1 //----------------------------------------------------------------------------
2 // Anti-Grain Geometry - Version 2.4 (Public License)
3 // Copyright (C) 2002-2005 Maxim Shemanarev (http://www.antigrain.com)
4 //
5 // Anti-Grain Geometry - Version 2.4 Release Milano 3 (AggPas 2.4 RM3)
6 // Pascal Port By: Milan Marusinec alias Milano
7 // milan@marusinec.sk
8 // http://www.aggpas.org
9 // Copyright (c) 2005-2006
10 //
11 // Permission to copy, use, modify, sell and distribute this software
12 // is granted provided this copyright notice appears in all copies.
13 // This software is provided "as is" without express or implied
14 // warranty, and with no claim as to its suitability for any purpose.
15 //
16 //----------------------------------------------------------------------------
17 // Contact: mcseem@antigrain.com
18 // mcseemagg@yahoo.com
19 // http://www.antigrain.com
20 //
21 //----------------------------------------------------------------------------
22 // Bessel function (besj) was adapted for use in AGG library by Andy Wilk
23 // Contact: castor.vulgaris@gmail.com
24 //
25 // [Pascal Port History] -----------------------------------------------------
26 //
27 // 23.06.2006-Milano: ptrcomp adjustments
28 // 18.12.2005-Milano: Unit port establishment
29 //
30 { agg_math.pas }
31 unit
32 agg_math ;
33
34 INTERFACE
35
36 {$I agg_mode.inc }
37 {$Q- }
38 {$R- }
39 uses
40 Math ,
41 agg_basics ,
42 agg_vertex_sequence ;
43
44 { GLOBAL VARIABLES & CONSTANTS }
45 type
46 double_xy_ptr = ^double_xy;
47 double_xy = record
48 x ,y : double;
49
50 end;
51
52 poly_xy_ptr = ^poly_xy;
53 poly_xy = array[0..0 ] of double_xy;
54
55 storage_xy_ptr = ^storage_xy;
56 storage_xy = record
57 poly : poly_xy_ptr;
58 size : unsigned;
59
60 end;
61
62 const
63 intersection_epsilon : double = 1.0e-30;
64
65 // Tables for fast sqrt
66 const
67 g_sqrt_table : array[0..1023 ] of int16u = (0 ,
68 2048,2896,3547,4096,4579,5017,5418,5793,6144,6476,6792,7094,7384,7663,7932,8192,8444,
69 8689,8927,9159,9385,9606,9822,10033,10240,10443,10642,10837,11029,11217,11403,11585,
70 11765,11942,12116,12288,12457,12625,12790,12953,13114,13273,13430,13585,13738,13890,
71 14040,14189,14336,14482,14626,14768,14910,15050,15188,15326,15462,15597,15731,15864,
72 15995,16126,16255,16384,16512,16638,16764,16888,17012,17135,17257,17378,17498,17618,
73 17736,17854,17971,18087,18203,18318,18432,18545,18658,18770,18882,18992,19102,19212,
74 19321,19429,19537,19644,19750,19856,19961,20066,20170,20274,20377,20480,20582,20684,
75 20785,20886,20986,21085,21185,21283,21382,21480,21577,21674,21771,21867,21962,22058,
76 22153,22247,22341,22435,22528,22621,22713,22806,22897,22989,23080,23170,23261,23351,
77 23440,23530,23619,23707,23796,23884,23971,24059,24146,24232,24319,24405,24491,24576,
78 24661,24746,24831,24915,24999,25083,25166,25249,25332,25415,25497,25580,25661,25743,
79 25824,25905,25986,26067,26147,26227,26307,26387,26466,26545,26624,26703,26781,26859,
80 26937,27015,27092,27170,27247,27324,27400,27477,27553,27629,27705,27780,27856,27931,
81 28006,28081,28155,28230,28304,28378,28452,28525,28599,28672,28745,28818,28891,28963,
82 29035,29108,29180,29251,29323,29394,29466,29537,29608,29678,29749,29819,29890,29960,
83 30030,30099,30169,30238,30308,30377,30446,30515,30583,30652,30720,30788,30856,30924,
84 30992,31059,31127,31194,31261,31328,31395,31462,31529,31595,31661,31727,31794,31859,
85 31925,31991,32056,32122,32187,32252,32317,32382,32446,32511,32575,32640,32704,32768,
86 32832,32896,32959,33023,33086,33150,33213,33276,33339,33402,33465,33527,33590,33652,
87 33714,33776,33839,33900,33962,34024,34086,34147,34208,34270,34331,34392,34453,34514,
88 34574,34635,34695,34756,34816,34876,34936,34996,35056,35116,35176,35235,35295,35354,
89 35413,35472,35531,35590,35649,35708,35767,35825,35884,35942,36001,36059,36117,36175,
90 36233,36291,36348,36406,36464,36521,36578,36636,36693,36750,36807,36864,36921,36978,
91 37034,37091,37147,37204,37260,37316,37372,37429,37485,37540,37596,37652,37708,37763,
92 37819,37874,37929,37985,38040,38095,38150,38205,38260,38315,38369,38424,38478,38533,
93 38587,38642,38696,38750,38804,38858,38912,38966,39020,39073,39127,39181,39234,39287,
94 39341,39394,39447,39500,39553,39606,39659,39712,39765,39818,39870,39923,39975,40028,
95 40080,40132,40185,40237,40289,40341,40393,40445,40497,40548,40600,40652,40703,40755,
96 40806,40857,40909,40960,41011,41062,41113,41164,41215,41266,41317,41368,41418,41469,
97 41519,41570,41620,41671,41721,41771,41821,41871,41922,41972,42021,42071,42121,42171,
98 42221,42270,42320,42369,42419,42468,42518,42567,42616,42665,42714,42763,42813,42861,
99 42910,42959,43008,43057,43105,43154,43203,43251,43300,43348,43396,43445,43493,43541,
100 43589,43637,43685,43733,43781,43829,43877,43925,43972,44020,44068,44115,44163,44210,
101 44258,44305,44352,44400,44447,44494,44541,44588,44635,44682,44729,44776,44823,44869,
102 44916,44963,45009,45056,45103,45149,45195,45242,45288,45334,45381,45427,45473,45519,
103 45565,45611,45657,45703,45749,45795,45840,45886,45932,45977,46023,46069,46114,46160,
104 46205,46250,46296,46341,46386,46431,46477,46522,46567,46612,46657,46702,46746,46791,
105 46836,46881,46926,46970,47015,47059,47104,47149,47193,47237,47282,47326,47370,47415,
106 47459,47503,47547,47591,47635,47679,47723,47767,47811,47855,47899,47942,47986,48030,
107 48074,48117,48161,48204,48248,48291,48335,48378,48421,48465,48508,48551,48594,48637,
108 48680,48723,48766,48809,48852,48895,48938,48981,49024,49067,49109,49152,49195,49237,
109 49280,49322,49365,49407,49450,49492,49535,49577,49619,49661,49704,49746,49788,49830,
110 49872,49914,49956,49998,50040,50082,50124,50166,50207,50249,50291,50332,50374,50416,
111 50457,50499,50540,50582,50623,50665,50706,50747,50789,50830,50871,50912,50954,50995,
112 51036,51077,51118,51159,51200,51241,51282,51323,51364,51404,51445,51486,51527,51567,
113 51608,51649,51689,51730,51770,51811,51851,51892,51932,51972,52013,52053,52093,52134,
114 52174,52214,52254,52294,52334,52374,52414,52454,52494,52534,52574,52614,52654,52694,
115 52734,52773,52813,52853,52892,52932,52972,53011,53051,53090,53130,53169,53209,53248,
116 53287,53327,53366,53405,53445,53484,53523,53562,53601,53640,53679,53719,53758,53797,
117 53836,53874,53913,53952,53991,54030,54069,54108,54146,54185,54224,54262,54301,54340,
118 54378,54417,54455,54494,54532,54571,54609,54647,54686,54724,54762,54801,54839,54877,
119 54915,54954,54992,55030,55068,55106,55144,55182,55220,55258,55296,55334,55372,55410,
120 55447,55485,55523,55561,55599,55636,55674,55712,55749,55787,55824,55862,55900,55937,
121 55975,56012,56049,56087,56124,56162,56199,56236,56273,56311,56348,56385,56422,56459,
122 56497,56534,56571,56608,56645,56682,56719,56756,56793,56830,56867,56903,56940,56977,
123 57014,57051,57087,57124,57161,57198,57234,57271,57307,57344,57381,57417,57454,57490,
124 57527,57563,57599,57636,57672,57709,57745,57781,57817,57854,57890,57926,57962,57999,
125 58035,58071,58107,58143,58179,58215,58251,58287,58323,58359,58395,58431,58467,58503,
126 58538,58574,58610,58646,58682,58717,58753,58789,58824,58860,58896,58931,58967,59002,
127 59038,59073,59109,59144,59180,59215,59251,59286,59321,59357,59392,59427,59463,59498,
128 59533,59568,59603,59639,59674,59709,59744,59779,59814,59849,59884,59919,59954,59989,
129 60024,60059,60094,60129,60164,60199,60233,60268,60303,60338,60373,60407,60442,60477,
130 60511,60546,60581,60615,60650,60684,60719,60753,60788,60822,60857,60891,60926,60960,
131 60995,61029,61063,61098,61132,61166,61201,61235,61269,61303,61338,61372,61406,61440,
132 61474,61508,61542,61576,61610,61644,61678,61712,61746,61780,61814,61848,61882,61916,
133 61950,61984,62018,62051,62085,62119,62153,62186,62220,62254,62287,62321,62355,62388,
134 62422,62456,62489,62523,62556,62590,62623,62657,62690,62724,62757,62790,62824,62857,
135 62891,62924,62957,62991,63024,63057,63090,63124,63157,63190,63223,63256,63289,63323,
136 63356,63389,63422,63455,63488,63521,63554,63587,63620,63653,63686,63719,63752,63785,
137 63817,63850,63883,63916,63949,63982,64014,64047,64080,64113,64145,64178,64211,64243,
138 64276,64309,64341,64374,64406,64439,64471,64504,64536,64569,64601,64634,64666,64699,
139 64731,64763,64796,64828,64861,64893,64925,64957,64990,65022,65054,65086,65119,65151,
140 65183,65215,65247,65279,65312,65344,65376,65408,65440,65472,65504 );
141
142 g_elder_bit_table : array[0..255 ] of int8 = (
143 0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,
144 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,
145 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,
146 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,
147 7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,
148 7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,
149 7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,
150 7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7 );
151
152 { GLOBAL PROCEDURES }
calc_point_locationnull153 function calc_point_location(x1 ,y1 ,x2 ,y2 ,x ,y : double ) : double;
154
point_in_trianglenull155 function point_in_triangle(x1 ,y1 ,x2 ,y2 ,x3 ,y3 ,x ,y : double ) : boolean;
156
calc_distancenull157 function calc_distance(x1 ,y1 ,x2 ,y2 : double ) : double;
158
calc_line_point_distancenull159 function calc_line_point_distance(x1 ,y1 ,x2 ,y2 ,x ,y : double ) : double;
160
calc_intersectionnull161 function calc_intersection(ax ,ay ,bx ,by ,cx ,cy ,dx ,dy : double; x ,y : double_ptr ) : boolean;
162
intersection_existsnull163 function intersection_exists(x1 ,y1 ,x2 ,y2 ,x3 ,y3 ,x4 ,y4 : double ) : boolean;
164
165 procedure calc_orthogonal(thickness ,x1 ,y1 ,x2 ,y2 : double; x ,y : double_ptr );
166
167 procedure dilate_triangle(x1 ,y1 ,x2 ,y2 ,x3 ,y3 : double; x ,y : double_ptr; d : double );
168
calc_triangle_areanull169 function calc_triangle_area(x1 ,y1 ,x2 ,y2 ,x3 ,y3 : double ) : double;
170
calc_polygon_areanull171 function calc_polygon_area(st : storage_xy_ptr ) : double;
172
calc_polygon_area_vsnull173 function calc_polygon_area_vs(st : vertex_sequence_ptr ) : double;
174
fast_sqrtnull175 function fast_sqrt(val : unsigned ) : unsigned;
176
besjnull177 function besj(x : double; n : int ) : double;
178
cross_productnull179 function cross_product(x1 ,y1 ,x2 ,y2 ,x ,y : double ) : double;
180
181
182 IMPLEMENTATION
183 { LOCAL VARIABLES & CONSTANTS }
184 { UNIT IMPLEMENTATION }
185 { calc_point_location }
calc_point_locationnull186 function calc_point_location;
187 begin
188 result:=(x - x2 ) * (y2 - y1 ) - (y - y2 ) * (x2 - x1 );
189
190 end;
191
192 { point_in_triangle }
point_in_trianglenull193 function point_in_triangle;
194 var
195 cp1 ,cp2 ,cp3 : boolean;
196
197 begin
198 cp1:=calc_point_location(x1 ,y1 ,x2 ,y2 ,x ,y ) < 0.0;
199 cp2:=calc_point_location(x2 ,y2 ,x3 ,y3 ,x ,y ) < 0.0;
200 cp3:=calc_point_location(x3 ,y3 ,x1 ,y1 ,x ,y ) < 0.0;
201
202 result:=(cp1 = cp2 ) and (cp2 = cp3 ) and (cp3 = cp1 );
203
204 end;
205
206 { calc_distance }
calc_distancenull207 function calc_distance;
208 var
209 dx ,dy : double;
210
211 begin
212 dx:=x2 - x1;
213 dy:=y2 - y1;
214
215 result:=sqrt(dx * dx + dy * dy );
216
217 end;
218
219 { calc_line_point_distance }
calc_line_point_distancenull220 function calc_line_point_distance;
221 var
222 d ,
223 dx ,
224 dy : double;
225
226 begin
227 dx:=x2 - x1;
228 dy:=y2 - y1;
229 d :=sqrt(dx * dx + dy * dy );
230
231 if d < intersection_epsilon then
232 result:=calc_distance(x1 ,y1 ,x ,y )
233 else
234 result:=((x - x2 ) * dy - (y - y2 ) * dx) / d;
235
236 end;
237
238 { calc_intersection }
calc_intersectionnull239 function calc_intersection;
240 var
241 r ,
242
243 num ,
244 den : double;
245
246 begin
247 num:=(ay - cy ) * (dx - cx ) - (ax - cx ) * (dy - cy );
248 den:=(bx - ax ) * (dy - cy ) - (by - ay ) * (dx - cx );
249
250 if Abs(den ) < intersection_epsilon then
251 result:=false
252
253 else
254 begin
255 r :=num / den;
256 x^:=ax + r * (bx - ax );
257 y^:=ay + r * (by - ay );
258
259 result:=true;
260
261 end;
262
263 end;
264
265 { intersection_exists }
intersection_existsnull266 function intersection_exists;
267 var
268 dx1 ,dy1 ,
269 dx2 ,dy2 : double;
270
271 begin
272 dx1:=x2 - x1;
273 dy1:=y2 - y1;
274 dx2:=x4 - x3;
275 dy2:=y4 - y3;
276
277 result:=
278 (((x3 - x2 ) * dy1 - (y3 - y2 ) * dx1 < 0.0 ) <>
279 ((x4 - x2 ) * dy1 - (y4 - y2 ) * dx1 < 0.0 ) ) and
280 (((x1 - x4 ) * dy2 - (y1 - y4 ) * dx2 < 0.0 ) <>
281 ((x2 - x4 ) * dy2 - (y2 - y4 ) * dx2 < 0.0 ) );
282
283 end;
284
285 { calc_orthogonal }
286 procedure calc_orthogonal;
287 var
288 d ,dx ,dy : double;
289
290 begin
291 dx:=x2 - x1;
292 dy:=y2 - y1;
293 d :=sqrt(dx * dx + dy * dy );
294 x^:=thickness * dy / d;
295 y^:=thickness * dx / d;
296
297 end;
298
299 { dilate_triangle }
300 procedure dilate_triangle;
301 var
302 loc ,
303 dx1 ,dy1 ,
304 dx2 ,dy2 ,
305 dx3 ,dy3 : double;
306
307 begin
308 dx1:=0.0;
309 dy1:=0.0;
310 dx2:=0.0;
311 dy2:=0.0;
312 dx3:=0.0;
313 dy3:=0.0;
314 loc:=calc_point_location(x1 ,y1 ,x2 ,y2 ,x3 ,y3 );
315
316 if Abs(loc ) > intersection_epsilon then
317 begin
318 if calc_point_location(x1 ,y1 ,x2 ,y2 ,x3 ,y3 ) > 0.0 then
319 d:=-d;
320
321 calc_orthogonal(d ,x1 ,y1 ,x2 ,y2 ,@dx1 ,@dy1 );
322 calc_orthogonal(d ,x2 ,y2 ,x3 ,y3 ,@dx2 ,@dy2 );
323 calc_orthogonal(d ,x3 ,y3 ,x1 ,y1 ,@dx3 ,@dy3 );
324
325 end;
326
327 x^:=x1 + dx1; inc(ptrcomp(x ) ,sizeof(double ) );
328 y^:=y1 - dy1; inc(ptrcomp(y ) ,sizeof(double ) );
329 x^:=x2 + dx1; inc(ptrcomp(x ) ,sizeof(double ) );
330 y^:=y2 - dy1; inc(ptrcomp(y ) ,sizeof(double ) );
331 x^:=x2 + dx2; inc(ptrcomp(x ) ,sizeof(double ) );
332 y^:=y2 - dy2; inc(ptrcomp(y ) ,sizeof(double ) );
333 x^:=x3 + dx2; inc(ptrcomp(x ) ,sizeof(double ) );
334 y^:=y3 - dy2; inc(ptrcomp(y ) ,sizeof(double ) );
335 x^:=x3 + dx3; inc(ptrcomp(x ) ,sizeof(double ) );
336 y^:=y3 - dy3; inc(ptrcomp(y ) ,sizeof(double ) );
337 x^:=x1 + dx3; inc(ptrcomp(x ) ,sizeof(double ) );
338 y^:=y1 - dy3; inc(ptrcomp(y ) ,sizeof(double ) );
339
340 end;
341
342 { calc_triangle_area }
calc_triangle_areanull343 function calc_triangle_area;
344 begin
345 result:=(x1 * y2 - x2 * y1 + x2 * y3 - x3 * y2 + x3 * y1 - x1 * y3 ) * 0.5;
346
347 end;
348
349 { calc_polygon_area }
calc_polygon_areanull350 function calc_polygon_area;
351 var
352 i : unsigned;
353 v : double_xy_ptr;
354
355 x ,y ,sum ,xs ,ys : double;
356
357 begin
358 sum:=0.0;
359 x :=st.poly[0 ].x;
360 y :=st.poly[0 ].y;
361 xs :=x;
362 ys :=y;
363
364 if st.size > 0 then
365 for i:=1 to st.size - 1 do
366 begin
367 v:=@st.poly[i ];
368
369 sum:=sum + (x * v.y - y * v.x );
370
371 x:=v.x;
372 y:=v.y;
373
374 end;
375
376 result:=(sum + x * ys - y * xs ) * 0.5;
377
378 end;
379
380 { calc_polygon_area_vs }
calc_polygon_area_vsnull381 function calc_polygon_area_vs;
382 var
383 i : unsigned;
384 v : vertex_dist_ptr;
385
386 x ,y ,sum ,xs ,ys : double;
387
388 begin
389 sum:=0.0;
390 x :=vertex_dist_ptr(st.array_operator(0 ) ).x;
391 y :=vertex_dist_ptr(st.array_operator(0 ) ).y;
392 xs :=x;
393 ys :=y;
394
395 if st.size > 0 then
396 for i:=1 to st.size - 1 do
397 begin
398 v:=st.array_operator(i );
399
400 sum:=sum + (x * v.y - y * v.x );
401
402 x:=v.x;
403 y:=v.y;
404
405 end;
406
407 result:=(sum + x * ys - y * xs ) * 0.5;
408
409 end;
410
411 { fast_sqrt }
fast_sqrtnull412 function fast_sqrt;
413 var
414 bit : int;
415
416 t ,shift : unsigned;
417
418 begin
419 t :=val;
420 bit:=0;
421
422 shift:=11;
423
424 //The following piece of code is just an emulation of the
425 //Ix86 assembler command "bsr" (see below). However on old
426 //Intels (like Intel MMX 233MHz) this code is about twice
427 //faster (sic!) then just one "bsr". On PIII and PIV the
428 //bsr is optimized quite well.
429 bit:=t shr 24;
430
431 if bit <> 0 then
432 bit:=g_elder_bit_table[bit ] + 24
433
434 else
435 begin
436 bit:=(t shr 16 ) and $FF;
437
438 if bit <> 0 then
439 bit:=g_elder_bit_table[bit ] + 16
440 else
441 begin
442 bit:=(t shr 8 ) and $FF;
443
444 if bit <> 0 then
445 bit:=g_elder_bit_table[bit ] + 8
446 else
447 bit:=g_elder_bit_table[t ];
448
449 end;
450
451 end;
452
453 // This is calculation sqrt itself.
454 bit:=bit - 9;
455
456 if bit > 0 then
457 begin
458 bit :=(shr_int32(bit ,1 ) ) + (bit and 1 );
459 shift:=shift - bit;
460 val :=val shr (bit shl 1 );
461
462 end;
463
464 result:=g_sqrt_table[val ] shr shift;
465
466 end;
467
468 //--------------------------------------------------------------------besj
BESJnull469 // Function BESJ calculates Bessel function of first kind of order n
470 // Arguments:
471 // n - an integer (>=0), the order
472 // x - value at which the Bessel function is required
473 //--------------------
474 // C++ Mathematical Library
475 // Convereted from equivalent FORTRAN library
476 // Converetd by Gareth Walker for use by course 392 computational project
477 // All functions tested and yield the same results as the corresponding
478 // FORTRAN versions.
479 //
480 // If you have any problems using these functions please report them to
481 // M.Muldoon@UMIST.ac.uk
482 //
483 // Documentation available on the web
484 // http://www.ma.umist.ac.uk/mrm/Teaching/392/libs/392.html
485 // Version 1.0 8/98
486 // 29 October, 1999
487 //--------------------
488 // Adapted for use in AGG library by Andy Wilk (castor.vulgaris@gmail.com)
489 //------------------------------------------------------------------------
490 { besj }
491 function besj;
492 var
493 i ,m1 ,m2 ,m8 ,imax : int;
494
495 d ,b ,b1 ,c2 ,c3 ,c4 ,c6 : double;
496
497 begin
498 if n < 0 then
499 begin
500 result:=0;
501
502 exit;
503
504 end;
505
506 d:=1E-6;
507 b:=0;
508
509 if Abs(x ) <= d then
510 begin
511 if n <> 0 then
512 result:=0
513 else
514 result:=1;
515
516 exit;
517
518 end;
519
520 b1:=0; // b1 is the value from the previous iteration
521
522 // Set up a starting order for recurrence
523 m1:=trunc(Abs(x ) + 6 );
524
525 if Abs(x ) > 5 then
526 m1:=trunc(Abs(1.4 * x + 60 / x ) );
527
528 m2:=trunc(n + 2 + Abs(x ) / 4 );
529
530 if m1 > m2 then
531 m2:=m1;
532
533 // Apply recurrence down from curent max order
534 repeat
535 c3:=0;
536 c2:=1E-30;
537 c4:=0;
538 m8:=1;
539
540 if m2 div 2 * 2 = m2 then
541 m8:=-1;
542
543 imax:=m2 - 2;
544
545 for i:=1 to imax do
546 begin
547 c6:=2 * (m2 - i ) * c2 / x - c3;
548 c3:=c2;
549 c2:=c6;
550
551 if m2 - i - 1 = n then
552 b:=c6;
553
554 m8:=-1 * m8;
555
556 if m8 > 0 then
557 c4:=c4 + 2 * c6;
558
559 end;
560
561 c6:=2 * c2 / x - c3;
562
563 if n = 0 then
564 b:=c6;
565
566 c4:=c4 + c6;
567 b :=b / c4;
568
569 if Abs(b - b1 ) < d then
570 begin
571 result:=b;
572
573 exit;
574
575 end;
576
577 b1:=b;
578
579 inc(m2 ,3 );
580
581 until false;
582
583 end;
584
585 { CROSS_PRODUCT }
cross_productnull586 function cross_product(x1 ,y1 ,x2 ,y2 ,x ,y : double ) : double;
587 begin
588 result:=(x - x2 ) * (y2 - y1 ) - (y - y2 ) * (x2 - x1 );
589
590 end;
591
592 END.
593
594