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  6. <title>FreeType-2.5.0 API Reference</title>
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  34.  
  35. <table align=center><tr><td><font size=-1>[<a href="ft2-index.html">Index</a>]</font></td>
  36. <td width="100%"></td>
  37. <td><font size=-1>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  38. <center><h1>FreeType-2.5.0 API Reference</h1></center>
  39.  
  40. <center><h1>
  41. Computations
  42. </h1></center>
  43. <h2>Synopsis</h2>
  44. <table align=center cellspacing=5 cellpadding=0 border=0>
  45. <tr><td></td><td><a href="#FT_MulDiv">FT_MulDiv</a></td><td></td><td><a href="#FT_Matrix_Invert">FT_Matrix_Invert</a></td><td></td><td><a href="#FT_Tan">FT_Tan</a></td></tr>
  46. <tr><td></td><td><a href="#FT_MulFix">FT_MulFix</a></td><td></td><td><a href="#FT_Angle">FT_Angle</a></td><td></td><td><a href="#FT_Atan2">FT_Atan2</a></td></tr>
  47. <tr><td></td><td><a href="#FT_DivFix">FT_DivFix</a></td><td></td><td><a href="#FT_ANGLE_PI">FT_ANGLE_PI</a></td><td></td><td><a href="#FT_Angle_Diff">FT_Angle_Diff</a></td></tr>
  48. <tr><td></td><td><a href="#FT_RoundFix">FT_RoundFix</a></td><td></td><td><a href="#FT_ANGLE_2PI">FT_ANGLE_2PI</a></td><td></td><td><a href="#FT_Vector_Unit">FT_Vector_Unit</a></td></tr>
  49. <tr><td></td><td><a href="#FT_CeilFix">FT_CeilFix</a></td><td></td><td><a href="#FT_ANGLE_PI2">FT_ANGLE_PI2</a></td><td></td><td><a href="#FT_Vector_Rotate">FT_Vector_Rotate</a></td></tr>
  50. <tr><td></td><td><a href="#FT_FloorFix">FT_FloorFix</a></td><td></td><td><a href="#FT_ANGLE_PI4">FT_ANGLE_PI4</a></td><td></td><td><a href="#FT_Vector_Length">FT_Vector_Length</a></td></tr>
  51. <tr><td></td><td><a href="#FT_Vector_Transform">FT_Vector_Transform</a></td><td></td><td><a href="#FT_Sin">FT_Sin</a></td><td></td><td><a href="#FT_Vector_Polarize">FT_Vector_Polarize</a></td></tr>
  52. <tr><td></td><td><a href="#FT_Matrix_Multiply">FT_Matrix_Multiply</a></td><td></td><td><a href="#FT_Cos">FT_Cos</a></td><td></td><td><a href="#FT_Vector_From_Polar">FT_Vector_From_Polar</a></td></tr>
  53. </table><br><br>
  54.  
  55. <table align=center width="87%"><tr><td>
  56. <p>This section contains various functions used to perform computations on 16.16 fixed-float numbers or 2d vectors.</p>
  57. </td></tr></table><br>
  58. <table align=center width="75%"><tr><td>
  59. <h4><a name="FT_MulDiv">FT_MulDiv</a></h4>
  60. <table align=center width="87%"><tr><td>
  61. Defined in FT_FREETYPE_H (freetype/freetype.h).
  62. </td></tr></table><br>
  63. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  64.  
  65.   FT_EXPORT( <a href="ft2-basic_types.html#FT_Long">FT_Long</a> )
  66.   <b>FT_MulDiv</b>( <a href="ft2-basic_types.html#FT_Long">FT_Long</a>  a,
  67.              <a href="ft2-basic_types.html#FT_Long">FT_Long</a>  b,
  68.              <a href="ft2-basic_types.html#FT_Long">FT_Long</a>  c );
  69.  
  70. </pre></table><br>
  71. <table align=center width="87%"><tr><td>
  72. <p>A very simple function used to perform the computation &lsquo;(a*b)/c&rsquo; with maximum accuracy (it uses a 64-bit intermediate integer whenever necessary).</p>
  73. <p>This function isn't necessarily as fast as some processor specific operations, but is at least completely portable.</p>
  74. </td></tr></table><br>
  75. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>input</b></em></td></tr><tr><td>
  76. <p></p>
  77. <table cellpadding=3 border=0>
  78. <tr valign=top><td><b>a</b></td><td>
  79. <p>The first multiplier.</p>
  80. </td></tr>
  81. <tr valign=top><td><b>b</b></td><td>
  82. <p>The second multiplier.</p>
  83. </td></tr>
  84. <tr valign=top><td><b>c</b></td><td>
  85. <p>The divisor.</p>
  86. </td></tr>
  87. </table>
  88. </td></tr></table>
  89. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>return</b></em></td></tr><tr><td>
  90. <p>The result of &lsquo;(a*b)/c&rsquo;. 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 &lsquo;a&rsquo; and &lsquo;b&rsquo;.</p>
  91. </td></tr></table>
  92. </td></tr></table>
  93. <hr width="75%">
  94. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  95. <td width="100%"></td>
  96. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  97.  
  98. <table align=center width="75%"><tr><td>
  99. <h4><a name="FT_MulFix">FT_MulFix</a></h4>
  100. <table align=center width="87%"><tr><td>
  101. Defined in FT_FREETYPE_H (freetype/freetype.h).
  102. </td></tr></table><br>
  103. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  104.  
  105.  FT_EXPORT( <a href="ft2-basic_types.html#FT_Long">FT_Long</a> )
  106.  <b>FT_MulFix</b>( <a href="ft2-basic_types.html#FT_Long">FT_Long</a>  a,
  107.             <a href="ft2-basic_types.html#FT_Long">FT_Long</a>  b );
  108.  
  109. </pre></table><br>
  110. <table align=center width="87%"><tr><td>
  111. <p>A very simple function used to perform the computation &lsquo;(a*b)/0x10000&rsquo; with maximum accuracy. Most of the time this is used to multiply a given value by a 16.16 fixed-point factor.</p>
  112. </td></tr></table><br>
  113. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>input</b></em></td></tr><tr><td>
  114. <p></p>
  115. <table cellpadding=3 border=0>
  116. <tr valign=top><td><b>a</b></td><td>
  117. <p>The first multiplier.</p>
  118. </td></tr>
  119. <tr valign=top><td><b>b</b></td><td>
  120. <p>The second multiplier. Use a 16.16 factor here whenever possible (see note below).</p>
  121. </td></tr>
  122. </table>
  123. </td></tr></table>
  124. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>return</b></em></td></tr><tr><td>
  125. <p>The result of &lsquo;(a*b)/0x10000&rsquo;.</p>
  126. </td></tr></table>
  127. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>note</b></em></td></tr><tr><td>
  128. <p>This function has been optimized for the case where the absolute value of &lsquo;a&rsquo; is less than 2048, and &lsquo;b&rsquo; 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>
  129. <p>As a conclusion, always try to place a 16.16 factor as the <i>second</i> argument of this function; this can make a great difference.</p>
  130. </td></tr></table>
  131. </td></tr></table>
  132. <hr width="75%">
  133. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  134. <td width="100%"></td>
  135. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  136.  
  137. <table align=center width="75%"><tr><td>
  138. <h4><a name="FT_DivFix">FT_DivFix</a></h4>
  139. <table align=center width="87%"><tr><td>
  140. Defined in FT_FREETYPE_H (freetype/freetype.h).
  141. </td></tr></table><br>
  142. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  143.  
  144.  FT_EXPORT( <a href="ft2-basic_types.html#FT_Long">FT_Long</a> )
  145.  <b>FT_DivFix</b>( <a href="ft2-basic_types.html#FT_Long">FT_Long</a>  a,
  146.             <a href="ft2-basic_types.html#FT_Long">FT_Long</a>  b );
  147.  
  148. </pre></table><br>
  149. <table align=center width="87%"><tr><td>
  150. <p>A very simple function used to perform the computation &lsquo;(a*0x10000)/b&rsquo; with maximum accuracy. Most of the time, this is used to divide a given value by a 16.16 fixed-point factor.</p>
  151. </td></tr></table><br>
  152. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>input</b></em></td></tr><tr><td>
  153. <p></p>
  154. <table cellpadding=3 border=0>
  155. <tr valign=top><td><b>a</b></td><td>
  156. <p>The first multiplier.</p>
  157. </td></tr>
  158. <tr valign=top><td><b>b</b></td><td>
  159. <p>The second multiplier. Use a 16.16 factor here whenever possible (see note below).</p>
  160. </td></tr>
  161. </table>
  162. </td></tr></table>
  163. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>return</b></em></td></tr><tr><td>
  164. <p>The result of &lsquo;(a*0x10000)/b&rsquo;.</p>
  165. </td></tr></table>
  166. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>note</b></em></td></tr><tr><td>
  167. <p>The optimization for FT_DivFix() is simple: If (a&nbsp;&lt;&lt;&nbsp;16) fits in 32&nbsp;bits, then the division is computed directly. Otherwise, we use a specialized version of <a href="ft2-computations.html#FT_MulDiv">FT_MulDiv</a>.</p>
  168. </td></tr></table>
  169. </td></tr></table>
  170. <hr width="75%">
  171. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  172. <td width="100%"></td>
  173. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  174.  
  175. <table align=center width="75%"><tr><td>
  176. <h4><a name="FT_RoundFix">FT_RoundFix</a></h4>
  177. <table align=center width="87%"><tr><td>
  178. Defined in FT_FREETYPE_H (freetype/freetype.h).
  179. </td></tr></table><br>
  180. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  181.  
  182.  FT_EXPORT( <a href="ft2-basic_types.html#FT_Fixed">FT_Fixed</a> )
  183.  <b>FT_RoundFix</b>( <a href="ft2-basic_types.html#FT_Fixed">FT_Fixed</a>  a );
  184.  
  185. </pre></table><br>
  186. <table align=center width="87%"><tr><td>
  187. <p>A very simple function used to round a 16.16 fixed number.</p>
  188. </td></tr></table><br>
  189. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>input</b></em></td></tr><tr><td>
  190. <p></p>
  191. <table cellpadding=3 border=0>
  192. <tr valign=top><td><b>a</b></td><td>
  193. <p>The number to be rounded.</p>
  194. </td></tr>
  195. </table>
  196. </td></tr></table>
  197. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>return</b></em></td></tr><tr><td>
  198. <p>The result of &lsquo;(a + 0x8000) &amp; -0x10000&rsquo;.</p>
  199. </td></tr></table>
  200. </td></tr></table>
  201. <hr width="75%">
  202. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  203. <td width="100%"></td>
  204. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  205.  
  206. <table align=center width="75%"><tr><td>
  207. <h4><a name="FT_CeilFix">FT_CeilFix</a></h4>
  208. <table align=center width="87%"><tr><td>
  209. Defined in FT_FREETYPE_H (freetype/freetype.h).
  210. </td></tr></table><br>
  211. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  212.  
  213.  FT_EXPORT( <a href="ft2-basic_types.html#FT_Fixed">FT_Fixed</a> )
  214.  <b>FT_CeilFix</b>( <a href="ft2-basic_types.html#FT_Fixed">FT_Fixed</a>  a );
  215.  
  216. </pre></table><br>
  217. <table align=center width="87%"><tr><td>
  218. <p>A very simple function used to compute the ceiling function of a 16.16 fixed number.</p>
  219. </td></tr></table><br>
  220. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>input</b></em></td></tr><tr><td>
  221. <p></p>
  222. <table cellpadding=3 border=0>
  223. <tr valign=top><td><b>a</b></td><td>
  224. <p>The number for which the ceiling function is to be computed.</p>
  225. </td></tr>
  226. </table>
  227. </td></tr></table>
  228. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>return</b></em></td></tr><tr><td>
  229. <p>The result of &lsquo;(a + 0x10000 - 1) &amp; -0x10000&rsquo;.</p>
  230. </td></tr></table>
  231. </td></tr></table>
  232. <hr width="75%">
  233. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  234. <td width="100%"></td>
  235. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  236.  
  237. <table align=center width="75%"><tr><td>
  238. <h4><a name="FT_FloorFix">FT_FloorFix</a></h4>
  239. <table align=center width="87%"><tr><td>
  240. Defined in FT_FREETYPE_H (freetype/freetype.h).
  241. </td></tr></table><br>
  242. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  243.  
  244.  FT_EXPORT( <a href="ft2-basic_types.html#FT_Fixed">FT_Fixed</a> )
  245.  <b>FT_FloorFix</b>( <a href="ft2-basic_types.html#FT_Fixed">FT_Fixed</a>  a );
  246.  
  247. </pre></table><br>
  248. <table align=center width="87%"><tr><td>
  249. <p>A very simple function used to compute the floor function of a 16.16 fixed number.</p>
  250. </td></tr></table><br>
  251. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>input</b></em></td></tr><tr><td>
  252. <p></p>
  253. <table cellpadding=3 border=0>
  254. <tr valign=top><td><b>a</b></td><td>
  255. <p>The number for which the floor function is to be computed.</p>
  256. </td></tr>
  257. </table>
  258. </td></tr></table>
  259. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>return</b></em></td></tr><tr><td>
  260. <p>The result of &lsquo;a &amp; -0x10000&rsquo;.</p>
  261. </td></tr></table>
  262. </td></tr></table>
  263. <hr width="75%">
  264. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  265. <td width="100%"></td>
  266. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  267.  
  268. <table align=center width="75%"><tr><td>
  269. <h4><a name="FT_Vector_Transform">FT_Vector_Transform</a></h4>
  270. <table align=center width="87%"><tr><td>
  271. Defined in FT_FREETYPE_H (freetype/freetype.h).
  272. </td></tr></table><br>
  273. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  274.  
  275.  FT_EXPORT( <span class="keyword">void</span> )
  276.  <b>FT_Vector_Transform</b>( <a href="ft2-basic_types.html#FT_Vector">FT_Vector</a>*        vec,
  277.                       <span class="keyword">const</span> <a href="ft2-basic_types.html#FT_Matrix">FT_Matrix</a>*  matrix );
  278.  
  279. </pre></table><br>
  280. <table align=center width="87%"><tr><td>
  281. <p>Transform a single vector through a 2x2 matrix.</p>
  282. </td></tr></table><br>
  283. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>inout</b></em></td></tr><tr><td>
  284. <p></p>
  285. <table cellpadding=3 border=0>
  286. <tr valign=top><td><b>vector</b></td><td>
  287. <p>The target vector to transform.</p>
  288. </td></tr>
  289. </table>
  290. </td></tr></table>
  291. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>input</b></em></td></tr><tr><td>
  292. <p></p>
  293. <table cellpadding=3 border=0>
  294. <tr valign=top><td><b>matrix</b></td><td>
  295. <p>A pointer to the source 2x2 matrix.</p>
  296. </td></tr>
  297. </table>
  298. </td></tr></table>
  299. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>note</b></em></td></tr><tr><td>
  300. <p>The result is undefined if either &lsquo;vector&rsquo; or &lsquo;matrix&rsquo; is invalid.</p>
  301. </td></tr></table>
  302. </td></tr></table>
  303. <hr width="75%">
  304. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  305. <td width="100%"></td>
  306. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  307.  
  308. <table align=center width="75%"><tr><td>
  309. <h4><a name="FT_Matrix_Multiply">FT_Matrix_Multiply</a></h4>
  310. <table align=center width="87%"><tr><td>
  311. Defined in FT_GLYPH_H (freetype/ftglyph.h).
  312. </td></tr></table><br>
  313. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  314.  
  315.  FT_EXPORT( <span class="keyword">void</span> )
  316.  <b>FT_Matrix_Multiply</b>( <span class="keyword">const</span> <a href="ft2-basic_types.html#FT_Matrix">FT_Matrix</a>*  a,
  317.                      <a href="ft2-basic_types.html#FT_Matrix">FT_Matrix</a>*        b );
  318.  
  319. </pre></table><br>
  320. <table align=center width="87%"><tr><td>
  321. <p>Perform the matrix operation &lsquo;b = a*b&rsquo;.</p>
  322. </td></tr></table><br>
  323. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>input</b></em></td></tr><tr><td>
  324. <p></p>
  325. <table cellpadding=3 border=0>
  326. <tr valign=top><td><b>a</b></td><td>
  327. <p>A pointer to matrix &lsquo;a&rsquo;.</p>
  328. </td></tr>
  329. </table>
  330. </td></tr></table>
  331. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>inout</b></em></td></tr><tr><td>
  332. <p></p>
  333. <table cellpadding=3 border=0>
  334. <tr valign=top><td><b>b</b></td><td>
  335. <p>A pointer to matrix &lsquo;b&rsquo;.</p>
  336. </td></tr>
  337. </table>
  338. </td></tr></table>
  339. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>note</b></em></td></tr><tr><td>
  340. <p>The result is undefined if either &lsquo;a&rsquo; or &lsquo;b&rsquo; is zero.</p>
  341. </td></tr></table>
  342. </td></tr></table>
  343. <hr width="75%">
  344. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  345. <td width="100%"></td>
  346. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  347.  
  348. <table align=center width="75%"><tr><td>
  349. <h4><a name="FT_Matrix_Invert">FT_Matrix_Invert</a></h4>
  350. <table align=center width="87%"><tr><td>
  351. Defined in FT_GLYPH_H (freetype/ftglyph.h).
  352. </td></tr></table><br>
  353. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  354.  
  355.  FT_EXPORT( <a href="ft2-basic_types.html#FT_Error">FT_Error</a> )
  356.  <b>FT_Matrix_Invert</b>( <a href="ft2-basic_types.html#FT_Matrix">FT_Matrix</a>*  matrix );
  357.  
  358. </pre></table><br>
  359. <table align=center width="87%"><tr><td>
  360. <p>Invert a 2x2 matrix. Return an error if it can't be inverted.</p>
  361. </td></tr></table><br>
  362. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>inout</b></em></td></tr><tr><td>
  363. <p></p>
  364. <table cellpadding=3 border=0>
  365. <tr valign=top><td><b>matrix</b></td><td>
  366. <p>A pointer to the target matrix. Remains untouched in case of error.</p>
  367. </td></tr>
  368. </table>
  369. </td></tr></table>
  370. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>return</b></em></td></tr><tr><td>
  371. <p>FreeType error code. 0&nbsp;means success.</p>
  372. </td></tr></table>
  373. </td></tr></table>
  374. <hr width="75%">
  375. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  376. <td width="100%"></td>
  377. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  378.  
  379. <table align=center width="75%"><tr><td>
  380. <h4><a name="FT_Angle">FT_Angle</a></h4>
  381. <table align=center width="87%"><tr><td>
  382. Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).
  383. </td></tr></table><br>
  384. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  385.  
  386.   <span class="keyword">typedef</span> <a href="ft2-basic_types.html#FT_Fixed">FT_Fixed</a>  <b>FT_Angle</b>;
  387.  
  388. </pre></table><br>
  389. <table align=center width="87%"><tr><td>
  390. <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>
  391. </td></tr></table><br>
  392. </td></tr></table>
  393. <hr width="75%">
  394. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  395. <td width="100%"></td>
  396. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  397.  
  398. <table align=center width="75%"><tr><td>
  399. <h4><a name="FT_ANGLE_PI">FT_ANGLE_PI</a></h4>
  400. <table align=center width="87%"><tr><td>
  401. Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).
  402. </td></tr></table><br>
  403. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  404.  
  405. #define <b>FT_ANGLE_PI</b>  ( 180L &lt;&lt; 16 )
  406.  
  407. </pre></table><br>
  408. <table align=center width="87%"><tr><td>
  409. <p>The angle pi expressed in <a href="ft2-computations.html#FT_Angle">FT_Angle</a> units.</p>
  410. </td></tr></table><br>
  411. </td></tr></table>
  412. <hr width="75%">
  413. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  414. <td width="100%"></td>
  415. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  416.  
  417. <table align=center width="75%"><tr><td>
  418. <h4><a name="FT_ANGLE_2PI">FT_ANGLE_2PI</a></h4>
  419. <table align=center width="87%"><tr><td>
  420. Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).
  421. </td></tr></table><br>
  422. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  423.  
  424. #define <b>FT_ANGLE_2PI</b>  ( <a href="ft2-computations.html#FT_ANGLE_PI">FT_ANGLE_PI</a> * 2 )
  425.  
  426. </pre></table><br>
  427. <table align=center width="87%"><tr><td>
  428. <p>The angle 2*pi expressed in <a href="ft2-computations.html#FT_Angle">FT_Angle</a> units.</p>
  429. </td></tr></table><br>
  430. </td></tr></table>
  431. <hr width="75%">
  432. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  433. <td width="100%"></td>
  434. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  435.  
  436. <table align=center width="75%"><tr><td>
  437. <h4><a name="FT_ANGLE_PI2">FT_ANGLE_PI2</a></h4>
  438. <table align=center width="87%"><tr><td>
  439. Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).
  440. </td></tr></table><br>
  441. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  442.  
  443. #define <b>FT_ANGLE_PI2</b>  ( <a href="ft2-computations.html#FT_ANGLE_PI">FT_ANGLE_PI</a> / 2 )
  444.  
  445. </pre></table><br>
  446. <table align=center width="87%"><tr><td>
  447. <p>The angle pi/2 expressed in <a href="ft2-computations.html#FT_Angle">FT_Angle</a> units.</p>
  448. </td></tr></table><br>
  449. </td></tr></table>
  450. <hr width="75%">
  451. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  452. <td width="100%"></td>
  453. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  454.  
  455. <table align=center width="75%"><tr><td>
  456. <h4><a name="FT_ANGLE_PI4">FT_ANGLE_PI4</a></h4>
  457. <table align=center width="87%"><tr><td>
  458. Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).
  459. </td></tr></table><br>
  460. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  461.  
  462. #define <b>FT_ANGLE_PI4</b>  ( <a href="ft2-computations.html#FT_ANGLE_PI">FT_ANGLE_PI</a> / 4 )
  463.  
  464. </pre></table><br>
  465. <table align=center width="87%"><tr><td>
  466. <p>The angle pi/4 expressed in <a href="ft2-computations.html#FT_Angle">FT_Angle</a> units.</p>
  467. </td></tr></table><br>
  468. </td></tr></table>
  469. <hr width="75%">
  470. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  471. <td width="100%"></td>
  472. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  473.  
  474. <table align=center width="75%"><tr><td>
  475. <h4><a name="FT_Sin">FT_Sin</a></h4>
  476. <table align=center width="87%"><tr><td>
  477. Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).
  478. </td></tr></table><br>
  479. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  480.  
  481.   FT_EXPORT( <a href="ft2-basic_types.html#FT_Fixed">FT_Fixed</a> )
  482.   <b>FT_Sin</b>( <a href="ft2-computations.html#FT_Angle">FT_Angle</a>  angle );
  483.  
  484. </pre></table><br>
  485. <table align=center width="87%"><tr><td>
  486. <p>Return the sinus of a given angle in fixed-point format.</p>
  487. </td></tr></table><br>
  488. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>input</b></em></td></tr><tr><td>
  489. <p></p>
  490. <table cellpadding=3 border=0>
  491. <tr valign=top><td><b>angle</b></td><td>
  492. <p>The input angle.</p>
  493. </td></tr>
  494. </table>
  495. </td></tr></table>
  496. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>return</b></em></td></tr><tr><td>
  497. <p>The sinus value.</p>
  498. </td></tr></table>
  499. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>note</b></em></td></tr><tr><td>
  500. <p>If you need both the sinus and cosinus for a given angle, use the function <a href="ft2-computations.html#FT_Vector_Unit">FT_Vector_Unit</a>.</p>
  501. </td></tr></table>
  502. </td></tr></table>
  503. <hr width="75%">
  504. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  505. <td width="100%"></td>
  506. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  507.  
  508. <table align=center width="75%"><tr><td>
  509. <h4><a name="FT_Cos">FT_Cos</a></h4>
  510. <table align=center width="87%"><tr><td>
  511. Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).
  512. </td></tr></table><br>
  513. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  514.  
  515.   FT_EXPORT( <a href="ft2-basic_types.html#FT_Fixed">FT_Fixed</a> )
  516.   <b>FT_Cos</b>( <a href="ft2-computations.html#FT_Angle">FT_Angle</a>  angle );
  517.  
  518. </pre></table><br>
  519. <table align=center width="87%"><tr><td>
  520. <p>Return the cosinus of a given angle in fixed-point format.</p>
  521. </td></tr></table><br>
  522. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>input</b></em></td></tr><tr><td>
  523. <p></p>
  524. <table cellpadding=3 border=0>
  525. <tr valign=top><td><b>angle</b></td><td>
  526. <p>The input angle.</p>
  527. </td></tr>
  528. </table>
  529. </td></tr></table>
  530. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>return</b></em></td></tr><tr><td>
  531. <p>The cosinus value.</p>
  532. </td></tr></table>
  533. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>note</b></em></td></tr><tr><td>
  534. <p>If you need both the sinus and cosinus for a given angle, use the function <a href="ft2-computations.html#FT_Vector_Unit">FT_Vector_Unit</a>.</p>
  535. </td></tr></table>
  536. </td></tr></table>
  537. <hr width="75%">
  538. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  539. <td width="100%"></td>
  540. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  541.  
  542. <table align=center width="75%"><tr><td>
  543. <h4><a name="FT_Tan">FT_Tan</a></h4>
  544. <table align=center width="87%"><tr><td>
  545. Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).
  546. </td></tr></table><br>
  547. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  548.  
  549.   FT_EXPORT( <a href="ft2-basic_types.html#FT_Fixed">FT_Fixed</a> )
  550.   <b>FT_Tan</b>( <a href="ft2-computations.html#FT_Angle">FT_Angle</a>  angle );
  551.  
  552. </pre></table><br>
  553. <table align=center width="87%"><tr><td>
  554. <p>Return the tangent of a given angle in fixed-point format.</p>
  555. </td></tr></table><br>
  556. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>input</b></em></td></tr><tr><td>
  557. <p></p>
  558. <table cellpadding=3 border=0>
  559. <tr valign=top><td><b>angle</b></td><td>
  560. <p>The input angle.</p>
  561. </td></tr>
  562. </table>
  563. </td></tr></table>
  564. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>return</b></em></td></tr><tr><td>
  565. <p>The tangent value.</p>
  566. </td></tr></table>
  567. </td></tr></table>
  568. <hr width="75%">
  569. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  570. <td width="100%"></td>
  571. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  572.  
  573. <table align=center width="75%"><tr><td>
  574. <h4><a name="FT_Atan2">FT_Atan2</a></h4>
  575. <table align=center width="87%"><tr><td>
  576. Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).
  577. </td></tr></table><br>
  578. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  579.  
  580.   FT_EXPORT( <a href="ft2-computations.html#FT_Angle">FT_Angle</a> )
  581.   <b>FT_Atan2</b>( <a href="ft2-basic_types.html#FT_Fixed">FT_Fixed</a>  x,
  582.             <a href="ft2-basic_types.html#FT_Fixed">FT_Fixed</a>  y );
  583.  
  584. </pre></table><br>
  585. <table align=center width="87%"><tr><td>
  586. <p>Return the arc-tangent corresponding to a given vector (x,y) in the 2d plane.</p>
  587. </td></tr></table><br>
  588. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>input</b></em></td></tr><tr><td>
  589. <p></p>
  590. <table cellpadding=3 border=0>
  591. <tr valign=top><td><b>x</b></td><td>
  592. <p>The horizontal vector coordinate.</p>
  593. </td></tr>
  594. <tr valign=top><td><b>y</b></td><td>
  595. <p>The vertical vector coordinate.</p>
  596. </td></tr>
  597. </table>
  598. </td></tr></table>
  599. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>return</b></em></td></tr><tr><td>
  600. <p>The arc-tangent value (i.e. angle).</p>
  601. </td></tr></table>
  602. </td></tr></table>
  603. <hr width="75%">
  604. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  605. <td width="100%"></td>
  606. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  607.  
  608. <table align=center width="75%"><tr><td>
  609. <h4><a name="FT_Angle_Diff">FT_Angle_Diff</a></h4>
  610. <table align=center width="87%"><tr><td>
  611. Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).
  612. </td></tr></table><br>
  613. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  614.  
  615.   FT_EXPORT( <a href="ft2-computations.html#FT_Angle">FT_Angle</a> )
  616.   <b>FT_Angle_Diff</b>( <a href="ft2-computations.html#FT_Angle">FT_Angle</a>  angle1,
  617.                  <a href="ft2-computations.html#FT_Angle">FT_Angle</a>  angle2 );
  618.  
  619. </pre></table><br>
  620. <table align=center width="87%"><tr><td>
  621. <p>Return the difference between two angles. The result is always constrained to the ]-PI..PI] interval.</p>
  622. </td></tr></table><br>
  623. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>input</b></em></td></tr><tr><td>
  624. <p></p>
  625. <table cellpadding=3 border=0>
  626. <tr valign=top><td><b>angle1</b></td><td>
  627. <p>First angle.</p>
  628. </td></tr>
  629. <tr valign=top><td><b>angle2</b></td><td>
  630. <p>Second angle.</p>
  631. </td></tr>
  632. </table>
  633. </td></tr></table>
  634. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>return</b></em></td></tr><tr><td>
  635. <p>Constrained value of &lsquo;value2-value1&rsquo;.</p>
  636. </td></tr></table>
  637. </td></tr></table>
  638. <hr width="75%">
  639. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  640. <td width="100%"></td>
  641. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  642.  
  643. <table align=center width="75%"><tr><td>
  644. <h4><a name="FT_Vector_Unit">FT_Vector_Unit</a></h4>
  645. <table align=center width="87%"><tr><td>
  646. Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).
  647. </td></tr></table><br>
  648. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  649.  
  650.   FT_EXPORT( <span class="keyword">void</span> )
  651.   <b>FT_Vector_Unit</b>( <a href="ft2-basic_types.html#FT_Vector">FT_Vector</a>*  vec,
  652.                   <a href="ft2-computations.html#FT_Angle">FT_Angle</a>    angle );
  653.  
  654. </pre></table><br>
  655. <table align=center width="87%"><tr><td>
  656. <p>Return the unit vector corresponding to a given angle. After the call, the value of &lsquo;vec.x&rsquo; will be &lsquo;sin(angle)&rsquo;, and the value of &lsquo;vec.y&rsquo; will be &lsquo;cos(angle)&rsquo;.</p>
  657. <p>This function is useful to retrieve both the sinus and cosinus of a given angle quickly.</p>
  658. </td></tr></table><br>
  659. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>output</b></em></td></tr><tr><td>
  660. <p></p>
  661. <table cellpadding=3 border=0>
  662. <tr valign=top><td><b>vec</b></td><td>
  663. <p>The address of target vector.</p>
  664. </td></tr>
  665. </table>
  666. </td></tr></table>
  667. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>input</b></em></td></tr><tr><td>
  668. <p></p>
  669. <table cellpadding=3 border=0>
  670. <tr valign=top><td><b>angle</b></td><td>
  671. <p>The address of angle.</p>
  672. </td></tr>
  673. </table>
  674. </td></tr></table>
  675. </td></tr></table>
  676. <hr width="75%">
  677. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  678. <td width="100%"></td>
  679. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  680.  
  681. <table align=center width="75%"><tr><td>
  682. <h4><a name="FT_Vector_Rotate">FT_Vector_Rotate</a></h4>
  683. <table align=center width="87%"><tr><td>
  684. Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).
  685. </td></tr></table><br>
  686. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  687.  
  688.   FT_EXPORT( <span class="keyword">void</span> )
  689.   <b>FT_Vector_Rotate</b>( <a href="ft2-basic_types.html#FT_Vector">FT_Vector</a>*  vec,
  690.                     <a href="ft2-computations.html#FT_Angle">FT_Angle</a>    angle );
  691.  
  692. </pre></table><br>
  693. <table align=center width="87%"><tr><td>
  694. <p>Rotate a vector by a given angle.</p>
  695. </td></tr></table><br>
  696. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>inout</b></em></td></tr><tr><td>
  697. <p></p>
  698. <table cellpadding=3 border=0>
  699. <tr valign=top><td><b>vec</b></td><td>
  700. <p>The address of target vector.</p>
  701. </td></tr>
  702. </table>
  703. </td></tr></table>
  704. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>input</b></em></td></tr><tr><td>
  705. <p></p>
  706. <table cellpadding=3 border=0>
  707. <tr valign=top><td><b>angle</b></td><td>
  708. <p>The address of angle.</p>
  709. </td></tr>
  710. </table>
  711. </td></tr></table>
  712. </td></tr></table>
  713. <hr width="75%">
  714. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  715. <td width="100%"></td>
  716. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  717.  
  718. <table align=center width="75%"><tr><td>
  719. <h4><a name="FT_Vector_Length">FT_Vector_Length</a></h4>
  720. <table align=center width="87%"><tr><td>
  721. Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).
  722. </td></tr></table><br>
  723. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  724.  
  725.   FT_EXPORT( <a href="ft2-basic_types.html#FT_Fixed">FT_Fixed</a> )
  726.   <b>FT_Vector_Length</b>( <a href="ft2-basic_types.html#FT_Vector">FT_Vector</a>*  vec );
  727.  
  728. </pre></table><br>
  729. <table align=center width="87%"><tr><td>
  730. <p>Return the length of a given vector.</p>
  731. </td></tr></table><br>
  732. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>input</b></em></td></tr><tr><td>
  733. <p></p>
  734. <table cellpadding=3 border=0>
  735. <tr valign=top><td><b>vec</b></td><td>
  736. <p>The address of target vector.</p>
  737. </td></tr>
  738. </table>
  739. </td></tr></table>
  740. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>return</b></em></td></tr><tr><td>
  741. <p>The vector length, expressed in the same units that the original vector coordinates.</p>
  742. </td></tr></table>
  743. </td></tr></table>
  744. <hr width="75%">
  745. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  746. <td width="100%"></td>
  747. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  748.  
  749. <table align=center width="75%"><tr><td>
  750. <h4><a name="FT_Vector_Polarize">FT_Vector_Polarize</a></h4>
  751. <table align=center width="87%"><tr><td>
  752. Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).
  753. </td></tr></table><br>
  754. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  755.  
  756.   FT_EXPORT( <span class="keyword">void</span> )
  757.   <b>FT_Vector_Polarize</b>( <a href="ft2-basic_types.html#FT_Vector">FT_Vector</a>*  vec,
  758.                       <a href="ft2-basic_types.html#FT_Fixed">FT_Fixed</a>   *length,
  759.                       <a href="ft2-computations.html#FT_Angle">FT_Angle</a>   *angle );
  760.  
  761. </pre></table><br>
  762. <table align=center width="87%"><tr><td>
  763. <p>Compute both the length and angle of a given vector.</p>
  764. </td></tr></table><br>
  765. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>input</b></em></td></tr><tr><td>
  766. <p></p>
  767. <table cellpadding=3 border=0>
  768. <tr valign=top><td><b>vec</b></td><td>
  769. <p>The address of source vector.</p>
  770. </td></tr>
  771. </table>
  772. </td></tr></table>
  773. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>output</b></em></td></tr><tr><td>
  774. <p></p>
  775. <table cellpadding=3 border=0>
  776. <tr valign=top><td><b>length</b></td><td>
  777. <p>The vector length.</p>
  778. </td></tr>
  779. <tr valign=top><td><b>angle</b></td><td>
  780. <p>The vector angle.</p>
  781. </td></tr>
  782. </table>
  783. </td></tr></table>
  784. </td></tr></table>
  785. <hr width="75%">
  786. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  787. <td width="100%"></td>
  788. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  789.  
  790. <table align=center width="75%"><tr><td>
  791. <h4><a name="FT_Vector_From_Polar">FT_Vector_From_Polar</a></h4>
  792. <table align=center width="87%"><tr><td>
  793. Defined in FT_TRIGONOMETRY_H (freetype/fttrigon.h).
  794. </td></tr></table><br>
  795. <table align=center width="87%"><tr bgcolor="#D6E8FF"><td><pre>
  796.  
  797.   FT_EXPORT( <span class="keyword">void</span> )
  798.   <b>FT_Vector_From_Polar</b>( <a href="ft2-basic_types.html#FT_Vector">FT_Vector</a>*  vec,
  799.                         <a href="ft2-basic_types.html#FT_Fixed">FT_Fixed</a>    length,
  800.                         <a href="ft2-computations.html#FT_Angle">FT_Angle</a>    angle );
  801.  
  802. </pre></table><br>
  803. <table align=center width="87%"><tr><td>
  804. <p>Compute vector coordinates from a length and angle.</p>
  805. </td></tr></table><br>
  806. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>output</b></em></td></tr><tr><td>
  807. <p></p>
  808. <table cellpadding=3 border=0>
  809. <tr valign=top><td><b>vec</b></td><td>
  810. <p>The address of source vector.</p>
  811. </td></tr>
  812. </table>
  813. </td></tr></table>
  814. <table align=center width="87%" cellpadding=5><tr bgcolor="#EEEEFF"><td><em><b>input</b></em></td></tr><tr><td>
  815. <p></p>
  816. <table cellpadding=3 border=0>
  817. <tr valign=top><td><b>length</b></td><td>
  818. <p>The vector length.</p>
  819. </td></tr>
  820. <tr valign=top><td><b>angle</b></td><td>
  821. <p>The vector angle.</p>
  822. </td></tr>
  823. </table>
  824. </td></tr></table>
  825. </td></tr></table>
  826. <hr width="75%">
  827. <table align=center width="75%"><tr><td><font size=-2>[<a href="ft2-index.html">Index</a>]</font></td>
  828. <td width="100%"></td>
  829. <td><font size=-2>[<a href="ft2-toc.html">TOC</a>]</font></td></tr></table>
  830.  
  831. </body>
  832. </html>
  833.