x3 matrix</tr>
<tr><td><tt>quat</tt> <td>non-commutative quaternion</tr>
<tr><td><tt>cquat</tt><td>commutative quaternion</tr>
<tr><td><tt>hyper</tt><td>hypercomplex number</tr>
</table>
<p>In order for the tag system to make sense, operators and
functions are overloaded based on the types of their
arguments. For this to work at compile time, types
must be statically determined. That means that any expression
must have only one possible type, e.g. the expression <tt>if x
then abc:[1,2,3] else xyz:[4,5] end</tt> is not valid.
Furthermore, all assignments to the same variable must have
the same type. Hence, <tt>v=[1,2];v=[1,2,3]</tt> is not
valid.
<p>Tuple types are expressed in this manual in the form
<tt>T:L</tt> where <tt>T</tt> is the tag and <tt>L</tt> is the
length, e.g. <tt>rgba:[1,0,0,1]</tt> has the type
<tt>rbga:4</tt>.
<p>Elements of tuples can be extracted by the indexing
operator <tt>[]</tt>. The expression <tt>p[0]</tt>, for
example, extracts the first element of <tt>p</tt> and is of
type <tt>nil:1</tt>.
</blockquote>
<h2>Constants and Internal Variables</h2>
<blockquote>
A few variables are set for the position of the pixel to be
calculated:
<dl>
<dt><tt>xy</tt> (<tt>xy:2</tt>)
<dd>The cartesian coordinates of the pixel.
<dt><tt>x</tt> (<tt>nil:1</tt>)
<dd>The first component of the cartesian coordinates of the pixel.
<dt><tt>y</tt> (<tt>nil:1</tt>)
<dd>The second component of the cartesian coordinates of the pixel.
<dt><tt>ra</tt> (<tt>ra:2</tt>)
<dd>The polar coordinates of the pixel.
<dt><tt>r</tt> (<tt>nil:1</tt>)
<dd>The first component of the polar coordinates of the pixel
(the distance from the center).
<dt><tt>a</tt> (<tt>nil:1</tt>)
<dd>The second component of the polar coordinates of the pixel
(<tt>0 <= r < 2*pi</tt>).
</dl>
To make it easier to write expressions which depend on the
image size, a few additional variables are set:
<dl>
<dt><tt>WH</tt> (<tt>xy:2</tt>)
<dd>The size of the image.
<dt><tt>W</tt> (<tt>nil:1</tt>)
<dd>The width of the image.
<dt><tt>H</tt> (<tt>nil:1</tt>)
<dd>The height of the image.
<dt><tt>R</tt> (<tt>nil:1</tt>)
<dd>The biggest possible value for <tt>r</tt> in the image, which
it has at the four corners.
<dt><tt>XY</tt> (<tt>xy:2</tt>)
<dd>The biggest possible value (in both components) for
<tt>xy</tt>.
<dt><tt>X</tt> (<tt>nil:1</tt>)
<dd>The biggest possible value for <tt>x</tt> for the image.
<dt><tt>Y</tt> (<tt>nil:1</tt>)
<dd>The biggest possible value for <tt>y</tt> for the image.
</dl>
For the purpose of animations two additional variable are set:
<dl>
<dt><tt>t</tt> (<tt>nil:1</tt>)
<dd>The time which is <tt>0 <= t < 1</tt>. If animation
is disabled, the value of <tt>t</tt> can be chosen in the
Settings tab. If you want to make animations loop, set the
'Periodic' check-box in the Settings tab and make sure that the
images at <tt>t == 0</tt> and <tt>t == 1</tt> are the same.
<dt><tt>frame</tt> (<tt>nil:1</tt>)
<dd>The number of the current frame, beginning with <tt>0</tt>
for the first frame.
</dl>
</blockquote>
<h2>Mathematical Constants</h2>
<blockquote>
MathMap defines a few mathematical constants to make your life easier:
<dl>
<dt><tt>pi</tt> (<tt>nil:1</tt>)
<dd>3.1415926535...</dd>
<dt><tt>e</tt> (<tt>nil:1</tt>)
<dd>Euler's constant 2.7182818284...</dd>
<dt><tt>I</tt> (<tt>ri:2</tt>)
<dd>The imaginary unit <tt>ri:[0,1]</tt>
</dl>
</blockquote>
<h2>Conditionals and Loops</h2>
<blockquote>
<p>Conditions and invariants are always expected to evaluate to tuples of
length 1.
<blockquote>
<dl>
<dt><tt>if</tt> <em>condition</em> <tt>then</tt><br>
<em>consequence</em><br>
<tt>end</tt>
<dd>Returns the value of <em>consequence</em> if the value of
<em>condition</em> is not <tt>0</tt>, <tt>0</tt> otherwise.
</dl>
<dl>
<dt><tt>if</tt> <em>condition</em> <tt>then</tt><br>
<em>consequence</em><br>
<tt>else</tt><br>
<em>alternative</em><br>
<tt>end</tt>
<dd>Returns the value of <em>consequence</em> if the value of
<em>condition</em> is not <tt>0</tt>, otherwise the value of
<em>alternative</em>.
</dl>
<dl>
<dt><tt>while</tt> <em>invariant</em> <tt>do</tt><br>
<em>body</em><br>
<tt>end</tt>
<dd>While <em>invariant</em> is not <tt>0</tt>, executes
<em>body</em>, then returns <tt>0</tt>.
</dl>
<dl>
<dt><tt>do</tt><br>
<em>body</em><br>
<tt>while</tt> <em>invariant</em> <tt>end</tt>
<dd>Executes <em>body</em> until <em>invariant</em> is not equal
<tt>0</tt>, then returns <tt>0</tt>.
</dl>
<dl>
<dt><tt>for</tt> <em>counter</em> <tt>=</tt> <em>start</em> <tt>..</tt> <em>end</em> <tt>do</tt><br>
<em>body</em><br>
<tt>end</tt>
<dd>Executes <em>body</em> with the variable <em>counter</em> going from
<em>start</em> to <em>end</em>, then returns 0. Semantically equivalent to<br>
<br>
<em>counter</em> <tt>=</tt> <em>start</em><tt>;</tt><br>
<tt>__tmp</tt> <tt>=</tt> <em>end</em><tt>;</tt><br>
<tt>while</tt> <em>counter</em> <tt><=</tt> <tt>__tmp</tt> <tt>do</tt><br>
<em>body</em><tt>;</tt><br>
<em>counter</em> <tt>=</tt> <em>counter</em> <tt>+</tt> <tt>1</tt><tt>;</tt><br>
<tt>end</tt>
</dl>
</blockquote>
</blockquote>
<h2>Operators and Functions</h2>
<blockquote>
<p>Operators and functions in MathMap are overloaded based on
the number and tuple types of their arguments. The reference
enumerates all the ways in which operators and functions are
overloaded by giving the argument and return types like this:
<tt>(v3:3, m3x3:9) -> v3:3</tt>. This particular example
describes a function taking two arguments, the first of length
3 with the tag <tt>v3</tt> and the second of length 9 with the
tag <tt>m3x3</tt>, and returning a tuple of length 3 with the
tag <tt>v3</tt>.
<p>Most overloading specifications are parametric, though, in
that the length and/or tag of the tuples is not fixed, like
here: <tt>(?t:1, ?t:1) -> ?t:1</tt>. The question mark
indicates a parametric variable (in this case <tt>?t</tt>,
whose value is arbitrary, but which must have the same value
wherever it occurs. Here the function takes two arguments,
both of length 1 and both having the same, but arbitrary, tag.
It returns a tuple of length 1 with that very same tag.
Parametric variables denoted just by a question mark, without
any symbolic name, are arbitrary and independent of all other
parametric variables, i.e. two sole question marks can have
different values.
<a name="func___add"></a><h2><tt>a + b</tt></h2>
<blockquote><blockquote><tt>(ri:2, ri:2) -> ri:2</tt><br>
<tt>(ri:2, ?:1) -> ri:2</tt><br>
<tt>(?:1, ri:2) -> ri:2</tt><br>
<tt>(?t:1, ?t:1) -> ?t:1</tt><br>
<tt>(?t:?l, ?:1) -> ?t:?l</tt><br>
<tt>(?t:?l, ?t:?l) -> ?t:?l</tt><br>
</blockquote>
Addition. Works on real numbers, complex numbers and tuples.
Tuples can be added element-wise or the same real number can be added
to each element of a tuples.</blockquote>
<a name="func___and"></a><h2><tt>a && b</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1) -> ?t:1</tt><br>
</blockquote>The logical conjunction of the two arguments.</blockquote>
<a name="func___div"></a><h2><tt>a / b</tt></h2>
<blockquote><blockquote><tt>(ri:2, ri:2) -> ri:2</tt><br>
<tt>(?t:1, ri:2) -> ri:2</tt><br>
<tt>(?:2, m2x2:4) -> v2:2</tt><br>
<tt>(?:3, m3x3:9) -> v3:3</tt><br>
<tt>(?t:1, ?t:1) -> ?t:1</tt><br>
<tt>(?t:?l, ?:1) -> ?t:?l</tt><br>
<tt>(?t:?l, ?t:?l) -> ?t:?l</tt><br>
</blockquote>
Division. Works on real numbers, complex numbers, tuples, vectors
and matrices. A tuple can be divided by another element-wise or by
the same number for each element. Vectors can be divided by
matrices.</blockquote>
<a name="func___equal"></a><h2><tt>a == b</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1) -> ?t:1</tt><br>
</blockquote>Returns 1 if the arguments are equal, otherwise 0.</blockquote>
<a name="func___greater"></a><h2><tt>a > b</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1) -> ?t:1</tt><br>
</blockquote>Returns 1 if <tt>a</tt> is greater than <tt>b</tt>, otherwise 0.</blockquote>
<a name="func___greaterequal"></a><h2><tt>a >= b</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1) -> ?t:1</tt><br>
</blockquote>
Returns 1 if <tt>a</tt> is greater or equal than <tt>b</tt>,
otherwise 0.</blockquote>
<a name="func___less"></a><h2><tt>a < b</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1) -> ?t:1</tt><br>
</blockquote>Returns 1 if <tt>a</tt> is less than <tt>b</tt>, otherwise 0.</blockquote>
<a name="func___lessequal"></a><h2><tt>a <= b</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1) -> ?t:1</tt><br>
</blockquote>
Returns 1 if <tt>a</tt> is less or equal than <tt>b</tt>, otherwise
0.</blockquote>
<a name="func___mod"></a><h2><tt>a % b</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1) -> ?t:1</tt><br>
<tt>(?t:?l, ?:1) -> ?t:?l</tt><br>
<tt>(?t:?l, ?t:?l) -> ?t:?l</tt><br>
</blockquote>
Remainder. Calculates the remainder of a division. Works on real
numbers and tuples. The remainder can be calculated for two tuples
element-wise or for one tuple and the same number for each element of
the tuple.</blockquote>
<a name="func___mul"></a><h2><tt>a * b</tt></h2>
<blockquote><blockquote><tt>(ri:2, ri:2) -> ri:2</tt><br>
<tt>(?:1, ri:2) -> ri:2</tt><br>
<tt>(m2x2:4, m2x2:4) -> m2x2:4</tt><br>
<tt>(m3x3:9, m3x3:9) -> m3x3:9</tt><br>
<tt>(v2:2, m2x2:4) -> v2:2</tt><br>
<tt>(v3:3, m3x3:9) -> v3:3</tt><br>
<tt>(m2x2:4, v2:2) -> v2:2</tt><br>
<tt>(m3x3:9, v3:3) -> v3:3</tt><br>
<tt>(quat:4, quat:4) -> quat:4</tt><br>
<tt>(cquat:4, cquat:4) -> cquat:4</tt><br>
<tt>(hyper:4, hyper:4) -> hyper:4</tt><br>
<tt>(?t:1, ?t:1) -> ?t:1</tt><br>
<tt>(?t:?l, ?:1) -> ?t:?l</tt><br>
<tt>(?t:?l, ?t:?l) -> ?t:?l</tt><br>
</blockquote>
Multiplication. Works on real numbers, complex numbers,
quaternions, hypercomplex numbers, tuples, vectors and matrices. Two
tuples can be multiplied element-wise or a tuple can be multipled by a
single number for each element. Vectors and matrices can be multipled
in both directions and two matrices can be multipled as well.</blockquote>
<a name="func___neg"></a><h2><tt>-x</tt></h2>
<blockquote><blockquote><tt>(?t:?l) -> ?t:?l</tt><br>
</blockquote>Negation.</blockquote>
<a name="func___not"></a><h2><tt>!a</tt></h2>
<blockquote><blockquote><tt>(?t:1) -> ?t:1</tt><br>
</blockquote>The logical negation of the argument.</blockquote>
<a name="func___notequal"></a><h2><tt>a != b</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1) -> ?t:1</tt><br>
</blockquote>Returns 1 if the arguments are not equal, otherwise 0.</blockquote>
<a name="func___or"></a><h2><tt>a || b</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1) -> ?t:1</tt><br>
</blockquote>The logical disjunction of the two arguments.</blockquote>
<a name="func___pow"></a><h2><tt>a ^ b</tt></h2>
<blockquote><blockquote><tt>(ri:2, ?t:1) -> ri:2</tt><br>
<tt>(ri:2, ri:2) -> ri:2</tt><br>
<tt>(?t:1, ri:2) -> ri:2</tt><br>
<tt>(?t:1, ?t:1) -> ?t:1</tt><br>
<tt>(?t:?l, ?:1) -> ?t:?l</tt><br>
</blockquote>
Exponentiation of real and complex numbers and tuples. A tuple can
be exponentiated for each element by a single number.</blockquote>
<a name="func___sub"></a><h2><tt>a - b</tt></h2>
<blockquote><blockquote><tt>(ri:2, ri:2) -> ri:2</tt><br>
<tt>(ri:2, ?:1) -> ri:2</tt><br>
<tt>(?:1, ri:2) -> ri:2</tt><br>
<tt>(?t:1, ?t:1) -> ?t:1</tt><br>
<tt>(?t:?l, ?:1) -> ?t:?l</tt><br>
<tt>(?t:?l, ?t:?l) -> ?t:?l</tt><br>
</blockquote>
Subtraction. Works on real numbers, complex numbers and tuples.
One tuple can be subtracted from another element-wise or the same real
number can be subtracted from each element of a tuple.</blockquote>
<a name="func___xor"></a><h2><tt>a xor b</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1) -> ?t:1</tt><br>
</blockquote>The logical exclusive disjunction of the two arguments.</blockquote>
<a name="func_abs"></a><h2><tt>abs(a)</tt></h2>
<blockquote><blockquote><tt>(ri:2) -> nil:1</tt><br>
<tt>(quat:4) -> nil:1</tt><br>
<tt>(cquat:4) -> nil:1</tt><br>
<tt>(hyper:4) -> nil:1</tt><br>
<tt>(?t:1) -> ?t:1</tt><br>
<tt>(?t:?l) -> ?t:?l</tt><br>
</blockquote>
Absolute value of real numbers, complex numbers (magnitude),
quaternions, hypercomplex numbers and vectors (Euclidian norm).</blockquote>
<a name="func_acos"></a><h2><tt>acos(a)</tt></h2>
<blockquote><blockquote><tt>(ri:2) -> ri:2</tt><br>
<tt>(?t:1) -> ?t:1</tt><br>
</blockquote>Arccosine of real and complex numbers.</blockquote>
<a name="func_acosh"></a><h2><tt>acosh(a)</tt></h2>
<blockquote><blockquote><tt>(ri:2) -> ri:2</tt><br>
<tt>(?t:1) -> ?t:1</tt><br>
</blockquote>Hyperbolic arccosine of real and complex numbers.</blockquote>
<a name="func_alpha"></a><h2><tt>alpha(c)</tt></h2>
<blockquote><blockquote><tt>(rgba:4) -> nil:1</tt><br>
</blockquote>The alpha (opacity) component of the color <tt>c</tt>.</blockquote>
<a name="func_arg"></a><h2><tt>arg(a)</tt></h2>
<blockquote><blockquote><tt>(ri:2) -> nil:1</tt><br>
</blockquote>The argument of a complex number.</blockquote>
<a name="func_asin"></a><h2><tt>asin(a)</tt></h2>
<blockquote><blockquote><tt>(ri:2) -> ri:2</tt><br>
<tt>(?t:1) -> ?t:1</tt><br>
</blockquote>Arcsine of real and complex numbers.</blockquote>
<a name="func_asinh"></a><h2><tt>asinh(a)</tt></h2>
<blockquote><blockquote><tt>(ri:2) -> ri:2</tt><br>
<tt>(?t:1) -> ?t:1</tt><br>
</blockquote>Hyperbolic arcsine of real and complex numbers.</blockquote>
<a name="func_atan"></a><h2><tt>atan(y, x)</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1) -> ?t:1</tt><br>
</blockquote>
Arctangent of <tt>y/x</tt>, with the signs of the arguments taken
into account to determine the correct quadrant of the result.</blockquote>
<a name="func_atan"></a><h2><tt>atan(a)</tt></h2>
<blockquote><blockquote><tt>(ri:2) -> ri:2</tt><br>
<tt>(?t:1) -> ?t:1</tt><br>
</blockquote>Arctangent of real and complex numbers.</blockquote>
<a name="func_atanh"></a><h2><tt>atanh(a)</tt></h2>
<blockquote><blockquote><tt>(ri:2) -> ri:2</tt><br>
<tt>(?t:1) -> ?t:1</tt><br>
</blockquote>Hyperbolic arctangent of real and complex numbers.</blockquote>
<a name="func_beta"></a><h2><tt>beta(a, b)</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1) -> ?t:1</tt><br>
</blockquote>The complete beta function for positive real arguments.</blockquote>
<a name="func_blue"></a><h2><tt>blue(c)</tt></h2>
<blockquote><blockquote><tt>(rgba:4) -> nil:1</tt><br>
</blockquote>The blue component of the color <tt>c</tt>.</blockquote>
<a name="func_ceil"></a><h2><tt>ceil(a)</tt></h2>
<blockquote><blockquote><tt>(?t:1) -> ?t:1</tt><br>
</blockquote>
The ceiling of a number, defined as the smallest integer not
smaller than that number.</blockquote>
<a name="func_clamp"></a><h2><tt>clamp(a, l, u)</tt></h2>
<blockquote><blockquote><tt>(?t:?l, ?t:?l, ?t:?l) -> ?t:?l</tt><br>
</blockquote>
Clamp each element of tuple <tt>a</tt> to be not less than the
corresponding element in <tt>l</tt> and not greater than the
corresponding element in <tt>u</tt>.</blockquote>
<a name="func_conj"></a><h2><tt>conj(a)</tt></h2>
<blockquote><blockquote><tt>(ri:2) -> ri:2</tt><br>
</blockquote>The complex conjugate.</blockquote>
<a name="func_cos"></a><h2><tt>cos(a)</tt></h2>
<blockquote><blockquote><tt>(ri:2) -> ri:2</tt><br>
<tt>(?t:1) -> ?t:1</tt><br>
</blockquote>Cosine of real and complex numbers.</blockquote>
<a name="func_cosh"></a><h2><tt>cosh(a)</tt></h2>
<blockquote><blockquote><tt>(ri:2) -> ri:2</tt><br>
<tt>(?t:1) -> ?t:1</tt><br>
</blockquote>Hyperbolic cosine of real and complex numbers.</blockquote>
<a name="func_crossp"></a><h2><tt>crossp(a, b)</tt></h2>
<blockquote><blockquote><tt>(?t:3, ?t:3) -> ?t:3</tt><br>
</blockquote>Cross product of two tuples/vectors with three elements.</blockquote>
<a name="func_deg2rad"></a><h2><tt>deg2rad(a)</tt></h2>
<blockquote><blockquote><tt>(?:1) -> nil:1</tt><br>
</blockquote>Convert degrees to radians.</blockquote>
<a name="func_det"></a><h2><tt>det(a)</tt></h2>
<blockquote><blockquote><tt>(m2x2:4) -> nil:1</tt><br>
<tt>(m3x3:9) -> nil:1</tt><br>
</blockquote>Determinant of a matrix.</blockquote>
<a name="func_dotp"></a><h2><tt>dotp(a, b)</tt></h2>
<blockquote><blockquote><tt>(?t:?l, ?t:?l) -> nil:1</tt><br>
</blockquote>Dot product of two tuples/vectors.</blockquote>
<a name="func_ell_int_D"></a><h2><tt>ell_int_D(phi, k, n)</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1, ?t:1) -> ?t:1</tt><br>
</blockquote>Incomplete elliptic integral D in Legendre form.</blockquote>
<a name="func_ell_int_E"></a><h2><tt>ell_int_E(phi, k)</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1) -> ?t:1</tt><br>
</blockquote>Incomplete elliptic integral E in Legendre form.</blockquote>
<a name="func_ell_int_Ecomp"></a><h2><tt>ell_int_Ecomp(k)</tt></h2>
<blockquote><blockquote><tt>(?t:1) -> ?t:1</tt><br>
</blockquote>Complete elliptic integral E in Legendre form.</blockquote>
<a name="func_ell_int_F"></a><h2><tt>ell_int_F(phi, k)</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1) -> ?t:1</tt><br>
</blockquote>Incomplete elliptic integral F in Legendre form.</blockquote>
<a name="func_ell_int_Kcomp"></a><h2><tt>ell_int_Kcomp(k)</tt></h2>
<blockquote><blockquote><tt>(?t:1) -> ?t:1</tt><br>
</blockquote>Complete elliptic integral K in Legendre form.</blockquote>
<a name="func_ell_int_P"></a><h2><tt>ell_int_P(phi, k, n)</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1, ?t:1) -> ?t:1</tt><br>
</blockquote>Incomplete elliptic integral P in Legendre form.</blockquote>
<a name="func_ell_int_RC"></a><h2><tt>ell_int_RC(x, y)</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1) -> ?t:1</tt><br>
</blockquote>Incomplete elliptic integral RC in Carlson form.</blockquote>
<a name="func_ell_int_RD"></a><h2><tt>ell_int_RD(x, y, z)</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1, ?t:1) -> ?t:1</tt><br>
</blockquote>Incomplete elliptic integral RD in Carlson form.</blockquote>
<a name="func_ell_int_RF"></a><h2><tt>ell_int_RF(x, y, z)</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1, ?t:1) -> ?t:1</tt><br>
</blockquote>Incomplete elliptic integral RF in Carlson form.</blockquote>
<a name="func_ell_int_RJ"></a><h2><tt>ell_int_RJ(x, y, z, p)</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1, ?t:1, ?t:1) -> ?t:1</tt><br>
</blockquote>Incomplete elliptic integral RJ in Carlson form.</blockquote>
<a name="func_ell_jac_cn"></a><h2><tt>ell_jac_cn(u, m)</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1) -> ?t:1</tt><br>
<tt>(ri:2, ?:1) -> ri:2</tt><br>
</blockquote>Jacobian elliptic function cn for real and complex arguments.</blockquote>
<a name="func_ell_jac_dn"></a><h2><tt>ell_jac_dn(u, m)</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1) -> ?t:1</tt><br>
<tt>(ri:2, ?:1) -> ri:2</tt><br>
</blockquote>Jacobian elliptic function dn for real and complex arguments.</blockquote>
<a name="func_ell_jac_sn"></a><h2><tt>ell_jac_sn(u, m)</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1) -> ?t:1</tt><br>
<tt>(ri:2, ?:1) -> ri:2</tt><br>
</blockquote>Jacobian elliptic function sn for real and complex arguments.</blockquote>
<a name="func_exp"></a><h2><tt>exp(a)</tt></h2>
<blockquote><blockquote><tt>(ri:2) -> ri:2</tt><br>
<tt>(?t:1) -> ?t:1</tt><br>
</blockquote>
The natural exponential function <b>e^x</b> for real and complex
numbers.</blockquote>
<a name="func_floor"></a><h2><tt>floor(a)</tt></h2>
<blockquote><blockquote><tt>(?t:1) -> ?t:1</tt><br>
</blockquote>
The floor of a number, defined as the largest integer not greater
than that number.</blockquote>
<a name="func_gamma"></a><h2><tt>gamma(a)</tt></h2>
<blockquote><blockquote><tt>(ri:2) -> ri:2</tt><br>
<tt>(?t:1) -> ?t:1</tt><br>
</blockquote>The (logarithm of the) gamma function for real and complex numbers.</blockquote>
<a name="func_gray"></a><h2><tt>gray(c)</tt></h2>
<blockquote><blockquote><tt>(rgba:4) -> nil:1</tt><br>
</blockquote>The luminance value of the color <tt>c</tt>.</blockquote>
<a name="func_grayColor"></a><h2><tt>grayColor(g)</tt></h2>
<blockquote><blockquote><tt>(?t:1) -> rgba:4</tt><br>
</blockquote>
Returns a fully opaque gray RGBA color with luminance <tt>g</tt>,
i.e. <tt>rgba:[g,g,g,1]</tt>.</blockquote>
<a name="func_grayaColor"></a><h2><tt>grayaColor(g, a)</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1) -> rgba:4</tt><br>
</blockquote>
Returns a gray RGBA color with luminance <tt>g</tt> and alpha
component <tt>a</tt>, i.e. <tt>rgba:[g,g,g,a]</tt>.</blockquote>
<a name="func_green"></a><h2><tt>green(c)</tt></h2>
<blockquote><blockquote><tt>(rgba:4) -> nil:1</tt><br>
</blockquote>The green component of the color <tt>c</tt>.</blockquote>
<a name="func_inintv"></a><h2><tt>inintv(a, l, u)</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1, ?t:1) -> ?t:1</tt><br>
</blockquote>
Returns 1 if <tt>a</tt> lies in the interval defined by the lower
bound <tt>l</tt> and the upper bound <tt>u</tt>, otherwise 0.</blockquote>
<a name="func_lerp"></a><h2><tt>lerp(p, a, b)</tt></h2>
<blockquote><blockquote><tt>(?:1, ?t:?l, ?t:?l) -> ?t:?l</tt><br>
<tt>(?t:?l, ?t:?l, ?t:?l) -> ?t:?l</tt><br>
</blockquote>
Linear interpolation between <tt>a</tt> and <tt>b</tt>, done
element-wise. The result is <tt>a</tt> if <tt>p</tt> is 0, <tt>b</tt>
if <tt>p</tt> is 1, and linearly interpolated in between. More
formally, the result is <tt>a*(1-t)+b*t</tt>.</blockquote>
<a name="func_log"></a><h2><tt>log(a)</tt></h2>
<blockquote><blockquote><tt>(ri:2) -> ri:2</tt><br>
<tt>(?t:1) -> ?t:1</tt><br>
</blockquote>The natural logarithm for real and complex numbers.</blockquote>
<a name="func_max"></a><h2><tt>max(a, b)</tt></h2>
<blockquote><blockquote><tt>(?t:?l, ?t:?l) -> ?t:?l</tt><br>
</blockquote>
The larger of two numbers. For tuples, the larger number for
each pair of elements is determined.</blockquote>
<a name="func_min"></a><h2><tt>min(a, b)</tt></h2>
<blockquote><blockquote><tt>(?t:?l, ?t:?l) -> ?t:?l</tt><br>
</blockquote>
The smaller of two numbers. For tuples, the smaller number for
each pair of elements is determined.</blockquote>
<a name="func_noise"></a><h2><tt>noise(a)</tt></h2>
<blockquote><blockquote><tt>(?:3) -> nil:1</tt><br>
</blockquote>
A solid noise function defined in three-dimensional space. Its
values lie between -1 and 1.</blockquote>
<a name="func_normalize"></a><h2><tt>normalize(a)</tt></h2>
<blockquote><blockquote><tt>(?t:?l) -> ?t:?l</tt><br>
</blockquote>Normalize a vector to Euclidian length 1.</blockquote>
<a name="func_pmod"></a><h2><tt>pmod(a, b)</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1) -> ?t:1</tt><br>
</blockquote>
The remainder of a division, made positive if the dividend is
negative by adding the divisor.</blockquote>
<a name="func_print"></a><h2><tt>print(val)</tt></h2>
<blockquote><blockquote><tt>(?:?) -> nil:1</tt><br>
</blockquote>Print a tuple to standard output. Useful for debugging a script.</blockquote>
<a name="func_rad2deg"></a><h2><tt>rad2deg(a)</tt></h2>
<blockquote><blockquote><tt>(?:1) -> deg:1</tt><br>
</blockquote>Convert radians to degrees.</blockquote>
<a name="func_rand"></a><h2><tt>rand(a, b)</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1) -> ?t:1</tt><br>
</blockquote>A random number between <tt>a</tt> and <tt>b</tt>.</blockquote>
<a name="func_red"></a><h2><tt>red(c)</tt></h2>
<blockquote><blockquote><tt>(rgba:4) -> nil:1</tt><br>
</blockquote>The red component of the color <tt>c</tt>.</blockquote>
<a name="func_rgbColor"></a><h2><tt>rgbColor(r, g, b)</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1, ?t:1) -> rgba:4</tt><br>
</blockquote>
Returns a fully opaque RGBA color with red component <tt>r</tt>,
green component <tt>g</tt> and blue component <tt>b</tt>,
i.e. <tt>rgba:[r,g,b,1]</tt>.</blockquote>
<a name="func_rgbaColor"></a><h2><tt>rgbaColor(r, g, b, a)</tt></h2>
<blockquote><blockquote><tt>(?t:1, ?t:1, ?t:1, ?t:1) -> rgba:4</tt><br>
</blockquote>
Returns an RGBA color with red component <tt>r</tt>, green
component <tt>g</tt>, blue component <tt>b</tt> and alpha component
<tt>a</tt>, i.e. <tt>rgba:[r,g,b,a]</tt>.</blockquote>
<a name="func_scale"></a><h2><tt>scale(a, fl, fu, tl, tu)</tt></h2>
<blockquote><blockquote><tt>(?t:?l, ?t:?l, ?t:?l, ?t:?l, ?t:?l) -> ?t:?l</tt><br>
</blockquote>
Scale each element of <tt>a</tt> which is supposed to lie between
the corresponding elements of <tt>fl</tt> and <tt>fu</tt> to lie at
the same point between <tt>tl</tt> and <tt>tu</tt>, proportionately.
More formally, computes <tt>((a-fl)/(fu-fl))*(tu-tl)+tl</tt>.</blockquote>
<a name="func_sign"></a><h2><tt>sign(a)</tt></h2>
<blockquote><blockquote><tt>(?t:?l) -> ?t:?l</tt><br>
</blockquote>
The sign of a number or tuple. The sign of a number is -1 if the
number is negative, 1 if the number is positive and 0 if the number is
0. For a tuple, calculates the sign element-wise.</blockquote>
<a name="func_sin"></a><h2><tt>sin(a)</tt></h2>
<blockquote><blockquote><tt>(ri:2) -> ri:2</tt><br>
<tt>(?t:1) -> ?t:1</tt><br>
</blockquote>Sine of real and complex numbers.</blockquote>
<a name="func_sinh"></a><h2><tt>sinh(a)</tt></h2>
<blockquote><blockquote><tt>(ri:2) -> ri:2</tt><br>
<tt>(?t:1) -> ?t:1</tt><br>
</blockquote>Hyperbolic sine of real and complex numbers.</blockquote>
<a name="func_sqrt"></a><h2><tt>sqrt(a)</tt></h2>
<blockquote><blockquote><tt>(ri:2) -> ri:2</tt><br>
<tt>(?t:1) -> ?t:1</tt><br>
</blockquote>
The square root of a complex or real number. A real argument must
be positive, otherwise the result will not be definied.</blockquote>
<a name="func_sum"></a><h2><tt>sum(a)</tt></h2>
<blockquote><blockquote><tt>(?t:?l) -> ?t:1</tt><br>
</blockquote>The sum of all elements of a tuple.</blockquote>
<a name="func_tan"></a><h2><tt>tan(a)</tt></h2>
<blockquote><blockquote><tt>(ri:2) -> ri:2</tt><br>
<tt>(?t:1) -> ?t:1</tt><br>
</blockquote>Tangent of real and complex numbers.</blockquote>
<a name="func_tanh"></a><h2><tt>tanh(a)</tt></h2>
<blockquote><blockquote><tt>(ri:2) -> ri:2</tt><br>
<tt>(?t:1) -> ?t:1</tt><br>
</blockquote>Hyperbolic tangent of real and complex numbers.</blockquote>
<a name="func_toHSVA"></a><h2><tt>toHSVA(a)</tt></h2>
<blockquote><blockquote><tt>(rgba:4) -> hsva:4</tt><br>
</blockquote>Conversion of an RGBA color value to HSVA.</blockquote>
<a name="func_toRA"></a><h2><tt>toRA(arg)</tt></h2>
<blockquote><blockquote><tt>(xy:2) -> ra:2</tt><br>
<tt>(ra:2) -> ra:2</tt><br>
</blockquote>Conversion of rectangular coordinates to polar coordinates.</blockquote>
<a name="func_toRGBA"></a><h2><tt>toRGBA(a)</tt></h2>
<blockquote><blockquote><tt>(hsva:4) -> rgba:4</tt><br>
</blockquote>Conversion of an HSVA color value to RGBA.</blockquote>
<a name="func_toXY"></a><h2><tt>toXY(a)</tt></h2>
<blockquote><blockquote><tt>(ra:2) -> xy:2</tt><br>
<tt>(xy:2) -> xy:2</tt><br>
</blockquote>Conversion of polar coordinates to rectangular coordinates.</blockquote>
</blockquote>
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