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+ <li class="md-nav__item">
+ <a href="ft2-mac_specific.html" title="Mac Specific Interface" class="md-nav__link">
+ Mac Specific Interface
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+ <li class="md-nav__item">
+ <a href="ft2-sizes_management.html" title="Size Management" class="md-nav__link">
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+ Format-Specific API
+ </label>
+ <nav class="md-nav" data-md-component="collapsible" data-md-level="1">
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+ Format-Specific API
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+ <ul class="md-nav__list" data-md-scrollfix>
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+ <a href="ft2-multiple_masters.html" title="Multiple Masters" class="md-nav__link">
+ Multiple Masters
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+ <a href="ft2-truetype_tables.html" title="TrueType Tables" class="md-nav__link">
+ TrueType Tables
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+ <li class="md-nav__item">
+ <a href="ft2-type1_tables.html" title="Type 1 Tables" class="md-nav__link">
+ Type 1 Tables
+ </a>
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+ <li class="md-nav__item">
+ <a href="ft2-sfnt_names.html" title="SFNT Names" class="md-nav__link">
+ SFNT Names
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+ <label class="md-nav__link" for="nav-6">
+ Controlling FreeType Modules
+ </label>
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+ Controlling FreeType Modules
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+ <a href="ft2-auto_hinter.html" title="The auto-hinter" class="md-nav__link">
+ The auto-hinter
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+ <a href="ft2-cff_driver.html" title="The CFF driver" class="md-nav__link">
+ The CFF driver
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+ <a href="ft2-t1_cid_driver.html" title="The Type 1 and CID drivers" class="md-nav__link">
+ The Type 1 and CID drivers
+ </a>
+ </li>
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+ <li class="md-nav__item">
+ <a href="ft2-tt_driver.html" title="The TrueType driver" class="md-nav__link">
+ The TrueType driver
+ </a>
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+ <li class="md-nav__item">
+ <a href="ft2-pcf_driver.html" title="The PCF driver" class="md-nav__link">
+ The PCF driver
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+ Driver properties
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+ Parameter Tags
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+ <a href="ft2-lcd_rendering.html" title="Subpixel Rendering" class="md-nav__link">
+ Subpixel Rendering
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+ Cache Sub-System
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+ <a href="ft2-cache_subsystem.html" title="Cache Sub-System" class="md-nav__link">
+ Cache Sub-System
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+ Support API
+ </label>
+ <nav class="md-nav" data-md-component="collapsible" data-md-level="1">
+ <label class="md-nav__title" for="nav-8">
+ Support API
+ </label>
+ <ul class="md-nav__list" data-md-scrollfix>
+
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+ <li class="md-nav__item md-nav__item--active">
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+
+
+
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+ <label class="md-nav__link md-nav__link--active" for="__toc">
+ Computations
+ </label>
+
+ <a href="ft2-computations.html" title="Computations" class="md-nav__link md-nav__link--active">
+ Computations
+ </a>
+
+
+<nav class="md-nav md-nav--secondary">
+
+
+
+
+
+ <label class="md-nav__title" for="__toc">Table of contents</label>
+ <ul class="md-nav__list" data-md-scrollfix>
+
+ <li class="md-nav__item">
+ <a href="#synopsis" class="md-nav__link">
+ Synopsis
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_muldiv" class="md-nav__link">
+ FT_MulDiv
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_mulfix" class="md-nav__link">
+ FT_MulFix
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_divfix" class="md-nav__link">
+ FT_DivFix
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_roundfix" class="md-nav__link">
+ FT_RoundFix
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_ceilfix" class="md-nav__link">
+ FT_CeilFix
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_floorfix" class="md-nav__link">
+ FT_FloorFix
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_vector_transform" class="md-nav__link">
+ FT_Vector_Transform
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_matrix_multiply" class="md-nav__link">
+ FT_Matrix_Multiply
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_matrix_invert" class="md-nav__link">
+ FT_Matrix_Invert
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_angle" class="md-nav__link">
+ FT_Angle
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_angle_pi" class="md-nav__link">
+ FT_ANGLE_PI
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_angle_2pi" class="md-nav__link">
+ FT_ANGLE_2PI
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_angle_pi2" class="md-nav__link">
+ FT_ANGLE_PI2
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_angle_pi4" class="md-nav__link">
+ FT_ANGLE_PI4
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_sin" class="md-nav__link">
+ FT_Sin
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_cos" class="md-nav__link">
+ FT_Cos
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_tan" class="md-nav__link">
+ FT_Tan
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_atan2" class="md-nav__link">
+ FT_Atan2
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_angle_diff" class="md-nav__link">
+ FT_Angle_Diff
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_vector_unit" class="md-nav__link">
+ FT_Vector_Unit
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_vector_rotate" class="md-nav__link">
+ FT_Vector_Rotate
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_vector_length" class="md-nav__link">
+ FT_Vector_Length
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_vector_polarize" class="md-nav__link">
+ FT_Vector_Polarize
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_vector_from_polar" class="md-nav__link">
+ FT_Vector_From_Polar
+ </a>
+
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+ <a href="ft2-list_processing.html" title="List Processing" class="md-nav__link">
+ List Processing
+ </a>
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+ <li class="md-nav__item">
+ <a href="ft2-outline_processing.html" title="Outline Processing" class="md-nav__link">
+ Outline Processing
+ </a>
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+
+ <li class="md-nav__item">
+ <a href="ft2-quick_advance.html" title="Quick retrieval of advance values" class="md-nav__link">
+ Quick retrieval of advance values
+ </a>
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+ <li class="md-nav__item">
+ <a href="ft2-bitmap_handling.html" title="Bitmap Handling" class="md-nav__link">
+ Bitmap Handling
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+ <li class="md-nav__item">
+ <a href="ft2-raster.html" title="Scanline Converter" class="md-nav__link">
+ Scanline Converter
+ </a>
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+ <li class="md-nav__item">
+ <a href="ft2-glyph_stroker.html" title="Glyph Stroker" class="md-nav__link">
+ Glyph Stroker
+ </a>
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+ <li class="md-nav__item">
+ <a href="ft2-system_interface.html" title="System Interface" class="md-nav__link">
+ System Interface
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+ Module Management
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+ <li class="md-nav__item">
+ <a href="ft2-gzip.html" title="GZIP Streams" class="md-nav__link">
+ GZIP Streams
+ </a>
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+
+
+ <li class="md-nav__item">
+ <a href="ft2-lzw.html" title="LZW Streams" class="md-nav__link">
+ LZW Streams
+ </a>
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+ Error Codes
+ </label>
+ <nav class="md-nav" data-md-component="collapsible" data-md-level="1">
+ <label class="md-nav__title" for="nav-9">
+ Error Codes
+ </label>
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+
+
+
+
+
+
+
+ <li class="md-nav__item">
+ <a href="ft2-error_enumerations.html" title="Error Enumerations" class="md-nav__link">
+ Error Enumerations
+ </a>
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+
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+
+
+
+
+ <li class="md-nav__item">
+ <a href="ft2-error_code_values.html" title="Error Code Values" class="md-nav__link">
+ Error Code Values
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+
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+
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+ <li class="md-nav__item">
+ <a href="ft2-gx_validation.html" title="TrueTypeGX/AAT Validation" class="md-nav__link">
+ TrueTypeGX/AAT Validation
+ </a>
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+
+
+
+
+
+
+
+ <li class="md-nav__item">
+ <a href="ft2-incremental.html" title="Incremental Loading" class="md-nav__link">
+ Incremental Loading
+ </a>
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+
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+
+
+
+
+
+ <li class="md-nav__item">
+ <a href="ft2-truetype_engine.html" title="The TrueType Engine" class="md-nav__link">
+ The TrueType Engine
+ </a>
+ </li>
+
+
+
+
+
+
+
+ <li class="md-nav__item">
+ <a href="ft2-ot_validation.html" title="OpenType Validation" class="md-nav__link">
+ OpenType Validation
+ </a>
+ </li>
+
+
+ </ul>
+ </nav>
+ </li>
+
+
+ </ul>
+</nav>
+ </div>
+ </div>
+ </div>
+
+
+ <div class="md-sidebar md-sidebar--secondary" data-md-component="toc">
+ <div class="md-sidebar__scrollwrap">
+ <div class="md-sidebar__inner">
+
+<nav class="md-nav md-nav--secondary">
+
+
+
+
+
+ <label class="md-nav__title" for="__toc">Table of contents</label>
+ <ul class="md-nav__list" data-md-scrollfix>
+
+ <li class="md-nav__item">
+ <a href="#synopsis" class="md-nav__link">
+ Synopsis
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_muldiv" class="md-nav__link">
+ FT_MulDiv
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_mulfix" class="md-nav__link">
+ FT_MulFix
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_divfix" class="md-nav__link">
+ FT_DivFix
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_roundfix" class="md-nav__link">
+ FT_RoundFix
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_ceilfix" class="md-nav__link">
+ FT_CeilFix
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_floorfix" class="md-nav__link">
+ FT_FloorFix
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_vector_transform" class="md-nav__link">
+ FT_Vector_Transform
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_matrix_multiply" class="md-nav__link">
+ FT_Matrix_Multiply
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_matrix_invert" class="md-nav__link">
+ FT_Matrix_Invert
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_angle" class="md-nav__link">
+ FT_Angle
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_angle_pi" class="md-nav__link">
+ FT_ANGLE_PI
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_angle_2pi" class="md-nav__link">
+ FT_ANGLE_2PI
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_angle_pi2" class="md-nav__link">
+ FT_ANGLE_PI2
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_angle_pi4" class="md-nav__link">
+ FT_ANGLE_PI4
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_sin" class="md-nav__link">
+ FT_Sin
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_cos" class="md-nav__link">
+ FT_Cos
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_tan" class="md-nav__link">
+ FT_Tan
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_atan2" class="md-nav__link">
+ FT_Atan2
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_angle_diff" class="md-nav__link">
+ FT_Angle_Diff
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_vector_unit" class="md-nav__link">
+ FT_Vector_Unit
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_vector_rotate" class="md-nav__link">
+ FT_Vector_Rotate
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_vector_length" class="md-nav__link">
+ FT_Vector_Length
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_vector_polarize" class="md-nav__link">
+ FT_Vector_Polarize
+ </a>
+
+</li>
+
+ <li class="md-nav__item">
+ <a href="#ft_vector_from_polar" class="md-nav__link">
+ FT_Vector_From_Polar
+ </a>
+
+</li>
+
+
+
+
+
+ </ul>
+
+</nav>
+ </div>
+ </div>
+ </div>
+
+
+ <div class="md-content">
+ <article class="md-content__inner md-typeset">
+
+
+
+ <p><a href="https://www.freetype.org">FreeType</a> &raquo; <a href="../">Docs</a> &raquo; <a href="index.html#support-api">Support API</a> &raquo; Computations</p>
+<hr />
+<h1 id="computations">Computations<a class="headerlink" href="#computations" title="Permanent link">&para;</a></h1>
+<h2 id="synopsis">Synopsis<a class="headerlink" href="#synopsis" title="Permanent link">&para;</a></h2>
+<p>This section contains various functions used to perform computations on 16.16 fixed-float numbers or 2d vectors.</p>
+<p><strong>Attention</strong>: Most arithmetic functions take <code>FT_Long</code> as arguments. For historical reasons, FreeType was designed under the assumption that <code>FT_Long</code> is a 32-bit integer; results can thus be undefined if the arguments don't fit into 32 bits.</p>
+<h2 id="ft_muldiv">FT_MulDiv<a class="headerlink" href="#ft_muldiv" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_FREETYPE_H (freetype/freetype.h).</p>
+<div class = "codehilite"><pre><code> FT_EXPORT( <a href="ft2-basic_types.html#ft_long">FT_Long</a> )
+ <b>FT_MulDiv</b>( <a href="ft2-basic_types.html#ft_long">FT_Long</a> a,
+ <a href="ft2-basic_types.html#ft_long">FT_Long</a> b,
+ <a href="ft2-basic_types.html#ft_long">FT_Long</a> c );
+</code></pre></div>
+
+<p>Compute <code>(a*b)/c</code> with maximum accuracy, using a 64-bit intermediate integer whenever necessary.</p>
+<p>This function isn't necessarily as fast as some processor-specific operations, but is at least completely portable.</p>
+<h4>input</h4>
+
+<table class="fields">
+<tr><td class="val" id="a">a</td><td class="desc">
+<p>The first multiplier.</p>
+</td></tr>
+<tr><td class="val" id="b">b</td><td class="desc">
+<p>The second multiplier.</p>
+</td></tr>
+<tr><td class="val" id="c">c</td><td class="desc">
+<p>The divisor.</p>
+</td></tr>
+</table>
+
+<h4>return</h4>
+
+<p>The result of <code>(a*b)/c</code>. This function never traps when trying to divide by zero; it simply returns &lsquo;MaxInt&rsquo; or &lsquo;MinInt&rsquo; depending on the signs of <code>a</code> and <code>b</code>.</p>
+<hr>
+
+<h2 id="ft_mulfix">FT_MulFix<a class="headerlink" href="#ft_mulfix" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_FREETYPE_H (freetype/freetype.h).</p>
+<div class = "codehilite"><pre><code> FT_EXPORT( <a href="ft2-basic_types.html#ft_long">FT_Long</a> )
+ <b>FT_MulFix</b>( <a href="ft2-basic_types.html#ft_long">FT_Long</a> a,
+ <a href="ft2-basic_types.html#ft_long">FT_Long</a> b );
+</code></pre></div>
+
+<p>Compute <code>(a*b)/0x10000</code> with maximum accuracy. Its main use is to multiply a given value by a 16.16 fixed-point factor.</p>
+<h4>input</h4>
+
+<table class="fields">
+<tr><td class="val" id="a">a</td><td class="desc">
+<p>The first multiplier.</p>
+</td></tr>
+<tr><td class="val" id="b">b</td><td class="desc">
+<p>The second multiplier. Use a 16.16 factor here whenever possible (see note below).</p>
+</td></tr>
+</table>
+
+<h4>return</h4>
+
+<p>The result of <code>(a*b)/0x10000</code>.</p>
+<h4>note</h4>
+
+<p>This function has been optimized for the case where the absolute value of <code>a</code> is less than 2048, and <code>b</code> is a 16.16 scaling factor. As this happens mainly when scaling from notional units to fractional pixels in FreeType, it resulted in noticeable speed improvements between versions 2.x and 1.x.</p>
+<p>As a conclusion, always try to place a 16.16 factor as the <em>second</em> argument of this function; this can make a great difference.</p>
+<hr>
+
+<h2 id="ft_divfix">FT_DivFix<a class="headerlink" href="#ft_divfix" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_FREETYPE_H (freetype/freetype.h).</p>
+<div class = "codehilite"><pre><code> FT_EXPORT( <a href="ft2-basic_types.html#ft_long">FT_Long</a> )
+ <b>FT_DivFix</b>( <a href="ft2-basic_types.html#ft_long">FT_Long</a> a,
+ <a href="ft2-basic_types.html#ft_long">FT_Long</a> b );
+</code></pre></div>
+
+<p>Compute <code>(a*0x10000)/b</code> with maximum accuracy. Its main use is to divide a given value by a 16.16 fixed-point factor.</p>
+<h4>input</h4>
+
+<table class="fields">
+<tr><td class="val" id="a">a</td><td class="desc">
+<p>The numerator.</p>
+</td></tr>
+<tr><td class="val" id="b">b</td><td class="desc">
+<p>The denominator. Use a 16.16 factor here.</p>
+</td></tr>
+</table>
+
+<h4>return</h4>
+
+<p>The result of <code>(a*0x10000)/b</code>.</p>
+<hr>
+
+<h2 id="ft_roundfix">FT_RoundFix<a class="headerlink" href="#ft_roundfix" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_FREETYPE_H (freetype/freetype.h).</p>
+<div class = "codehilite"><pre><code> FT_EXPORT( <a href="ft2-basic_types.html#ft_fixed">FT_Fixed</a> )
+ <b>FT_RoundFix</b>( <a href="ft2-basic_types.html#ft_fixed">FT_Fixed</a> a );
+</code></pre></div>
+
+<p>Round a 16.16 fixed number.</p>
+<h4>input</h4>
+
+<table class="fields">
+<tr><td class="val" id="a">a</td><td class="desc">
+<p>The number to be rounded.</p>
+</td></tr>
+</table>
+
+<h4>return</h4>
+
+<p><code>a</code> rounded to the nearest 16.16 fixed integer, halfway cases away from zero.</p>
+<h4>note</h4>
+
+<p>The function uses wrap-around arithmetic.</p>
+<hr>
+
+<h2 id="ft_ceilfix">FT_CeilFix<a class="headerlink" href="#ft_ceilfix" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_FREETYPE_H (freetype/freetype.h).</p>
+<div class = "codehilite"><pre><code> FT_EXPORT( <a href="ft2-basic_types.html#ft_fixed">FT_Fixed</a> )
+ <b>FT_CeilFix</b>( <a href="ft2-basic_types.html#ft_fixed">FT_Fixed</a> a );
+</code></pre></div>
+
+<p>Compute the smallest following integer of a 16.16 fixed number.</p>
+<h4>input</h4>
+
+<table class="fields">
+<tr><td class="val" id="a">a</td><td class="desc">
+<p>The number for which the ceiling function is to be computed.</p>
+</td></tr>
+</table>
+
+<h4>return</h4>
+
+<p><code>a</code> rounded towards plus infinity.</p>
+<h4>note</h4>
+
+<p>The function uses wrap-around arithmetic.</p>
+<hr>
+
+<h2 id="ft_floorfix">FT_FloorFix<a class="headerlink" href="#ft_floorfix" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_FREETYPE_H (freetype/freetype.h).</p>
+<div class = "codehilite"><pre><code> FT_EXPORT( <a href="ft2-basic_types.html#ft_fixed">FT_Fixed</a> )
+ <b>FT_FloorFix</b>( <a href="ft2-basic_types.html#ft_fixed">FT_Fixed</a> a );
+</code></pre></div>
+
+<p>Compute the largest previous integer of a 16.16 fixed number.</p>
+<h4>input</h4>
+
+<table class="fields">
+<tr><td class="val" id="a">a</td><td class="desc">
+<p>The number for which the floor function is to be computed.</p>
+</td></tr>
+</table>
+
+<h4>return</h4>
+
+<p><code>a</code> rounded towards minus infinity.</p>
+<hr>
+
+<h2 id="ft_vector_transform">FT_Vector_Transform<a class="headerlink" href="#ft_vector_transform" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_FREETYPE_H (freetype/freetype.h).</p>
+<div class = "codehilite"><pre><code> FT_EXPORT( <span class="keyword">void</span> )
+ <b>FT_Vector_Transform</b>( <a href="ft2-basic_types.html#ft_vector">FT_Vector</a>* vector,
+ <span class="keyword">const</span> <a href="ft2-basic_types.html#ft_matrix">FT_Matrix</a>* matrix );
+</code></pre></div>
+
+<p>Transform a single vector through a 2x2 matrix.</p>
+<h4>inout</h4>
+
+<table class="fields">
+<tr><td class="val" id="vector">vector</td><td class="desc">
+<p>The target vector to transform.</p>
+</td></tr>
+</table>
+
+<h4>input</h4>
+
+<table class="fields">
+<tr><td class="val" id="matrix">matrix</td><td class="desc">
+<p>A pointer to the source 2x2 matrix.</p>
+</td></tr>
+</table>
+
+<h4>note</h4>
+
+<p>The result is undefined if either <code>vector</code> or <code>matrix</code> is invalid.</p>
+<hr>
+
+<h2 id="ft_matrix_multiply">FT_Matrix_Multiply<a class="headerlink" href="#ft_matrix_multiply" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_GLYPH_H (freetype/ftglyph.h).</p>
+<div class = "codehilite"><pre><code> FT_EXPORT( <span class="keyword">void</span> )
+ <b>FT_Matrix_Multiply</b>( <span class="keyword">const</span> <a href="ft2-basic_types.html#ft_matrix">FT_Matrix</a>* a,
+ <a href="ft2-basic_types.html#ft_matrix">FT_Matrix</a>* b );
+</code></pre></div>
+
+<p>Perform the matrix operation <code>b = a*b</code>.</p>
+<h4>input</h4>
+
+<table class="fields">
+<tr><td class="val" id="a">a</td><td class="desc">
+<p>A pointer to matrix <code>a</code>.</p>
+</td></tr>
+</table>
+
+<h4>inout</h4>
+
+<table class="fields">
+<tr><td class="val" id="b">b</td><td class="desc">
+<p>A pointer to matrix <code>b</code>.</p>
+</td></tr>
+</table>
+
+<h4>note</h4>
+
+<p>The result is undefined if either <code>a</code> or <code>b</code> is zero.</p>
+<p>Since the function uses wrap-around arithmetic, results become meaningless if the arguments are very large.</p>
+<hr>
+
+<h2 id="ft_matrix_invert">FT_Matrix_Invert<a class="headerlink" href="#ft_matrix_invert" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_GLYPH_H (freetype/ftglyph.h).</p>
+<div class = "codehilite"><pre><code> FT_EXPORT( <a href="ft2-basic_types.html#ft_error">FT_Error</a> )
+ <b>FT_Matrix_Invert</b>( <a href="ft2-basic_types.html#ft_matrix">FT_Matrix</a>* matrix );
+</code></pre></div>
+
+<p>Invert a 2x2 matrix. Return an error if it can't be inverted.</p>
+<h4>inout</h4>
+
+<table class="fields">
+<tr><td class="val" id="matrix">matrix</td><td class="desc">
+<p>A pointer to the target matrix. Remains untouched in case of error.</p>
+</td></tr>
+</table>
+
+<h4>return</h4>
+
+<p>FreeType error code. 0&nbsp;means success.</p>
+<hr>
+
+<h2 id="ft_angle">FT_Angle<a class="headerlink" href="#ft_angle" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).</p>
+<div class = "codehilite"><pre><code> <span class="keyword">typedef</span> <a href="ft2-basic_types.html#ft_fixed">FT_Fixed</a> <b>FT_Angle</b>;
+</code></pre></div>
+
+<p>This type is used to model angle values in FreeType. Note that the angle is a 16.16 fixed-point value expressed in degrees.</p>
+<hr>
+
+<h2 id="ft_angle_pi">FT_ANGLE_PI<a class="headerlink" href="#ft_angle_pi" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).</p>
+<div class = "codehilite"><pre><code>#<span class="keyword">define</span> <b>FT_ANGLE_PI</b> ( 180L &lt;&lt; 16 )
+</code></pre></div>
+
+<p>The angle pi expressed in <code><a href="ft2-computations.html#ft_angle">FT_Angle</a></code> units.</p>
+<hr>
+
+<h2 id="ft_angle_2pi">FT_ANGLE_2PI<a class="headerlink" href="#ft_angle_2pi" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).</p>
+<div class = "codehilite"><pre><code>#<span class="keyword">define</span> <b>FT_ANGLE_2PI</b> ( <a href="ft2-computations.html#ft_angle_pi">FT_ANGLE_PI</a> * 2 )
+</code></pre></div>
+
+<p>The angle 2*pi expressed in <code><a href="ft2-computations.html#ft_angle">FT_Angle</a></code> units.</p>
+<hr>
+
+<h2 id="ft_angle_pi2">FT_ANGLE_PI2<a class="headerlink" href="#ft_angle_pi2" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).</p>
+<div class = "codehilite"><pre><code>#<span class="keyword">define</span> <b>FT_ANGLE_PI2</b> ( <a href="ft2-computations.html#ft_angle_pi">FT_ANGLE_PI</a> / 2 )
+</code></pre></div>
+
+<p>The angle pi/2 expressed in <code><a href="ft2-computations.html#ft_angle">FT_Angle</a></code> units.</p>
+<hr>
+
+<h2 id="ft_angle_pi4">FT_ANGLE_PI4<a class="headerlink" href="#ft_angle_pi4" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).</p>
+<div class = "codehilite"><pre><code>#<span class="keyword">define</span> <b>FT_ANGLE_PI4</b> ( <a href="ft2-computations.html#ft_angle_pi">FT_ANGLE_PI</a> / 4 )
+</code></pre></div>
+
+<p>The angle pi/4 expressed in <code><a href="ft2-computations.html#ft_angle">FT_Angle</a></code> units.</p>
+<hr>
+
+<h2 id="ft_sin">FT_Sin<a class="headerlink" href="#ft_sin" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).</p>
+<div class = "codehilite"><pre><code> FT_EXPORT( <a href="ft2-basic_types.html#ft_fixed">FT_Fixed</a> )
+ <b>FT_Sin</b>( <a href="ft2-computations.html#ft_angle">FT_Angle</a> angle );
+</code></pre></div>
+
+<p>Return the sinus of a given angle in fixed-point format.</p>
+<h4>input</h4>
+
+<table class="fields">
+<tr><td class="val" id="angle">angle</td><td class="desc">
+<p>The input angle.</p>
+</td></tr>
+</table>
+
+<h4>return</h4>
+
+<p>The sinus value.</p>
+<h4>note</h4>
+
+<p>If you need both the sinus and cosinus for a given angle, use the function <code><a href="ft2-computations.html#ft_vector_unit">FT_Vector_Unit</a></code>.</p>
+<hr>
+
+<h2 id="ft_cos">FT_Cos<a class="headerlink" href="#ft_cos" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).</p>
+<div class = "codehilite"><pre><code> FT_EXPORT( <a href="ft2-basic_types.html#ft_fixed">FT_Fixed</a> )
+ <b>FT_Cos</b>( <a href="ft2-computations.html#ft_angle">FT_Angle</a> angle );
+</code></pre></div>
+
+<p>Return the cosinus of a given angle in fixed-point format.</p>
+<h4>input</h4>
+
+<table class="fields">
+<tr><td class="val" id="angle">angle</td><td class="desc">
+<p>The input angle.</p>
+</td></tr>
+</table>
+
+<h4>return</h4>
+
+<p>The cosinus value.</p>
+<h4>note</h4>
+
+<p>If you need both the sinus and cosinus for a given angle, use the function <code><a href="ft2-computations.html#ft_vector_unit">FT_Vector_Unit</a></code>.</p>
+<hr>
+
+<h2 id="ft_tan">FT_Tan<a class="headerlink" href="#ft_tan" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).</p>
+<div class = "codehilite"><pre><code> FT_EXPORT( <a href="ft2-basic_types.html#ft_fixed">FT_Fixed</a> )
+ <b>FT_Tan</b>( <a href="ft2-computations.html#ft_angle">FT_Angle</a> angle );
+</code></pre></div>
+
+<p>Return the tangent of a given angle in fixed-point format.</p>
+<h4>input</h4>
+
+<table class="fields">
+<tr><td class="val" id="angle">angle</td><td class="desc">
+<p>The input angle.</p>
+</td></tr>
+</table>
+
+<h4>return</h4>
+
+<p>The tangent value.</p>
+<hr>
+
+<h2 id="ft_atan2">FT_Atan2<a class="headerlink" href="#ft_atan2" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).</p>
+<div class = "codehilite"><pre><code> FT_EXPORT( <a href="ft2-computations.html#ft_angle">FT_Angle</a> )
+ <b>FT_Atan2</b>( <a href="ft2-basic_types.html#ft_fixed">FT_Fixed</a> x,
+ <a href="ft2-basic_types.html#ft_fixed">FT_Fixed</a> y );
+</code></pre></div>
+
+<p>Return the arc-tangent corresponding to a given vector (x,y) in the 2d plane.</p>
+<h4>input</h4>
+
+<table class="fields">
+<tr><td class="val" id="x">x</td><td class="desc">
+<p>The horizontal vector coordinate.</p>
+</td></tr>
+<tr><td class="val" id="y">y</td><td class="desc">
+<p>The vertical vector coordinate.</p>
+</td></tr>
+</table>
+
+<h4>return</h4>
+
+<p>The arc-tangent value (i.e. angle).</p>
+<hr>
+
+<h2 id="ft_angle_diff">FT_Angle_Diff<a class="headerlink" href="#ft_angle_diff" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).</p>
+<div class = "codehilite"><pre><code> FT_EXPORT( <a href="ft2-computations.html#ft_angle">FT_Angle</a> )
+ <b>FT_Angle_Diff</b>( <a href="ft2-computations.html#ft_angle">FT_Angle</a> angle1,
+ <a href="ft2-computations.html#ft_angle">FT_Angle</a> angle2 );
+</code></pre></div>
+
+<p>Return the difference between two angles. The result is always constrained to the ]-PI..PI] interval.</p>
+<h4>input</h4>
+
+<table class="fields">
+<tr><td class="val" id="angle1">angle1</td><td class="desc">
+<p>First angle.</p>
+</td></tr>
+<tr><td class="val" id="angle2">angle2</td><td class="desc">
+<p>Second angle.</p>
+</td></tr>
+</table>
+
+<h4>return</h4>
+
+<p>Constrained value of <code>angle2-angle1</code>.</p>
+<hr>
+
+<h2 id="ft_vector_unit">FT_Vector_Unit<a class="headerlink" href="#ft_vector_unit" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).</p>
+<div class = "codehilite"><pre><code> FT_EXPORT( <span class="keyword">void</span> )
+ <b>FT_Vector_Unit</b>( <a href="ft2-basic_types.html#ft_vector">FT_Vector</a>* vec,
+ <a href="ft2-computations.html#ft_angle">FT_Angle</a> angle );
+</code></pre></div>
+
+<p>Return the unit vector corresponding to a given angle. After the call, the value of <code>vec.x</code> will be <code>cos(angle)</code>, and the value of <code>vec.y</code> will be <code>sin(angle)</code>.</p>
+<p>This function is useful to retrieve both the sinus and cosinus of a given angle quickly.</p>
+<h4>output</h4>
+
+<table class="fields">
+<tr><td class="val" id="vec">vec</td><td class="desc">
+<p>The address of target vector.</p>
+</td></tr>
+</table>
+
+<h4>input</h4>
+
+<table class="fields">
+<tr><td class="val" id="angle">angle</td><td class="desc">
+<p>The input angle.</p>
+</td></tr>
+</table>
+
+<hr>
+
+<h2 id="ft_vector_rotate">FT_Vector_Rotate<a class="headerlink" href="#ft_vector_rotate" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).</p>
+<div class = "codehilite"><pre><code> FT_EXPORT( <span class="keyword">void</span> )
+ <b>FT_Vector_Rotate</b>( <a href="ft2-basic_types.html#ft_vector">FT_Vector</a>* vec,
+ <a href="ft2-computations.html#ft_angle">FT_Angle</a> angle );
+</code></pre></div>
+
+<p>Rotate a vector by a given angle.</p>
+<h4>inout</h4>
+
+<table class="fields">
+<tr><td class="val" id="vec">vec</td><td class="desc">
+<p>The address of target vector.</p>
+</td></tr>
+</table>
+
+<h4>input</h4>
+
+<table class="fields">
+<tr><td class="val" id="angle">angle</td><td class="desc">
+<p>The input angle.</p>
+</td></tr>
+</table>
+
+<hr>
+
+<h2 id="ft_vector_length">FT_Vector_Length<a class="headerlink" href="#ft_vector_length" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).</p>
+<div class = "codehilite"><pre><code> FT_EXPORT( <a href="ft2-basic_types.html#ft_fixed">FT_Fixed</a> )
+ <b>FT_Vector_Length</b>( <a href="ft2-basic_types.html#ft_vector">FT_Vector</a>* vec );
+</code></pre></div>
+
+<p>Return the length of a given vector.</p>
+<h4>input</h4>
+
+<table class="fields">
+<tr><td class="val" id="vec">vec</td><td class="desc">
+<p>The address of target vector.</p>
+</td></tr>
+</table>
+
+<h4>return</h4>
+
+<p>The vector length, expressed in the same units that the original vector coordinates.</p>
+<hr>
+
+<h2 id="ft_vector_polarize">FT_Vector_Polarize<a class="headerlink" href="#ft_vector_polarize" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).</p>
+<div class = "codehilite"><pre><code> FT_EXPORT( <span class="keyword">void</span> )
+ <b>FT_Vector_Polarize</b>( <a href="ft2-basic_types.html#ft_vector">FT_Vector</a>* vec,
+ <a href="ft2-basic_types.html#ft_fixed">FT_Fixed</a> *length,
+ <a href="ft2-computations.html#ft_angle">FT_Angle</a> *angle );
+</code></pre></div>
+
+<p>Compute both the length and angle of a given vector.</p>
+<h4>input</h4>
+
+<table class="fields">
+<tr><td class="val" id="vec">vec</td><td class="desc">
+<p>The address of source vector.</p>
+</td></tr>
+</table>
+
+<h4>output</h4>
+
+<table class="fields">
+<tr><td class="val" id="length">length</td><td class="desc">
+<p>The vector length.</p>
+</td></tr>
+<tr><td class="val" id="angle">angle</td><td class="desc">
+<p>The vector angle.</p>
+</td></tr>
+</table>
+
+<hr>
+
+<h2 id="ft_vector_from_polar">FT_Vector_From_Polar<a class="headerlink" href="#ft_vector_from_polar" title="Permanent link">&para;</a></h2>
+<p>Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).</p>
+<div class = "codehilite"><pre><code> FT_EXPORT( <span class="keyword">void</span> )
+ <b>FT_Vector_From_Polar</b>( <a href="ft2-basic_types.html#ft_vector">FT_Vector</a>* vec,
+ <a href="ft2-basic_types.html#ft_fixed">FT_Fixed</a> length,
+ <a href="ft2-computations.html#ft_angle">FT_Angle</a> angle );
+</code></pre></div>
+
+<p>Compute vector coordinates from a length and angle.</p>
+<h4>output</h4>
+
+<table class="fields">
+<tr><td class="val" id="vec">vec</td><td class="desc">
+<p>The address of source vector.</p>
+</td></tr>
+</table>
+
+<h4>input</h4>
+
+<table class="fields">
+<tr><td class="val" id="length">length</td><td class="desc">
+<p>The vector length.</p>
+</td></tr>
+<tr><td class="val" id="angle">angle</td><td class="desc">
+<p>The vector angle.</p>
+</td></tr>
+</table>
+
+<hr>
+
+
+
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