-+-+-+-+-+-+-+-+ START OF PART 105 -+-+-+-+-+-+-+-+ XFq(,)j Fi(result)0 1325 y Fl(\254)h Fi(matrix1)h(*)f(matrix2)p XFq(.)p 1549 1422 318 2 v 1549 1501 2 80 v 1563 1475 a Fb(ptk_crossv3)p X1864 1501 V 1549 1503 318 2 v 0 1577 a Fq(Ppoint3)10 b Fp(ptk_cr)o(ossv3)h XFq(\()h(Ppoint3)d(*)p Fa(v1)p Fq(,)i(Ppoint3)f(*)p Fa(v2)g XFq(\))25 1666 y(IN)50 b(Ppoint3)9 b(*)50 b Fa(v1)f Fq(3D)11 Xb(vector)n(.)25 1715 y(IN)50 b(Ppoint3)9 b(*)50 b Fa(v2)f Fq(3D)11 Xb(vector)n(.)0 1791 y(This)16 b(function)e(evaluates)i(the)g(cross)g(product V) Xf(of)g(the)h(two)f(3D)g(vectors)h Fi(v1)g Fq(and)g Fi(v2)p XFq(,)i(returning)c(the)h(new)h(vector)g(as)g(the)0 1840 y(function)c(result. V) X22 b(Since)14 b(a)g(local)g(copy)f(is)h(made)h(statements)f(such)g(as)g XFi(v2)k(==)h(ptk_crossv\(v1,)h(v2\))14 b Fq(will)f(produce)g(the)0 X1890 y(correct)f(answer)n(.)p 1625 1987 241 2 v 1625 2066 2 X80 v 1639 2041 a Fb(ptk_dotv)p 1864 2066 V 1625 2068 241 2 Xv 0 2142 a Fq(P\257oat)f Fp(ptk_dotv)g Fq(\()g(Ppoint)f(*)p XFa(v1)p Fq(,)h(Ppoint)f(*)p Fa(v2)g Fq(\))25 2231 y(IN)50 b(Ppoint)9 Xb(*)50 b Fa(v1)f Fq(2D)11 b(vector)n(.)25 2281 y(IN)50 b(Ppoint)9 Xb(*)50 b Fa(v2)f Fq(2D)11 b(vector)n(.)0 2356 y(Evaluates)h(the)f(dot)f X(product)h(of)g(the)g(two)f(2D)i(vectors)f Fi(v1)h Fq(and)f XFi(v2)p Fq(,)h(returning)e(it)g(as)i(the)f(value)h(of)f(the)g(function.)0 X2956 y(13th)22 b(May)m(,)12 b(1992)636 b(69)839 b(2.01)p eop X%%Page: 70 71 Xbop 0 -137 a Fq(PHIGS)11 b(T)m(oolkit)f(User)i(Manual)1235 Xb(PTK/100)p 1595 -19 271 2 v 1595 60 2 80 v 1610 35 a Fb(ptk_dotv3)p X1864 60 V 1595 62 271 2 v 0 136 a Fq(P\257oat)11 b Fp(ptk_dotv3)g XFq(\()g(Ppoint3)e(*)p Fa(v1)p Fq(,)i(Ppoint3)f(*)p Fa(v2)g XFq(\))25 225 y(IN)50 b(Ppoint3)9 b(*)50 b Fa(v1)f Fq(3D)11 Xb(vector)n(.)25 275 y(IN)50 b(Ppoint3)9 b(*)50 b Fa(v2)f Fq(3D)11 Xb(vector)n(.)0 350 y(This)f(function)e(evaluates)j(the)f(dot)f(product)f(of) Vi X(the)g(two)f(3D)h(vectors)g Fi(v1)g Fq(and)g Fi(v2)p Fq(,)h(returning)e(it)g X(as)h(the)g(value)g(of)g(the)f(function.)p 1602 447 264 2 v X1602 526 2 80 v 1616 501 a Fb(ptk_equal)p 1864 526 V 1602 528 X264 2 v 0 602 a Fq(ptkboolean)h Fp(ptk_equal)h Fq(\()g(P\257oat)g XFa(one)p Fq(,)g(P\257oat)g Fa(two)h Fq(\))25 691 y(IN)50 b(P\257oat)f XFa(one)g Fq(\257oating)10 b(point)g(number)n(.)25 741 y(IN)50 Xb(P\257oat)f Fa(two)54 b Fq(\257oating)10 b(point)g(number)n(.)0 X816 y(This)k(function)f(returns)h(TRUE)h(if)f Fi(one)h Fq(and)f XFi(two)h Fq(are)h(equal,)f(or)f(their)g(dif)o(ference)h(is)f(less)h(than)f X(the)g(global)g(constant)f(tol-)0 866 y(erance)e Fi(ptkveps)p XFq(.)17 b(This)10 b(is)g(a)g(global)f(variable)h(of)f(type)h XFi(Pfloat)h Fq(which)f(may)g(be)h(changed)f(by)g(the)g(application.)j(The)e X(default)0 915 y(value)g(of)g Fi(ptkveps)i Fq(is)e(1.0e-7.)p X1307 1013 560 2 v 1307 1092 2 80 v 1321 1066 a Fb(ptk_evalviewm)o(appi)o(ng) Vp X1864 1092 V 1307 1094 560 2 v 0 1168 a Fq(void)f Fp(ptk_evalviewmapping)f XFq(\()19 b(Plimit)f(*)p Fa(wlimits)p Fq(,)j(Plimit)c(*)p Fa(vlimits)p XFq(,)j(Pcomptype)f Fa(operation)p Fq(,)486 1218 y(Pmatrix)11 Xb Fa(matrix)p Fq(,)g(Pint)f(*)p Fa(error)g Fq(\))25 1268 y(IN)91 Xb(Plimit)10 b(*)103 b Fa(wlimits)91 b Fq(window)10 b(limits.)25 X1318 y(IN)91 b(Plimit)10 b(*)103 b Fa(vlimits)98 b Fq(viewport)10 Xb(limits.)25 1368 y(IN)91 b(Pcomptype)50 b Fa(operation)d Fq(concatenation)1 V1 Xb(operation.)25 1418 y(OUT)50 b(Pmatrix)105 b Fa(matrix)100 Xb Fq(3x3)10 b(matrix.)25 1467 y(OUT)50 b(Pint)10 b(*)138 b XFa(error)121 b Fq(error)11 b(code.)0 1536 y(This)h(function)e(evaluates)j(a) Vf X(2d)f(PHIGS)h(view)g(mapping)f(matrix.)17 b(If)12 b(the)f(function)g X(succeeds,)j Fi(error)e Fq(is)g(set)g(to)f Fi(ptkcpcok)p Fq(.)0 X1586 y(Otherwise,)g Fi(error)i Fq(is)e(329)f(if)h(the)g(window)g(limits)f X(are)i(not)e(valid,)h(and)g(330)g(if)g(the)g(viewport)f(limits)g(are)i(not)e X(valid.)p 1277 1683 590 2 v 1277 1762 2 80 v 1291 1737 a Fb(ptk_evalviewm)o X(appi)o(ng3)p 1864 1762 V 1277 1764 590 2 v 0 1838 a Fq(void)g XFp(ptk_evalviewmapping3)f Fq(\()18 b(Plimit3)e(*)p Fa(wlimits)p XFq(,)k(Plimit3)c(*)p Fa(vlimits)p Fq(,)j(Pprojtype)d Fa(viewtype)p XFq(,)507 1888 y(Ppoint3)10 b(*)p Fa(refpoint)p Fq(,)f(P\257oat)i XFa(vplanedist)p Fq(,)d(Pcomptype)k Fa(operation)p Fq(,)507 X1938 y(Pmatrix3)f Fa(matrix)p Fq(,)g(Pint)f(*)p Fa(error)g XFq(\))25 1988 y(IN)91 b(Plimit3)10 b(*)82 b Fa(wlimits)102 Xb Fq(window)10 b(limits.)25 2038 y(IN)91 b(Plimit3)10 b(*)82 Xb Fa(vlimits)109 b Fq(viewport)10 b(limits.)25 2088 y(IN)91 Xb(Pprojtype)74 b Fa(viewtype)69 b Fq(projection)10 b(type.)25 X2138 y(IN)91 b(Ppoint3)10 b(*)75 b Fa(refpoint)88 b Fq(projection)10 Xb(reference)j(point.)25 2188 y(IN)91 b(P\257oat)140 b Fa(vplanedist)46 Xb Fq(view)11 b(plane)h(distance.)25 2238 y(IN)91 b(Pcomptype)50 Xb Fa(operation)58 b Fq(concatenation)11 b(operation.)25 2287 Xy(OUT)50 b(Pmatrix3)84 b Fa(matrix)111 b Fq(4x4)11 b(matrix.)25 X2337 y(OUT)50 b(Pint)10 b(*)138 b Fa(error)132 b Fq(error)11 Xb(code.)0 2406 y(This)g(function)f(evaluates)i(a)g(3D)f(PHIGS)h(view)f X(mapping)g(matrix.)k(If)c(the)g(function)f(succeeds,)j Fi(error)g XFq(is)e(set)h(to)e Fi(ptkcpcok)p Fq(.)0 2456 y(Otherwise,)i XFi(error)h Fq(is)e(329)h(if)f(the)h(window)e(limits)h(are)h(not)f(valid,)h X(336)f(if)g(the)h(back)g(plane)g(is)g(in)f(front)f(of)i(front)f(plane,)h(330 V) X0 2506 y(if)i(the)g(viewport)f(limits)g(are)i(not)f(valid,)g(335)g(if)g(the) Vg X(projection)f(reference)j(point)d(is)h(on)g(the)g(view)g(plane,)h(and)g(340) Vf X(if)f(the)0 2556 y(projection)d(reference)j(point)c(is)i(between)h(front)e X(and)i(back)f(planes.)0 2956 y(13th)22 b(May)m(,)12 b(1992)636 Xb(70)839 b(2.01)p eop X%%Page: 71 72 Xbop 0 -137 a Fq(PHIGS)11 b(T)m(oolkit)f(User)i(Manual)1235 Xb(PTK/100)p 1257 -19 610 2 v 1257 60 2 80 v 1271 35 a Fb(ptk_evalviewori)o X(ent)o(ati)o(on)p 1864 60 V 1257 62 610 2 v 0 136 a Fq(void)10 Xb Fp(ptk_evalvieworientation)g Fq(\()j(Ppoint)g(*)p Fa(viewrefpoint)p XFq(,)f(Ppoint)g(*)p Fa(viewupvector)p Fq(,)g(Pcomptype)526 X186 y Fa(operation)p Fq(,)c(Pmatrix)j Fa(matrix)p Fq(,)g(Pint)g(*)p XFa(error)e Fq(\))25 237 y(IN)91 b(Ppoint)10 b(*)96 b Fa(viewrefpoint)64 Xb Fq(view)11 b(reference)i(point.)25 287 y(IN)91 b(Ppoint)10 Xb(*)96 b Fa(viewupvector)46 b Fq(view)11 b(up)g(vector)n(.)25 X336 y(IN)91 b(Pcomptype)50 b Fa(operation)105 b Fq(concatenation)11 Xb(operation.)25 386 y(OUT)50 b(Pmatrix)105 b Fa(matrix)158 Xb Fq(3x3)11 b(matrix.)25 436 y(OUT)50 b(Pint)10 b(*)138 b Fa(error)179 Xb Fq(error)11 b(code.)0 505 y(This)j(function)e(computes)j(a)f(2D)g(PHIGS)g X(view)g(orientation)e(matrix)i(on)f(the)h(basis)g(of)g(a)h(speci\256ed)f X(view)g(reference)i(point)0 555 y(\()p Fi(viewrefpoint)p Fq(\))g(and)e(a)g X(view)g(up)g(vector)g(\()p Fi(viewupvector)p Fq(\).)25 b(If)14 Xb(the)g(function)e(succeeds,)17 b Fi(error)e Fq(is)f(set)g(to)f XFi(ptkcpcok)p Fq(.)0 605 y(Otherwise,)e Fi(error)i Fq(is)e(63)g(if)f(the)i X(view)f(up)g(vector)g(is)g(null.)p 1227 702 640 2 v 1227 781 X2 80 v 1241 755 a Fb(ptk_evalvieworie)o(nta)o(ti)o(on3)p 1864 X781 V 1227 783 640 2 v 0 857 a Fq(void)f Fp(ptk_evalvieworientation3)f XFq(\()329 b(Ppoint3)e(*)p Fa(viewref-)546 907 y(point)p Fq(,)16 Xb(Ppoint3)f(*)p Fa(viewplanenor)o(mal)p Fq(,)g(Ppoint3)f(*)p XFa(viewupvector)p Fq(,)546 956 y(Pcomptype)e Fa(operation)p XFq(,)c(Pmatrix3)j Fa(matrix)p Fq(,)g(Pint)f(*)p Fa(error)g XFq(\))25 1009 y(IN)91 b(Ppoint3)10 b(*)75 b Fa(viewrefpoint)123 Xb Fq(view)11 b(reference)i(point.)25 1058 y(IN)91 b(Ppoint3)10 Xb(*)75 b Fa(viewplanenor)o(mal)47 b Fq(view)11 b(plane)h(normal.)25 X1108 y(IN)91 b(Ppoint3)10 b(*)75 b Fa(viewupvector)105 b Fq(view)11 Xb(up)g(vector)n(.)25 1158 y(IN)91 b(Pcomptype)50 b Fa(operation)164 Xb Fq(concatenation)11 b(operation.)25 1208 y(OUT)50 b(Pmatrix3)84 Xb Fa(matrix)217 b Fq(4x4)11 b(matrix.)25 1258 y(OUT)50 b(Pint)10 Xb(*)138 b Fa(error)238 b Fq(error)11 b(code.)0 1327 y(This)j(function)e X(computes)j(a)f(3D)g(PHIGS)g(view)g(orientation)e(matrix)i(on)f(the)h(basis) Vg X(of)g(a)h(speci\256ed)f(view)g(reference)i(point)0 1376 y(\()p XFi(viewrefpoint)p Fq(\),)d(a)f(view)f(plane)g(normal)f(\()p XFi(viewplanenormal)p Fq(\))k(and)d(a)g(view)g(up)g(vector)g(\()p XFi(viewupvector)p Fq(\).)17 b(If)11 b(the)g(func-)0 1426 y(tion)e(succeeds,) Vk XFi(error)f Fq(is)e(set)h(to)f Fi(ptkcpcok)p Fq(.)17 b(Otherwise,)11 Xb Fi(error)g Fq(is)g(61)f(if)g(the)h(view)f(plane)h(normal)f(is)h(null,)f(63 V) Xg(if)g(the)g(view)0 1476 y(up)h(vector)g(is)g(null,)g(and)g(58)g(if)g(the)g X(cross)g(product)g(of)g(the)g(view)g(up)g(vector)g(and)g(the)h(view)f(plane) Vg X(normal)g(is)g(null.)p 1250 1573 616 2 v 1250 1652 2 80 v 1264 X1627 a Fb(ptk_examinestac)o(km)o(atri)o(x)p 1864 1652 V 1250 X1654 616 2 v 0 1728 a Fq(void)f Fp(ptk_examinestackmatrix)g XFq(\()h(Pmatrix)g Fa(matrix)g Fq(\))25 1828 y(IN)50 b(Pmatrix)f XFa(matrix)g Fq(3x3)10 b(matrix.)0 1928 y(This)h(function)f(returns)h(the)g X(top)f(entry)h(on)g(the)g(transformation)f(stack.)16 b(The)c(stack)g(is)f X(not)f(disturbed.)p 1220 2025 646 2 v 1220 2104 2 80 v 1234 X2078 a Fb(ptk_examinestac)o(kma)o(tri)o(x3)p 1864 2104 V 1220 X2106 646 2 v 0 2180 a Fq(void)g Fp(ptk_examinestackmatrix3)g XFq(\()h(Pmatrix3)g Fa(matrix)f Fq(\))25 2279 y(IN)50 b(Pmatrix3)f XFa(matrix)f Fq(4x4)11 b(matrix.)0 2379 y(This)g(function)f(returns)h(the)g X(top)f(entry)h(on)g(the)g(transformation)f(stack.)16 b(The)c(stack)g(is)f X(not)f(disturbed.)0 2956 y(13th)22 b(May)m(,)12 b(1992)636 Xb(71)839 b(2.01)p eop X%%Page: 72 73 Xbop 0 -137 a Fq(PHIGS)11 b(T)m(oolkit)f(User)i(Manual)1235 Xb(PTK/100)p 1436 -19 431 2 v 1436 60 2 80 v 1450 35 a Fb(ptk_invertmatri)o(x V) Xp 1864 60 V 1436 62 431 2 v 0 136 a Fq(void)10 b Fp(ptk_invertmatrix)g XFq(\()i(Pmatrix)e Fa(a)p Fq(,)i(Pmatrix)f Fa(ainverse)p Fq(,)e(Pint)h(*)p XFa(error)g Fq(\))25 225 y(IN)91 b(Pmatrix)49 b Fa(a)163 b Fq(3x3)11 Xb(matrix.)25 275 y(OUT)50 b(Pmatrix)f Fa(ainverse)e Fq(3x3)11 Xb(matrix.)25 324 y(OUT)50 b(Pint)10 b(*)82 b Fa(error)107 b XFq(error)12 b(code.)0 393 y(This)k(function)g(computes)g(the)h(inverse)f(of) Vg X(the)h(3)c Fl(\264)g Fq(3)k(matrix)f Fi(a)p Fq(,)i(returning)d(the)i(result) Vf X(in)g Fi(ainverse)p Fq(.)32 b(If)17 b(matrix)f Fi(a)g Fq(is)0 X443 y(singular)n(,)10 b(then)g Fi(error)h Fq(is)f(set)g(to)g XFl(-)p Fq(1,)g(and)g Fi(ainverse)i Fq(is)e(unde\256ned,)g(otherwise)g XFi(error)h Fq(is)f(set)h(to)e Fi(ptkcpcok)p Fq(.)17 b(This)10 Xb(function)0 493 y(uses)i(Crout')n(s)e(method,)h(a)h(modi\256cation)e(of)h X(Gaussian)h(elimination.)p 1406 590 460 2 v 1406 669 2 80 v X1420 644 a Fb(ptk_invertmatri)o(x3)p 1864 669 V 1406 671 460 X2 v 0 745 a Fq(void)e Fp(ptk_invertmatrix3)g Fq(\()h(Pmatrix3)g XFa(a)p Fq(,)g(Pmatrix3)g Fa(ainverse)p Fq(,)e(Pint)h(*)p Fa(error)g XFq(\))25 834 y(IN)91 b(Pmatrix3)49 b Fa(a)163 b Fq(4x4)11 b(matrix.)25 X884 y(OUT)50 b(Pmatrix3)f Fa(ainverse)e Fq(4x4)11 b(matrix.)25 X934 y(OUT)50 b(Pint)10 b(*)103 b Fa(error)k Fq(error)11 b(code.)0 X1003 y(This)16 b(function)g(computes)g(the)h(inverse)f(of)g(the)h(4)c XFl(\264)g Fq(4)k(matrix)f Fi(a)p Fq(,)i(returning)d(the)i(result)f(in)g XFi(ainverse)p Fq(.)32 b(If)17 b(matrix)f Fi(a)g Fq(is)0 1052 Xy(singular)n(,)10 b(then)g Fi(error)h Fq(is)f(set)g(to)g Fl(-)p XFq(1,)g(and)g Fi(ainverse)i Fq(is)e(unde\256ned,)g(otherwise)g XFi(error)h Fq(is)f(set)h(to)e Fi(ptkcpcok)p Fq(.)17 b(This)10 Xb(function)0 1102 y(uses)i(Crout')n(s)e(method,)h(a)h(modi\256cation)e(of)h X(Gaussian)h(elimination.)p 1619 1199 248 2 v 1619 1279 2 80 Xv 1633 1253 a Fb(ptk_limit)p 1864 1279 V 1619 1281 248 2 v X0 1355 a Fq(Plimit)e Fp(ptk_limit)f Fq(\()i(P\257oat)h Fa(xmin)p XFq(,)f(P\257oat)g Fa(xmax)p Fq(,)g(P\257oat)h Fa(ymin)p Fq(,)f(P\257oat)g XFa(ymax)g Fq(\))25 1443 y(IN)50 b(P\257oat)f Fa(xmin)59 b Fq(minimum)11 Xb(x)g(coordinate.)25 1493 y(IN)50 b(P\257oat)f Fa(xmax)g Fq(maximum)12 Xb(x)f(coordinate.)25 1543 y(IN)50 b(P\257oat)f Fa(ymin)59 b XFq(minimum)11 b(y)g(coordinate.)25 1593 y(IN)50 b(P\257oat)f XFa(ymax)g Fq(maximum)12 b(y)f(coordinate.)0 1662 y(This)d(function)f(returns V) Xh(a)h(structure)f(representing)g(the)g(2D)g(rectangle)h(whose)g(corners)f X(are)i(de\256ned)e(by)h Fi(xmin,)18 b(xmax,)h(ymin,)0 1712 Xy(ymax)p Fq(.)p 1589 1809 278 2 v 1589 1888 2 80 v 1603 1862 Xa Fb(ptk_limit3)p 1864 1888 V 1589 1890 278 2 v 0 1964 a Fq(Plimit3)10 Xb Fp(ptk_limit3)f Fq(\()g(P\257oat)g Fa(xmin)p Fq(,)h(P\257oat)g XFa(xmax)p Fq(,)f(P\257oat)h Fa(ymin)p Fq(,)g(P\257oat)f Fa(ymax)p XFq(,)h(P\257oat)330 2014 y Fa(zmin)p Fq(,)h(P\257oat)g Fa(zmax)g XFq(\))25 2064 y(IN)50 b(P\257oat)f Fa(xmin)59 b Fq(minimum)11 Xb(x)g(coordinate.)25 2114 y(IN)50 b(P\257oat)f Fa(xmax)g Fq(maximum)12 Xb(x)f(coordinate.)25 2164 y(IN)50 b(P\257oat)f Fa(ymin)59 b XFq(minimum)11 b(y)g(coordinate.)25 2214 y(IN)50 b(P\257oat)f XFa(ymax)g Fq(maximum)12 b(y)f(coordinate.)25 2263 y(IN)50 b(P\257oat)f XFa(zmin)59 b Fq(minimum)11 b(z)h(coordinate.)25 2313 y(IN)50 Xb(P\257oat)f Fa(zmax)g Fq(maximum)12 b(z)g(coordinate.)0 2382 Xy(This)k(function)f(returns)g(a)i(structure)e(representing)h(the)g(3D)g X(volume)g(de\256ned)g(by)g Fi(xmin,)j(xmax,)g(ymin,)f(ymax,)h(zmin,)0 X2432 y(zmax)p Fq(.)0 2956 y(13th)j(May)m(,)12 b(1992)636 b(72)839 Xb(2.01)p eop X%%Page: 73 74 Xbop 0 -137 a Fq(PHIGS)11 b(T)m(oolkit)f(User)i(Manual)1235 Xb(PTK/100)p 1343 -19 524 2 v 1343 60 2 80 v 1357 35 a Fb(ptk_matrixtoma)o X(tri)o(x3)p 1864 60 V 1343 62 524 2 v 0 136 a Fq(void)10 b XFp(ptk_matrixtomatrix3)e Fq(\()k(Pmatrix)f Fa(mat)p Fq(,)g(Pmatrix3)g XFa(mat3)g Fq(\))25 225 y(IN)91 b(Pmatrix)70 b Fa(mat)f Fq(3x3)11 Xb(matrix.)25 275 y(OUT)50 b(Pmatrix3)f Fa(mat3)g Fq(4x4)11 Xb(matrix.)0 350 y(This)g(function)f(converts)h(the)g(3)e Fl(\264)g XFq(3)i(matrix)g Fi(mat)h Fq(to)f(the)g(4)e Fl(\264)g Fq(4)i(matrix)g XFi(mat3)p Fq(,)h(as)g(follows:)610 428 y Fg(0)610 503 y(@)667 X461 y Fk(a)42 b(b)g(c)667 511 y(d)h(e)g Fl(\246)667 561 y Fk(g)f(h)k(j)834 X428 y Fg(1)834 503 y(A)881 511 y Fl(\256)934 403 y Fg(0)934 X476 y(B)934 501 y(B)934 528 y(@)991 436 y Fk(a)c(b)f Fq(0)i XFk(c)991 486 y(d)g(e)g Fq(0)f Fl(\246)991 536 y Fq(0)g(0)f(1)h(0)991 X586 y Fk(g)g(h)f Fq(0)46 b Fk(j)1220 403 y Fg(1)1220 476 y(C)1220 X501 y(C)1220 528 y(A)p 1595 743 271 2 v 1595 822 2 80 v 1610 X796 a Fb(ptk_modv)p 1864 822 V 1595 824 271 2 v 0 898 a Fq(P\257oat)11 Xb Fp(ptk_modv)g Fq(\()g(Ppoint)f(*)p Fa(vec)g Fq(\))25 997 Xy(IN)50 b(Ppoint)9 b(*)50 b Fa(vec)f Fq(2D)11 b(vector)n(.)0 X1097 y(This)g(function)f(returns)h(the)g(modulus)f(of)h(the)g(2D)h(vector)f XFi(vec)p Fq(.)p 1566 1193 301 2 v 1566 1272 2 80 v 1580 1247 Xa Fb(ptk_modv3)p 1864 1272 V 1566 1274 301 2 v 0 1348 a Fq(P\257oat)g XFp(ptk_modv3)f Fq(\()i(Ppoint3)d(*)p Fa(vec)h Fq(\))25 1448 Xy(IN)50 b(Ppoint3)9 b(*)50 b Fa(vec)e Fq(3D)12 b(vector)n(.)0 X1548 y(Returns)f(the)g(modulus)f(of)h(the)g(vector)g Fi(vec)p XFq(.)p 1373 1644 494 2 v 1373 1723 2 80 v 1387 1698 a Fb(ptk_multiplym)o +-+-+-+-+-+-+-+- END OF PART 105 +-+-+-+-+-+-+-+-