| 1 |
unit affine; |
| 2 |
{$IFDEF FPC} |
| 3 |
{$MODE DELPHI}{$H+} |
| 4 |
{$ENDIF} |
| 5 |
|
| 6 |
(***************************************) |
| 7 |
(* Copyright (C) 2009, SHIRAISHI Kazuo *) |
| 8 |
(***************************************) |
| 9 |
|
| 10 |
{$T+} |
| 11 |
interface |
| 12 |
uses math2sub,mathc,Arrays,ArraysC; |
| 13 |
(* |
| 14 |
class Affine means: |
| 15 |
x'= xx * x + xy * y + xo |
| 16 |
y'= yx * x + yy * y + yo |
| 17 |
*) |
| 18 |
type |
| 19 |
TAffine=class |
| 20 |
{$A4} |
| 21 |
xx, yx, zx, ox:double; |
| 22 |
xy, yy, zy, oy:double; |
| 23 |
xz, yz, zz, oz:double; |
| 24 |
xo, yo, zo, oo:double; |
| 25 |
{$A+} |
| 26 |
next:TAffine; |
| 27 |
class function NewInstance: TObject;override; |
| 28 |
procedure FreeInstance;override; |
| 29 |
constructor Create;overload; |
| 30 |
constructor Create(a:TArray2N); overload; |
| 31 |
constructor Create(a:TArray2C); overload; |
| 32 |
//procedure make(transform:TObjectList); |
| 33 |
procedure scale1(a:double); overload; |
| 34 |
procedure scale1(a:complex); overload; |
| 35 |
procedure scale(a,b:double); |
| 36 |
procedure shift(a,b:double);overload; |
| 37 |
procedure shift(c:complex);overload; |
| 38 |
procedure rotate(t:double); |
| 39 |
procedure rotate2(ct,st:double); |
| 40 |
procedure shear(t:double); |
| 41 |
procedure revmlt(m:TAffine); |
| 42 |
procedure mlt(m:TAffine);overload; |
| 43 |
procedure mlt(a:TArray2N);overload; |
| 44 |
procedure mlt(a:TArray2C);overload; |
| 45 |
procedure store(a:TArray2N); overload; |
| 46 |
procedure store(a:TArray2C); overload; |
| 47 |
procedure cmlt(const c:complex); |
| 48 |
function transform(var x,y:extended):boolean; overload; |
| 49 |
function transform(var x,y:double):boolean; overload; |
| 50 |
function InvTransform(var x,y:extended):boolean; overload; |
| 51 |
function InvTransform(var x,y:double):boolean; overload; |
| 52 |
procedure push; |
| 53 |
function det:double; |
| 54 |
function IsAffine:boolean; |
| 55 |
function IsSimilarPositive:boolean; |
| 56 |
private |
| 57 |
end; |
| 58 |
|
| 59 |
var |
| 60 |
currenttransform:TAffine=nil; |
| 61 |
{currenttransformInv:TAffine=nil;} |
| 62 |
|
| 63 |
//procedure push(a:TAffine); |
| 64 |
procedure pop; |
| 65 |
|
| 66 |
|
| 67 |
implementation |
| 68 |
uses objlist,math,base,vstack; |
| 69 |
|
| 70 |
procedure push(a:TAffine); |
| 71 |
begin |
| 72 |
a.next:=currenttransform; |
| 73 |
currenttransform:=a; |
| 74 |
end; |
| 75 |
|
| 76 |
procedure pop; |
| 77 |
var |
| 78 |
temp:TAffine; |
| 79 |
begin |
| 80 |
with currenttransform do |
| 81 |
begin |
| 82 |
temp:=next ; |
| 83 |
free; |
| 84 |
end; |
| 85 |
currenttransform:=temp; |
| 86 |
end; |
| 87 |
|
| 88 |
procedure TAffine.push; |
| 89 |
begin |
| 90 |
next:=currenttransform; |
| 91 |
currenttransform:=self |
| 92 |
end; |
| 93 |
|
| 94 |
|
| 95 |
|
| 96 |
class function TAffine.NewInstance: TObject; |
| 97 |
begin |
| 98 |
result:=InitInstance(getMemory(instancesize)); |
| 99 |
end; |
| 100 |
|
| 101 |
procedure TAffine.FreeInstance; |
| 102 |
begin |
| 103 |
freeMemory(instanceSize) |
| 104 |
end; |
| 105 |
|
| 106 |
|
| 107 |
constructor TAffine.create; |
| 108 |
begin |
| 109 |
inherited create; |
| 110 |
xx := 1.0; |
| 111 |
yy := 1.0; |
| 112 |
zz := 1.0; |
| 113 |
oo := 1.0; |
| 114 |
end; |
| 115 |
|
| 116 |
{ |
| 117 |
procedure TAffine.make(transform:TObjectList); |
| 118 |
var |
| 119 |
i:integer; |
| 120 |
p:TTransformTerm; |
| 121 |
begin |
| 122 |
i:=0; |
| 123 |
while (i<transform.count) do |
| 124 |
begin |
| 125 |
TObject(p):=transform.items[i]; |
| 126 |
p.exec(self); |
| 127 |
inc(i) |
| 128 |
end; |
| 129 |
end; |
| 130 |
} |
| 131 |
|
| 132 |
function TAffine.transform(var x,y:extended):boolean; overload; |
| 133 |
var |
| 134 |
cx,cy, c:extended; |
| 135 |
begin |
| 136 |
result:=true; |
| 137 |
if self=nil then exit; |
| 138 |
try |
| 139 |
cx := xx * x + xy * y + xo; |
| 140 |
cy := yx * x + yy * y + yo; |
| 141 |
c := ox * x + oy * y + oo; |
| 142 |
x:=cx / c; |
| 143 |
y:=cy / c; |
| 144 |
except |
| 145 |
{$IFNDEF Windows} asm finit end; set8087cw(controlword); {$ENDIF} |
| 146 |
//x:=cx; |
| 147 |
//y:=cy; |
| 148 |
result:=false; |
| 149 |
//setexception(-3009) |
| 150 |
end; |
| 151 |
end; |
| 152 |
|
| 153 |
function TAffine.transform(var x,y:double):boolean; overload; |
| 154 |
var |
| 155 |
cx,cy, c:double; |
| 156 |
begin |
| 157 |
result:=true; |
| 158 |
if self=nil then exit; |
| 159 |
try |
| 160 |
cx := xx * x + xy * y + xo; |
| 161 |
cy := yx * x + yy * y + yo; |
| 162 |
c := ox * x + oy * y + oo; |
| 163 |
x:=cx / c; |
| 164 |
y:=cy / c; |
| 165 |
except |
| 166 |
{$IFNDEF Windows} asm finit end; set8087cw(controlword); {$ENDIF} |
| 167 |
//x:=cx; |
| 168 |
//y:=cy; |
| 169 |
result:=false; |
| 170 |
//setexception(-3009) |
| 171 |
end; |
| 172 |
end; |
| 173 |
|
| 174 |
function TAffine.InvTransform(var x,y:extended):boolean; overload; |
| 175 |
var |
| 176 |
cx,cy,c:extended; |
| 177 |
dx,dy,dd:extended; |
| 178 |
begin |
| 179 |
result:=true; |
| 180 |
if self=nil then exit; |
| 181 |
try |
| 182 |
{ |
| 183 |
cx:= x*(yy * oo - oy * yo) -y*(xy * oo - oy * xo) +(xy * yo - yy * xo); |
| 184 |
cy:=-x*(yx * oo - ox * yo) +y*(xx * oo - ox * xo) -(xx * yo - yx * xo); |
| 185 |
c:= x*(yx * oy - ox * yy) -y*(xx * oy - ox * xy) +(xx * yy - yx * xy); |
| 186 |
} |
| 187 |
|
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dx:= oo * x - xo; |
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dy:= oo * y - yo; |
| 190 |
dd:= x * yo - y * xo; |
| 191 |
cx:= yy * dx - xy * dy - oy * dd; |
| 192 |
cy:= xx * dy - yx * dx + ox * dd; |
| 193 |
c := (xx * yy - yx * xy) - ox * (x * yy - y * xy) + oy * ( x * yx - y * xx); |
| 194 |
|
| 195 |
x:=cx/c; |
| 196 |
y:=cy/c; |
| 197 |
|
| 198 |
except |
| 199 |
{$IFNDEF Windows} asm finit end; set8087cw(controlword); {$ENDIF} |
| 200 |
result:=false; |
| 201 |
end; |
| 202 |
end; |
| 203 |
|
| 204 |
function TAffine.InvTransform(var x,y:double):boolean; overload; |
| 205 |
var |
| 206 |
cx,cy,c:double; |
| 207 |
dx,dy,dd:double; |
| 208 |
begin |
| 209 |
result:=true; |
| 210 |
if self=nil then exit; |
| 211 |
try |
| 212 |
{ |
| 213 |
cx:= x*(yy * oo - oy * yo) -y*(xy * oo - oy * xo) +(xy * yo - yy * xo); |
| 214 |
cy:=-x*(yx * oo - ox * yo) +y*(xx * oo - ox * xo) -(xx * yo - yx * xo); |
| 215 |
c:= x*(yx * oy - ox * yy) -y*(xx * oy - ox * xy) +(xx * yy - yx * xy); |
| 216 |
} |
| 217 |
|
| 218 |
dx:= oo * x - xo; |
| 219 |
dy:= oo * y - yo; |
| 220 |
dd:= x * yo - y * xo; |
| 221 |
cx:= yy * dx - xy * dy - oy * dd; |
| 222 |
cy:= xx * dy - yx * dx + ox * dd; |
| 223 |
c := (xx * yy - yx * xy) - ox * (x * yy - y * xy) + oy * ( x * yx - y * xx); |
| 224 |
|
| 225 |
x:=cx/c; |
| 226 |
y:=cy/c; |
| 227 |
|
| 228 |
except |
| 229 |
{$IFNDEF Windows} asm finit end; set8087cw(controlword); {$ENDIF} |
| 230 |
result:=false; |
| 231 |
end; |
| 232 |
end; |
| 233 |
|
| 234 |
procedure TAffine.scale1(a:double); |
| 235 |
begin |
| 236 |
xx:=xx*a; |
| 237 |
xy:=xy*a; |
| 238 |
xz:=xz*a; |
| 239 |
xo:=xo*a; |
| 240 |
|
| 241 |
yx:=yx*a; |
| 242 |
yy:=yy*a; |
| 243 |
yz:=yz*a; |
| 244 |
yo:=yo*a; |
| 245 |
|
| 246 |
zx:=zx*a; |
| 247 |
zy:=zy*a; |
| 248 |
zz:=zz*a; |
| 249 |
zo:=zo*a; |
| 250 |
|
| 251 |
end; |
| 252 |
|
| 253 |
|
| 254 |
procedure TAffine.scale(a,b:double) ; |
| 255 |
begin |
| 256 |
xx:=xx*a; |
| 257 |
xy:=xy*a; |
| 258 |
xz:=xz*a; |
| 259 |
xo:=xo*a; |
| 260 |
|
| 261 |
yx:=yx*b; |
| 262 |
yy:=yy*b; |
| 263 |
yz:=yz*b; |
| 264 |
yo:=yo*b; |
| 265 |
end; |
| 266 |
|
| 267 |
|
| 268 |
procedure TAffine.shift(a,b:double) ; |
| 269 |
begin |
| 270 |
xx:= xx + ox * a; |
| 271 |
xy:= xy + oy * a; |
| 272 |
xz:= xz + oz * a; |
| 273 |
xo:= xo + oo * a; |
| 274 |
|
| 275 |
yx:= yx + ox * b; |
| 276 |
yy:= yy + oy * b; |
| 277 |
yz:= yz + oz * b; |
| 278 |
yo:= yo + oo * b; |
| 279 |
end; |
| 280 |
|
| 281 |
procedure TAffine.shift(c:complex);overload; |
| 282 |
begin |
| 283 |
shift(c.x, c.y) |
| 284 |
end; |
| 285 |
|
| 286 |
procedure TAffine.rotate(t:double); |
| 287 |
begin |
| 288 |
rotate2(cos(t),sin(t)) |
| 289 |
end; |
| 290 |
|
| 291 |
procedure TAffine.rotate2(ct,st:double); |
| 292 |
var |
| 293 |
Nxx,Nxy,Nxz,Nxo,Nyx,Nyy,Nyz,Nyo:double; |
| 294 |
begin |
| 295 |
Nxx := xx * ct - yx * st; |
| 296 |
Nxy := xy * ct - yy * st; |
| 297 |
Nxz := xz * ct - yz * st; |
| 298 |
Nxo := xo * ct - yo * st; |
| 299 |
|
| 300 |
Nyx := yx * ct + xx * st; |
| 301 |
Nyy := yy * ct + xy * st; |
| 302 |
Nyz := yz * ct + xz * st; |
| 303 |
Nyo := yo * ct + xo * st; |
| 304 |
|
| 305 |
xx := Nxx; |
| 306 |
xy := Nxy; |
| 307 |
xz := Nxz; |
| 308 |
xo := Nxo; |
| 309 |
|
| 310 |
yx := Nyx; |
| 311 |
yy := Nyy; |
| 312 |
yz := Nyz; |
| 313 |
yo := Nyo; |
| 314 |
end; |
| 315 |
|
| 316 |
procedure TAffine.cmlt(const c:complex); |
| 317 |
begin |
| 318 |
rotate2(c.x,c.y) |
| 319 |
end ; |
| 320 |
|
| 321 |
procedure TAffine.shear(t:double); |
| 322 |
var |
| 323 |
tt:double; |
| 324 |
begin |
| 325 |
tt:=math.tan(t); |
| 326 |
xx := xx + yx * tt; |
| 327 |
xy := xy + yy * tt; |
| 328 |
xz := xz + yz * tt; |
| 329 |
xo := xo + yo * tt; |
| 330 |
end; |
| 331 |
|
| 332 |
procedure TAffine.revmlt(m:TAffine); |
| 333 |
var |
| 334 |
Nxx,Nxy,Nxz,Nyx,Nyy,Nyz,Nzx,Nzy,Nzz,Nxo,Nyo,Nzo,Nox,Noy,Noz,Noo: double; |
| 335 |
begin |
| 336 |
Nxx := m.xx * xx + m.yx * xy + m.zx * xz + m.ox * xo; |
| 337 |
Nxy := m.xy * xx + m.yy * xy + m.zy * xz + m.oy * xo; |
| 338 |
Nxz := m.xz * xx + m.yz * xy + m.zz * xz + m.oz * xo; |
| 339 |
Nxo := m.xo * xx + m.yo * xy + m.zo * xz + m.oo * xo; |
| 340 |
|
| 341 |
Nyx := m.xx * yx + m.yx * yy + m.zx * yz + m.ox * yo; |
| 342 |
Nyy := m.xy * yx + m.yy * yy + m.zy * yz + m.oy * yo; |
| 343 |
Nyz := m.xz * yx + m.yz * yy + m.zz * yz + m.oz * yo; |
| 344 |
Nyo := m.xo * yx + m.yo * yy + m.zo * yz + m.oo * yo; |
| 345 |
|
| 346 |
Nzx := m.xx * zx + m.yx * zy + m.zx * zz + m.ox * zo; |
| 347 |
Nzy := m.xy * zx + m.yy * zy + m.zy * zz + m.oy * zo; |
| 348 |
Nzz := m.xz * zx + m.yz * zy + m.zz * zz + m.oz * zo; |
| 349 |
Nzo := m.xo * zx + m.yo * zy + m.zo * zz + m.oo * zo; |
| 350 |
|
| 351 |
Nox := m.xx * ox + m.yx * oy + m.zx * oz + m.ox * oo; |
| 352 |
Noy := m.xy * ox + m.yy * oy + m.zy * oz + m.oy * oo; |
| 353 |
Noz := m.xz * ox + m.yz * oy + m.zz * oz + m.oz * oo; |
| 354 |
Noo := m.xo * ox + m.yo * oy + m.zo * oz + m.oo * oo; |
| 355 |
|
| 356 |
xx := Nxx; |
| 357 |
xy := Nxy; |
| 358 |
xz := Nxz; |
| 359 |
xo := Nxo; |
| 360 |
|
| 361 |
yx := Nyx; |
| 362 |
yy := Nyy; |
| 363 |
yz := Nyz; |
| 364 |
yo := Nyo; |
| 365 |
|
| 366 |
zx := Nzx; |
| 367 |
zy := Nzy; |
| 368 |
zz := Nzz; |
| 369 |
zo := Nzo; |
| 370 |
|
| 371 |
ox := Nox; |
| 372 |
oy := Noy; |
| 373 |
oz := Noz; |
| 374 |
oo := Noo; |
| 375 |
end ; |
| 376 |
|
| 377 |
procedure TAffine.mlt(m:TAffine); |
| 378 |
var |
| 379 |
Nxx,Nxy,Nxz,Nyx,Nyy,Nyz,Nzx,Nzy,Nzz,Nxo,Nyo,Nzo,Nox,Noy,Noz,Noo:double; |
| 380 |
begin |
| 381 |
Nxx := xx * m.xx + yx * m.xy + zx * m.xz + ox * m.xo; |
| 382 |
Nxy := xy * m.xx + yy * m.xy + zy * m.xz + oy * m.xo; |
| 383 |
Nxz := xz * m.xx + yz * m.xy + zz * m.xz + oz * m.xo; |
| 384 |
Nxo := xo * m.xx + yo * m.xy + zo * m.xz + oo * m.xo; |
| 385 |
|
| 386 |
Nyx := xx * m.yx + yx * m.yy + zx * m.yz + ox * m.yo; |
| 387 |
Nyy := xy * m.yx + yy * m.yy + zy * m.yz + oy * m.yo; |
| 388 |
Nyz := xz * m.yx + yz * m.yy + zz * m.yz + oz * m.yo; |
| 389 |
Nyo := xo * m.yx + yo * m.yy + zo * m.yz + oo * m.yo; |
| 390 |
|
| 391 |
Nzx := xx * m.zx + yx * m.zy + zx * m.zz + ox * m.zo; |
| 392 |
Nzy := xy * m.zx + yy * m.zy + zy * m.zz + oy * m.zo; |
| 393 |
Nzz := xz * m.zx + yz * m.zy + zz * m.zz + oz * m.zo; |
| 394 |
Nzo := xo * m.zx + yo * m.zy + zo * m.zz + oo * m.zo; |
| 395 |
|
| 396 |
Nox := xx * m.ox + yx * m.oy + zx * m.oz + ox * m.oo; |
| 397 |
Noy := xy * m.ox + yy * m.oy + zy * m.oz + oy * m.oo; |
| 398 |
Noz := xz * m.ox + yz * m.oy + zz * m.oz + oz * m.oo; |
| 399 |
Noo := xo * m.ox + yo * m.oy + zo * m.oz + oo * m.oo; |
| 400 |
|
| 401 |
xx := Nxx; |
| 402 |
xy := Nxy; |
| 403 |
xz := Nxz; |
| 404 |
xo := Nxo; |
| 405 |
|
| 406 |
yx := Nyx; |
| 407 |
yy := Nyy; |
| 408 |
yz := Nyz; |
| 409 |
yo := Nyo; |
| 410 |
|
| 411 |
zx := Nzx; |
| 412 |
zy := Nzy; |
| 413 |
zz := Nzz; |
| 414 |
zo := Nzo; |
| 415 |
|
| 416 |
ox := Nox; |
| 417 |
oy := Noy; |
| 418 |
oz := Noz; |
| 419 |
oo := Noo; |
| 420 |
end ; |
| 421 |
|
| 422 |
constructor TAffine.create(a:Tarray2N);overload; |
| 423 |
begin |
| 424 |
inherited create; |
| 425 |
if (a.size1=4) and (a.size2=4) then |
| 426 |
with a do |
| 427 |
begin |
| 428 |
xx:=elements^[0]; |
| 429 |
xy:=elements^[1*size2]; |
| 430 |
xz:=elements^[2*size2]; |
| 431 |
xo:=elements^[3*size2]; |
| 432 |
yx:=elements^[ 1]; |
| 433 |
yy:=elements^[1*size2+1]; |
| 434 |
yz:=elements^[2*size2+1]; |
| 435 |
yo:=elements^[3*size2+1]; |
| 436 |
zx:=elements^[ 2]; |
| 437 |
zy:=elements^[1*size2+2]; |
| 438 |
zz:=elements^[2*size2+2]; |
| 439 |
zo:=elements^[3*size2+2]; |
| 440 |
ox:=elements^[ 3]; |
| 441 |
oy:=elements^[1*size2+3]; |
| 442 |
oz:=elements^[2*size2+3]; |
| 443 |
oo:=elements^[3*size2+3]; |
| 444 |
end |
| 445 |
else |
| 446 |
setexception(6201); |
| 447 |
end; |
| 448 |
|
| 449 |
procedure TAffine.store(a:TArray2N); |
| 450 |
begin |
| 451 |
if a.maxsize<4*4 then setexception(5002); |
| 452 |
with a do |
| 453 |
begin |
| 454 |
resize(4,4); |
| 455 |
|
| 456 |
elements^[0] := xx; |
| 457 |
elements^[1*size2] := xy; |
| 458 |
elements^[2*size2] := xz; |
| 459 |
elements^[3*size2] := xo; |
| 460 |
elements^[ 1] := yx; |
| 461 |
elements^[1*size2+1] := yy; |
| 462 |
elements^[2*size2+1] := yz; |
| 463 |
elements^[3*size2+1] := yo; |
| 464 |
elements^[ 2] := zx; |
| 465 |
elements^[1*size2+2] := zy; |
| 466 |
elements^[2*size2+2] := zz; |
| 467 |
elements^[3*size2+2] := zo; |
| 468 |
elements^[ 3] := ox; |
| 469 |
elements^[1*size2+3] := oy; |
| 470 |
elements^[2*size2+3] := oz; |
| 471 |
elements^[3*size2+3] := oo; |
| 472 |
end; |
| 473 |
|
| 474 |
end; |
| 475 |
|
| 476 |
procedure TAffine.mlt(a:TArray2N);overload; |
| 477 |
var |
| 478 |
m:TAffine; |
| 479 |
begin |
| 480 |
m:=Taffine.create(a); |
| 481 |
mlt(m); |
| 482 |
m.free; |
| 483 |
end; |
| 484 |
|
| 485 |
|
| 486 |
function TAffine.IsAffine:boolean; |
| 487 |
begin |
| 488 |
result:=(ox=0) and (oy=0) and (oz=0) |
| 489 |
end; |
| 490 |
|
| 491 |
function TAffine.det:double; |
| 492 |
begin |
| 493 |
det:=oo*(xx*yy - xy*yx) +ox*(xy*yo - yy*xo) + oy*(yx*xo - xx*yo) ; |
| 494 |
end ; |
| 495 |
|
| 496 |
function TAffine.IsSimilarPositive:boolean; |
| 497 |
var |
| 498 |
s,t,u:double; |
| 499 |
begin |
| 500 |
result:=true; |
| 501 |
if self = nil then exit; |
| 502 |
|
| 503 |
s:=xx*xx + yx*yx; |
| 504 |
t:=yy*yy + xy*xy; |
| 505 |
u:=xx*xy + yx*yy; |
| 506 |
result:=isAffine and (s=t)and (abs(u/t)<1e-2) and (det>0) |
| 507 |
|
| 508 |
end; |
| 509 |
|
| 510 |
Constructor TAffine.Create(a:TArray2C); overload; |
| 511 |
var |
| 512 |
b:TArray2N; |
| 513 |
begin |
| 514 |
b:=a.Array2N; |
| 515 |
try |
| 516 |
Create(b); |
| 517 |
finally |
| 518 |
b.free; |
| 519 |
end; |
| 520 |
end; |
| 521 |
|
| 522 |
|
| 523 |
procedure TAffine.scale1(a:complex); overload; |
| 524 |
begin |
| 525 |
scale1(abs(a)); |
| 526 |
rotate(arg(a)) |
| 527 |
end; |
| 528 |
|
| 529 |
procedure TAffine.mlt(a:TArray2C);overload; |
| 530 |
var |
| 531 |
m:TAffine; |
| 532 |
begin |
| 533 |
m:=Taffine.create(a); |
| 534 |
mlt(m); |
| 535 |
m.free; |
| 536 |
end; |
| 537 |
|
| 538 |
procedure TAffine.store(a:TArray2C); overload; |
| 539 |
var |
| 540 |
n:TArray2N; |
| 541 |
i:integer; |
| 542 |
begin |
| 543 |
with a do |
| 544 |
n:=TArray2N.create(Lbound(1), Ubound(1), Lbound(2), Ubound(2)); |
| 545 |
try |
| 546 |
store(n); |
| 547 |
with a do |
| 548 |
for i:=0 to size -1 do |
| 549 |
elements^[i]:=n.elements^[i]; |
| 550 |
finally |
| 551 |
n.free; |
| 552 |
end; |
| 553 |
end; |
| 554 |
|
| 555 |
end. |
| 556 |
|
| 557 |
|
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