_bezier.h

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00001 
00026 /* === S T A R T =========================================================== */
00027 
00028 #ifndef __ETL_BEZIER_H
00029 #define __ETL_BEZIER_H
00030 
00031 /* === H E A D E R S ======================================================= */
00032 
00033 #include "_curve_func.h"
00034 #include <cmath>                // for ldexp
00035 // #include <ETL/fixed>         // not used
00036 
00037 /* === M A C R O S ========================================================= */
00038 
00039 #define MAXDEPTH 64 /*  Maximum depth for recursion */
00040 
00041 /* take binary sign of a, either -1, or 1 if >= 0 */
00042 #define SGN(a)      (((a)<0) ? -1 : 1)
00043 
00044 /* find minimum of a and b */
00045 #ifndef MIN
00046 #define MIN(a,b)    (((a)<(b))?(a):(b))
00047 #endif
00048 
00049 /* find maximum of a and b */
00050 #ifndef MAX
00051 #define MAX(a,b)    (((a)>(b))?(a):(b))
00052 #endif
00053 
00054 #define BEZIER_EPSILON  (ldexp(1.0,-MAXDEPTH-1)) /*Flatness control value */
00055 //#define   BEZIER_EPSILON  0.00005 /*Flatness control value */
00056 #define DEGREE  3           /*  Cubic Bezier curve      */
00057 #define W_DEGREE 5          /*  Degree of eqn to find roots of */
00058 
00059 /* === T Y P E D E F S ===================================================== */
00060 
00061 /* === C L A S S E S & S T R U C T S ======================================= */
00062 
00063 _ETL_BEGIN_NAMESPACE
00064 
00065 template<typename V,typename T> class bezier;
00066 
00068 // This generic implementation uses the DeCasteljau algorithm.
00069 // Works for just about anything that has an affine combination function
00070 template <typename V,typename T=float>
00071 class bezier_base : public std::unary_function<T,V>
00072 {
00073 public:
00074     typedef V value_type;
00075     typedef T time_type;
00076 
00077 private:
00078     value_type a,b,c,d;
00079     time_type r,s;
00080 
00081 protected:
00082     affine_combo<value_type,time_type> affine_func;
00083 
00084 public:
00085     bezier_base():r(0.0),s(1.0) { }
00086     bezier_base(
00087         const value_type &a, const value_type &b, const value_type &c, const value_type &d,
00088         const time_type &r=0.0, const time_type &s=1.0):
00089         a(a),b(b),c(c),d(d),r(r),s(s) { sync(); }
00090 
00091     void sync()
00092     {
00093     }
00094 
00095     value_type
00096     operator()(time_type t)const
00097     {
00098         t=(t-r)/(s-r);
00099         return
00100         affine_func(
00101             affine_func(
00102                 affine_func(a,b,t),
00103                 affine_func(b,c,t)
00104             ,t),
00105             affine_func(
00106                 affine_func(b,c,t),
00107                 affine_func(c,d,t)
00108             ,t)
00109         ,t);
00110     }
00111 
00112     /*
00113     void evaluate(time_type t, value_type &f, value_type &df) const
00114     {
00115         t=(t-r)/(s-r);
00116 
00117         value_type p1 = affine_func(
00118                             affine_func(a,b,t),
00119                             affine_func(b,c,t)
00120                             ,t);
00121         value_type p2 = affine_func(
00122                             affine_func(b,c,t),
00123                             affine_func(c,d,t)
00124                         ,t);
00125 
00126         f = affine_func(p1,p2,t);
00127         df = (p2-p1)*3;
00128     }
00129     */
00130 
00131     void set_rs(time_type new_r, time_type new_s) { r=new_r; s=new_s; }
00132     void set_r(time_type new_r) { r=new_r; }
00133     void set_s(time_type new_s) { s=new_s; }
00134     const time_type &get_r()const { return r; }
00135     const time_type &get_s()const { return s; }
00136     time_type get_dt()const { return s-r; }
00137 
00138     bool intersect_hull(const bezier_base<value_type,time_type> &x)const
00139     {
00140         return 0;
00141     }
00142 
00144 
00164     time_type intersect(const bezier_base<value_type,time_type> &x, time_type near=0.0)const
00165     {
00166         return 0;
00167     }
00168 
00169     /* subdivide at some time t into 2 separate curves left and right
00170 
00171         b0 l1
00172         *       0+1 l2
00173         b1      *       1+2*1+2 l3
00174         *       1+2     *           0+3*1+3*2+3 l4,r1
00175         b2      *       1+2*2+2 r2  *
00176         *       2+3 r3  *
00177         b3 r4   *
00178         *
00179 
00180         0.1 2.3 ->  0.1 2 3 4 5.6
00181     */
00182 /*  void subdivide(bezier_base *left, bezier_base *right, const time_type &time = (time_type)0.5) const
00183     {
00184         time_type t = (time-r)/(s-r);
00185         bezier_base lt,rt;
00186 
00187         value_type temp;
00188 
00189         //1st stage points to keep
00190         lt.a = a;
00191         rt.d = d;
00192 
00193         //2nd stage calc
00194         lt.b = affine_func(a,b,t);
00195         temp = affine_func(b,c,t);
00196         rt.c = affine_func(c,d,t);
00197 
00198         //3rd stage calc
00199         lt.c = affine_func(lt.b,temp,t);
00200         rt.b = affine_func(temp,rt.c,t);
00201 
00202         //last stage calc
00203         lt.d = rt.a = affine_func(lt.c,rt.b,t);
00204 
00205         //set the time range for l,r (the inside values should be 1, 0 respectively)
00206         lt.r = r;
00207         rt.s = s;
00208 
00209         //give back the curves
00210         if(left) *left = lt;
00211         if(right) *right = rt;
00212     }
00213     */
00214     value_type &
00215     operator[](int i)
00216     { return (&a)[i]; }
00217 
00218     const value_type &
00219     operator[](int i) const
00220     { return (&a)[i]; }
00221 };
00222 
00223 
00224 #if 1
00225 // Fast float implementation of a cubic bezier curve
00226 template <>
00227 class bezier_base<float,float> : public std::unary_function<float,float>
00228 {
00229 public:
00230     typedef float value_type;
00231     typedef float time_type;
00232 private:
00233     affine_combo<value_type,time_type> affine_func;
00234     value_type a,b,c,d;
00235     time_type r,s;
00236 
00237     value_type _coeff[4];
00238     time_type drs; // reciprocal of (s-r)
00239 public:
00240     bezier_base():r(0.0),s(1.0),drs(1.0) { }
00241     bezier_base(
00242         const value_type &a, const value_type &b, const value_type &c, const value_type &d,
00243         const time_type &r=0.0, const time_type &s=1.0):
00244         a(a),b(b),c(c),d(d),r(r),s(s),drs(1.0/(s-r)) { sync(); }
00245 
00246     void sync()
00247     {
00248 //      drs=1.0/(s-r);
00249         _coeff[0]=                 a;
00250         _coeff[1]=           b*3 - a*3;
00251         _coeff[2]=     c*3 - b*6 + a*3;
00252         _coeff[3]= d - c*3 + b*3 - a;
00253     }
00254 
00255     // Cost Summary: 4 products, 3 sums, and 1 difference.
00256     inline value_type
00257     operator()(time_type t)const
00258     { t-=r; t*=drs; return _coeff[0]+(_coeff[1]+(_coeff[2]+(_coeff[3])*t)*t)*t; }
00259 
00260     void set_rs(time_type new_r, time_type new_s) { r=new_r; s=new_s; drs=1.0/(s-r); }
00261     void set_r(time_type new_r) { r=new_r; drs=1.0/(s-r); }
00262     void set_s(time_type new_s) { s=new_s; drs=1.0/(s-r); }
00263     const time_type &get_r()const { return r; }
00264     const time_type &get_s()const { return s; }
00265     time_type get_dt()const { return s-r; }
00266 
00268 
00271     time_type intersect(const bezier_base<value_type,time_type> &x, time_type t=0.0,int i=15)const
00272     {
00273         //BROKEN - the time values of the 2 curves should be independent
00274         value_type system[4];
00275         system[0]=_coeff[0]-x._coeff[0];
00276         system[1]=_coeff[1]-x._coeff[1];
00277         system[2]=_coeff[2]-x._coeff[2];
00278         system[3]=_coeff[3]-x._coeff[3];
00279 
00280         t-=r;
00281         t*=drs;
00282 
00283         // Newton's method
00284         // Inner loop cost summary: 7 products, 5 sums, 1 difference
00285         for(;i;i--)
00286             t-= (system[0]+(system[1]+(system[2]+(system[3])*t)*t)*t)/
00287                 (system[1]+(system[2]*2+(system[3]*3)*t)*t);
00288 
00289         t*=(s-r);
00290         t+=r;
00291 
00292         return t;
00293     }
00294 
00295     value_type &
00296     operator[](int i)
00297     { return (&a)[i]; }
00298 
00299     const value_type &
00300     operator[](int i) const
00301     { return (&a)[i]; }
00302 };
00303 
00304 
00305 // Fast double implementation of a cubic bezier curve
00306 template <>
00307 class bezier_base<double,float> : public std::unary_function<float,double>
00308 {
00309 public:
00310     typedef double value_type;
00311     typedef float time_type;
00312 private:
00313     affine_combo<value_type,time_type> affine_func;
00314     value_type a,b,c,d;
00315     time_type r,s;
00316 
00317     value_type _coeff[4];
00318     time_type drs; // reciprocal of (s-r)
00319 public:
00320     bezier_base():r(0.0),s(1.0),drs(1.0) { }
00321     bezier_base(
00322         const value_type &a, const value_type &b, const value_type &c, const value_type &d,
00323         const time_type &r=0.0, const time_type &s=1.0):
00324         a(a),b(b),c(c),d(d),r(r),s(s),drs(1.0/(s-r)) { sync(); }
00325 
00326     void sync()
00327     {
00328 //      drs=1.0/(s-r);
00329         _coeff[0]=                 a;
00330         _coeff[1]=           b*3 - a*3;
00331         _coeff[2]=     c*3 - b*6 + a*3;
00332         _coeff[3]= d - c*3 + b*3 - a;
00333     }
00334 
00335     // 4 products, 3 sums, and 1 difference.
00336     inline value_type
00337     operator()(time_type t)const
00338     { t-=r; t*=drs; return _coeff[0]+(_coeff[1]+(_coeff[2]+(_coeff[3])*t)*t)*t; }
00339 
00340     void set_rs(time_type new_r, time_type new_s) { r=new_r; s=new_s; drs=1.0/(s-r); }
00341     void set_r(time_type new_r) { r=new_r; drs=1.0/(s-r); }
00342     void set_s(time_type new_s) { s=new_s; drs=1.0/(s-r); }
00343     const time_type &get_r()const { return r; }
00344     const time_type &get_s()const { return s; }
00345     time_type get_dt()const { return s-r; }
00346 
00348 
00351     time_type intersect(const bezier_base<value_type,time_type> &x, time_type t=0.0,int i=15)const
00352     {
00353         //BROKEN - the time values of the 2 curves should be independent
00354         value_type system[4];
00355         system[0]=_coeff[0]-x._coeff[0];
00356         system[1]=_coeff[1]-x._coeff[1];
00357         system[2]=_coeff[2]-x._coeff[2];
00358         system[3]=_coeff[3]-x._coeff[3];
00359 
00360         t-=r;
00361         t*=drs;
00362 
00363         // Newton's method
00364         // Inner loop: 7 products, 5 sums, 1 difference
00365         for(;i;i--)
00366             t-= (system[0]+(system[1]+(system[2]+(system[3])*t)*t)*t)/
00367                 (system[1]+(system[2]*2+(system[3]*3)*t)*t);
00368 
00369         t*=(s-r);
00370         t+=r;
00371 
00372         return t;
00373     }
00374 
00375     value_type &
00376     operator[](int i)
00377     { return (&a)[i]; }
00378 
00379     const value_type &
00380     operator[](int i) const
00381     { return (&a)[i]; }
00382 };
00383 
00384 //#ifdef __FIXED__
00385 
00386 // Fast double implementation of a cubic bezier curve
00387 /*
00388 template <>
00389 template <class T,unsigned int FIXED_BITS>
00390 class bezier_base<fixed_base<T,FIXED_BITS> > : std::unary_function<fixed_base<T,FIXED_BITS>,fixed_base<T,FIXED_BITS> >
00391 {
00392 public:
00393     typedef fixed_base<T,FIXED_BITS> value_type;
00394     typedef fixed_base<T,FIXED_BITS> time_type;
00395 
00396 private:
00397     affine_combo<value_type,time_type> affine_func;
00398     value_type a,b,c,d;
00399     time_type r,s;
00400 
00401     value_type _coeff[4];
00402     time_type drs; // reciprocal of (s-r)
00403 public:
00404     bezier_base():r(0.0),s(1.0),drs(1.0) { }
00405     bezier_base(
00406         const value_type &a, const value_type &b, const value_type &c, const value_type &d,
00407         const time_type &r=0, const time_type &s=1):
00408         a(a),b(b),c(c),d(d),r(r),s(s),drs(1.0/(s-r)) { sync(); }
00409 
00410     void sync()
00411     {
00412         drs=time_type(1)/(s-r);
00413         _coeff[0]=                 a;
00414         _coeff[1]=           b*3 - a*3;
00415         _coeff[2]=     c*3 - b*6 + a*3;
00416         _coeff[3]= d - c*3 + b*3 - a;
00417     }
00418 
00419     // 4 products, 3 sums, and 1 difference.
00420     inline value_type
00421     operator()(time_type t)const
00422     { t-=r; t*=drs; return _coeff[0]+(_coeff[1]+(_coeff[2]+(_coeff[3])*t)*t)*t; }
00423 
00424     void set_rs(time_type new_r, time_type new_s) { r=new_r; s=new_s; drs=time_type(1)/(s-r); }
00425     void set_r(time_type new_r) { r=new_r; drs=time_type(1)/(s-r); }
00426     void set_s(time_type new_s) { s=new_s; drs=time_type(1)/(s-r); }
00427     const time_type &get_r()const { return r; }
00428     const time_type &get_s()const { return s; }
00429     time_type get_dt()const { return s-r; }
00430 
00433     //  for the calling curve.
00434     //
00435     time_type intersect(const bezier_base<value_type,time_type> &x, time_type t=0,int i=15)const
00436     {
00437         value_type system[4];
00438         system[0]=_coeff[0]-x._coeff[0];
00439         system[1]=_coeff[1]-x._coeff[1];
00440         system[2]=_coeff[2]-x._coeff[2];
00441         system[3]=_coeff[3]-x._coeff[3];
00442 
00443         t-=r;
00444         t*=drs;
00445 
00446         // Newton's method
00447         // Inner loop: 7 products, 5 sums, 1 difference
00448         for(;i;i--)
00449             t-=(time_type) ( (system[0]+(system[1]+(system[2]+(system[3])*t)*t)*t)/
00450                 (system[1]+(system[2]*2+(system[3]*3)*t)*t) );
00451 
00452         t*=(s-r);
00453         t+=r;
00454 
00455         return t;
00456     }
00457 
00458     value_type &
00459     operator[](int i)
00460     { return (&a)[i]; }
00461 
00462     const value_type &
00463     operator[](int i) const
00464     { return (&a)[i]; }
00465 };
00466 */
00467 //#endif
00468 
00469 #endif
00470 
00471 
00472 
00473 template <typename V, typename T>
00474 class bezier_iterator
00475 {
00476 public:
00477 
00478     struct iterator_category {};
00479     typedef V value_type;
00480     typedef T difference_type;
00481     typedef V reference;
00482 
00483 private:
00484     difference_type t;
00485     difference_type dt;
00486     bezier_base<V,T>    curve;
00487 
00488 public:
00489 
00490 /*
00491     reference
00492     operator*(void)const { return curve(t); }
00493     const surface_iterator&
00494 
00495     operator++(void)
00496     { t+=dt; return &this; }
00497 
00498     const surface_iterator&
00499     operator++(int)
00500     { hermite_iterator _tmp=*this; t+=dt; return _tmp; }
00501 
00502     const surface_iterator&
00503     operator--(void)
00504     { t-=dt; return &this; }
00505 
00506     const surface_iterator&
00507     operator--(int)
00508     { hermite_iterator _tmp=*this; t-=dt; return _tmp; }
00509 
00510 
00511     surface_iterator
00512     operator+(difference_type __n) const
00513     { return surface_iterator(data+__n[0]+__n[1]*pitch,pitch); }
00514 
00515     surface_iterator
00516     operator-(difference_type __n) const
00517     { return surface_iterator(data-__n[0]-__n[1]*pitch,pitch); }
00518 */
00519 
00520 };
00521 
00522 template <typename V,typename T=float>
00523 class bezier : public bezier_base<V,T>
00524 {
00525 public:
00526     typedef V value_type;
00527     typedef T time_type;
00528     typedef float distance_type;
00529     typedef bezier_iterator<V,T> iterator;
00530     typedef bezier_iterator<V,T> const_iterator;
00531 
00532     distance_func<value_type> dist;
00533 
00534     using bezier_base<V,T>::get_r;
00535     using bezier_base<V,T>::get_s;
00536     using bezier_base<V,T>::get_dt;
00537 
00538 public:
00539     bezier() { }
00540     bezier(const value_type &a, const value_type &b, const value_type &c, const value_type &d):
00541         bezier_base<V,T>(a,b,c,d) { }
00542 
00543 
00544     const_iterator begin()const;
00545     const_iterator end()const;
00546 
00547     time_type find_closest(bool fast, const value_type& x, int i=7)const
00548     {
00549         if (!fast)
00550         {
00551             value_type array[4] = {
00552                 bezier<V,T>::operator[](0),
00553                 bezier<V,T>::operator[](1),
00554                 bezier<V,T>::operator[](2),
00555                 bezier<V,T>::operator[](3)};
00556             float t = NearestPointOnCurve(x, array);
00557             return t > 0.999999 ? 0.999999 : t < 0.000001 ? 0.000001 : t;
00558         }
00559         else
00560         {
00561             time_type r(0), s(1);
00562             float t((r+s)*0.5); /* half way between r and s */
00563 
00564             for(;i;i--)
00565             {
00566                 // compare 33% of the way between r and s with 67% of the way between r and s
00567                 if(dist(operator()((s-r)*(1.0/3.0)+r), x) <
00568                    dist(operator()((s-r)*(2.0/3.0)+r), x))
00569                     s=t;
00570                 else
00571                     r=t;
00572                 t=((r+s)*0.5);
00573             }
00574             return t;
00575         }
00576     }
00577 
00578     distance_type find_distance(time_type r, time_type s, int steps=7)const
00579     {
00580         const time_type inc((s-r)/steps);
00581         distance_type ret(0);
00582         value_type last(operator()(r));
00583 
00584         for(r+=inc;r<s;r+=inc)
00585         {
00586             const value_type n(operator()(r));
00587             ret+=dist.uncook(dist(last,n));
00588             last=n;
00589         }
00590         ret+=dist.uncook(dist(last,operator()(r)))*(s-(r-inc))/inc;
00591 
00592         return ret;
00593     }
00594 
00595     distance_type length()const { return find_distance(get_r(),get_s()); }
00596 
00597     /* subdivide at some time t into 2 separate curves left and right
00598 
00599         b0 l1
00600         *       0+1 l2
00601         b1      *       1+2*1+2 l3
00602         *       1+2     *           0+3*1+3*2+3 l4,r1
00603         b2      *       1+2*2+2 r2  *
00604         *       2+3 r3  *
00605         b3 r4   *
00606         *
00607 
00608         0.1 2.3 ->  0.1 2 3 4 5.6
00609     */
00610     void subdivide(bezier *left, bezier *right, const time_type &time = (time_type)0.5) const
00611     {
00612         time_type t=(time-get_r())/get_dt();
00613         bezier lt,rt;
00614 
00615         value_type temp;
00616         const value_type& a((*this)[0]);
00617         const value_type& b((*this)[1]);
00618         const value_type& c((*this)[2]);
00619         const value_type& d((*this)[3]);
00620 
00621         //1st stage points to keep
00622         lt[0] = a;
00623         rt[3] = d;
00624 
00625         //2nd stage calc
00626         lt[1] = affine_func(a,b,t);
00627         temp = affine_func(b,c,t);
00628         rt[2] = affine_func(c,d,t);
00629 
00630         //3rd stage calc
00631         lt[2] = affine_func(lt[1],temp,t);
00632         rt[1] = affine_func(temp,rt[2],t);
00633 
00634         //last stage calc
00635         lt[3] = rt[0] = affine_func(lt[2],rt[1],t);
00636 
00637         //set the time range for l,r (the inside values should be 1, 0 respectively)
00638         lt.set_r(get_r());
00639         rt.set_s(get_s());
00640 
00641         lt.sync();
00642         rt.sync();
00643 
00644         //give back the curves
00645         if(left) *left = lt;
00646         if(right) *right = rt;
00647     }
00648 
00649 
00650     void evaluate(time_type t, value_type &f, value_type &df) const
00651     {
00652         t=(t-get_r())/get_dt();
00653 
00654         const value_type& a((*this)[0]);
00655         const value_type& b((*this)[1]);
00656         const value_type& c((*this)[2]);
00657         const value_type& d((*this)[3]);
00658 
00659         const value_type p1 = affine_func(
00660                             affine_func(a,b,t),
00661                             affine_func(b,c,t)
00662                             ,t);
00663         const value_type p2 = affine_func(
00664                             affine_func(b,c,t),
00665                             affine_func(c,d,t)
00666                         ,t);
00667 
00668         f = affine_func(p1,p2,t);
00669         df = (p2-p1)*3;
00670     }
00671 
00672 private:
00673     /*
00674      *  Bezier :
00675      *  Evaluate a Bezier curve at a particular parameter value
00676      *      Fill in control points for resulting sub-curves if "Left" and
00677      *  "Right" are non-null.
00678      *
00679      *    int           degree;     Degree of bezier curve
00680      *    value_type    *VT;        Control pts
00681      *    time_type     t;          Parameter value
00682      *    value_type    *Left;      RETURN left half ctl pts
00683      *    value_type    *Right;     RETURN right half ctl pts
00684      */
00685     static value_type Bezier(value_type *VT, int degree, time_type t, value_type *Left, value_type *Right)
00686     {
00687         int         i, j;       /* Index variables  */
00688         value_type  Vtemp[W_DEGREE+1][W_DEGREE+1];
00689 
00690         /* Copy control points  */
00691         for (j = 0; j <= degree; j++)
00692             Vtemp[0][j] = VT[j];
00693 
00694         /* Triangle computation */
00695         for (i = 1; i <= degree; i++)
00696             for (j =0 ; j <= degree - i; j++)
00697             {
00698                 Vtemp[i][j][0] = (1.0 - t) * Vtemp[i-1][j][0] + t * Vtemp[i-1][j+1][0];
00699                 Vtemp[i][j][1] = (1.0 - t) * Vtemp[i-1][j][1] + t * Vtemp[i-1][j+1][1];
00700             }
00701 
00702         if (Left != NULL)
00703             for (j = 0; j <= degree; j++)
00704                 Left[j]  = Vtemp[j][0];
00705 
00706         if (Right != NULL)
00707             for (j = 0; j <= degree; j++)
00708                 Right[j] = Vtemp[degree-j][j];
00709 
00710         return (Vtemp[degree][0]);
00711     }
00712 
00713     /*
00714      * CrossingCount :
00715      *  Count the number of times a Bezier control polygon
00716      *  crosses the 0-axis. This number is >= the number of roots.
00717      *
00718      *    value_type    *VT;            Control pts of Bezier curve
00719      */
00720     static int CrossingCount(value_type *VT)
00721     {
00722         int     i;
00723         int     n_crossings = 0;    /*  Number of zero-crossings    */
00724         int     sign, old_sign;     /*  Sign of coefficients        */
00725 
00726         sign = old_sign = SGN(VT[0][1]);
00727         for (i = 1; i <= W_DEGREE; i++)
00728         {
00729             sign = SGN(VT[i][1]);
00730             if (sign != old_sign) n_crossings++;
00731             old_sign = sign;
00732         }
00733 
00734         return n_crossings;
00735     }
00736 
00737     /*
00738      *  ControlPolygonFlatEnough :
00739      *  Check if the control polygon of a Bezier curve is flat enough
00740      *  for recursive subdivision to bottom out.
00741      *
00742      *    value_type    *VT;        Control points
00743      */
00744     static int ControlPolygonFlatEnough(value_type *VT)
00745     {
00746         int             i;                  /* Index variable                   */
00747         distance_type   distance[W_DEGREE]; /* Distances from pts to line       */
00748         distance_type   max_distance_above; /* maximum of these                 */
00749         distance_type   max_distance_below;
00750         time_type       intercept_1, intercept_2, left_intercept, right_intercept;
00751         distance_type   a, b, c;            /* Coefficients of implicit         */
00752         /* eqn for line from VT[0]-VT[deg]          */
00753         /* Find the  perpendicular distance         */
00754         /* from each interior control point to      */
00755         /* line connecting VT[0] and VT[W_DEGREE]   */
00756         {
00757             distance_type   abSquared;
00758 
00759             /* Derive the implicit equation for line connecting first *
00760              *  and last control points */
00761             a = VT[0][1] - VT[W_DEGREE][1];
00762             b = VT[W_DEGREE][0] - VT[0][0];
00763             c = VT[0][0] * VT[W_DEGREE][1] - VT[W_DEGREE][0] * VT[0][1];
00764 
00765             abSquared = (a * a) + (b * b);
00766 
00767             for (i = 1; i < W_DEGREE; i++)
00768             {
00769                 /* Compute distance from each of the points to that line    */
00770                 distance[i] = a * VT[i][0] + b * VT[i][1] + c;
00771                 if (distance[i] > 0.0) distance[i] =  (distance[i] * distance[i]) / abSquared;
00772                 if (distance[i] < 0.0) distance[i] = -(distance[i] * distance[i]) / abSquared;
00773             }
00774         }
00775 
00776         /* Find the largest distance */
00777         max_distance_above = max_distance_below = 0.0;
00778 
00779         for (i = 1; i < W_DEGREE; i++)
00780         {
00781             if (distance[i] < 0.0) max_distance_below = MIN(max_distance_below, distance[i]);
00782             if (distance[i] > 0.0) max_distance_above = MAX(max_distance_above, distance[i]);
00783         }
00784 
00785         /* Implicit equation for "above" line */
00786         intercept_1 = -(c + max_distance_above)/a;
00787 
00788         /*  Implicit equation for "below" line */
00789         intercept_2 = -(c + max_distance_below)/a;
00790 
00791         /* Compute intercepts of bounding box   */
00792         left_intercept = MIN(intercept_1, intercept_2);
00793         right_intercept = MAX(intercept_1, intercept_2);
00794 
00795         return 0.5 * (right_intercept-left_intercept) < BEZIER_EPSILON ? 1 : 0;
00796     }
00797 
00798     /*
00799      *  ComputeXIntercept :
00800      *  Compute intersection of chord from first control point to last
00801      *  with 0-axis.
00802      *
00803      *    value_type    *VT;            Control points
00804      */
00805     static time_type ComputeXIntercept(value_type *VT)
00806     {
00807         distance_type YNM = VT[W_DEGREE][1] - VT[0][1];
00808         return (YNM*VT[0][0] - (VT[W_DEGREE][0] - VT[0][0])*VT[0][1]) / YNM;
00809     }
00810 
00811     /*
00812      *  FindRoots :
00813      *  Given a 5th-degree equation in Bernstein-Bezier form, find
00814      *  all of the roots in the interval [0, 1].  Return the number
00815      *  of roots found.
00816      *
00817      *    value_type    *w;             The control points
00818      *    time_type     *t;             RETURN candidate t-values
00819      *    int           depth;          The depth of the recursion
00820      */
00821     static int FindRoots(value_type *w, time_type *t, int depth)
00822     {
00823         int         i;
00824         value_type  Left[W_DEGREE+1];   /* New left and right   */
00825         value_type  Right[W_DEGREE+1];  /* control polygons     */
00826         int         left_count;         /* Solution count from  */
00827         int         right_count;        /* children             */
00828         time_type   left_t[W_DEGREE+1]; /* Solutions from kids  */
00829         time_type   right_t[W_DEGREE+1];
00830 
00831         switch (CrossingCount(w))
00832         {
00833             case 0 :
00834             {   /* No solutions here    */
00835                 return 0;
00836             }
00837             case 1 :
00838             {   /* Unique solution  */
00839                 /* Stop recursion when the tree is deep enough      */
00840                 /* if deep enough, return 1 solution at midpoint    */
00841                 if (depth >= MAXDEPTH)
00842                 {
00843                     t[0] = (w[0][0] + w[W_DEGREE][0]) / 2.0;
00844                     return 1;
00845                 }
00846                 if (ControlPolygonFlatEnough(w))
00847                 {
00848                     t[0] = ComputeXIntercept(w);
00849                     return 1;
00850                 }
00851                 break;
00852             }
00853         }
00854 
00855         /* Otherwise, solve recursively after   */
00856         /* subdividing control polygon          */
00857         Bezier(w, W_DEGREE, 0.5, Left, Right);
00858         left_count  = FindRoots(Left,  left_t,  depth+1);
00859         right_count = FindRoots(Right, right_t, depth+1);
00860 
00861         /* Gather solutions together    */
00862         for (i = 0; i < left_count;  i++) t[i] = left_t[i];
00863         for (i = 0; i < right_count; i++) t[i+left_count] = right_t[i];
00864 
00865         /* Send back total number of solutions  */
00866         return (left_count+right_count);
00867     }
00868 
00869     /*
00870      *  ConvertToBezierForm :
00871      *      Given a point and a Bezier curve, generate a 5th-degree
00872      *      Bezier-format equation whose solution finds the point on the
00873      *      curve nearest the user-defined point.
00874      *
00875      *    value_type&   P;              The point to find t for
00876      *    value_type    *VT;            The control points
00877      */
00878     static void ConvertToBezierForm(const value_type& P, value_type *VT, value_type w[W_DEGREE+1])
00879     {
00880         int     i, j, k, m, n, ub, lb;
00881         int     row, column;                /* Table indices                */
00882         value_type  c[DEGREE+1];            /* VT(i)'s - P                  */
00883         value_type  d[DEGREE];              /* VT(i+1) - VT(i)              */
00884         distance_type   cdTable[3][4];      /* Dot product of c, d          */
00885         static distance_type z[3][4] = {    /* Precomputed "z" for cubics   */
00886             {1.0, 0.6, 0.3, 0.1},
00887             {0.4, 0.6, 0.6, 0.4},
00888             {0.1, 0.3, 0.6, 1.0}};
00889 
00890         /* Determine the c's -- these are vectors created by subtracting */
00891         /* point P from each of the control points                       */
00892         for (i = 0; i <= DEGREE; i++)
00893             c[i] = VT[i] - P;
00894 
00895         /* Determine the d's -- these are vectors created by subtracting */
00896         /* each control point from the next                              */
00897         for (i = 0; i <= DEGREE - 1; i++)
00898             d[i] = (VT[i+1] - VT[i]) * 3.0;
00899 
00900         /* Create the c,d table -- this is a table of dot products of the */
00901         /* c's and d's                                                    */
00902         for (row = 0; row <= DEGREE - 1; row++)
00903             for (column = 0; column <= DEGREE; column++)
00904                 cdTable[row][column] = d[row] * c[column];
00905 
00906         /* Now, apply the z's to the dot products, on the skew diagonal */
00907         /* Also, set up the x-values, making these "points"             */
00908         for (i = 0; i <= W_DEGREE; i++)
00909         {
00910             w[i][0] = (distance_type)(i) / W_DEGREE;
00911             w[i][1] = 0.0;
00912         }
00913 
00914         n = DEGREE;
00915         m = DEGREE-1;
00916         for (k = 0; k <= n + m; k++)
00917         {
00918             lb = MAX(0, k - m);
00919             ub = MIN(k, n);
00920             for (i = lb; i <= ub; i++)
00921             {
00922                 j = k - i;
00923                 w[i+j][1] += cdTable[j][i] * z[j][i];
00924             }
00925         }
00926     }
00927 
00928     /*
00929      *  NearestPointOnCurve :
00930      *      Compute the parameter value of the point on a Bezier
00931      *      curve segment closest to some arbitrary, user-input point.
00932      *      Return the point on the curve at that parameter value.
00933      *
00934      *    value_type&   P;          The user-supplied point
00935      *    value_type    *VT;        Control points of cubic Bezier
00936      */
00937     static time_type NearestPointOnCurve(const value_type& P, value_type VT[4])
00938     {
00939         value_type  w[W_DEGREE+1];          /* Ctl pts of 5th-degree curve  */
00940         time_type   t_candidate[W_DEGREE];  /* Possible roots                */
00941         int         n_solutions;            /* Number of roots found         */
00942         time_type   t;                      /* Parameter value of closest pt */
00943 
00944         /*  Convert problem to 5th-degree Bezier form */
00945         ConvertToBezierForm(P, VT, w);
00946 
00947         /* Find all possible roots of 5th-degree equation */
00948         n_solutions = FindRoots(w, t_candidate, 0);
00949 
00950         /* Compare distances of P to all candidates, and to t=0, and t=1 */
00951         {
00952             distance_type   dist, new_dist;
00953             value_type      p, v;
00954             int             i;
00955 
00956             /* Check distance to beginning of curve, where t = 0    */
00957             dist = (P - VT[0]).mag_squared();
00958             t = 0.0;
00959 
00960             /* Find distances for candidate points  */
00961             for (i = 0; i < n_solutions; i++)
00962             {
00963                 p = Bezier(VT, DEGREE, t_candidate[i], (value_type *)NULL, (value_type *)NULL);
00964                 new_dist = (P - p).mag_squared();
00965                 if (new_dist < dist)
00966                 {
00967                     dist = new_dist;
00968                     t = t_candidate[i];
00969                 }
00970             }
00971 
00972             /* Finally, look at distance to end point, where t = 1.0 */
00973             new_dist = (P - VT[DEGREE]).mag_squared();
00974             if (new_dist < dist)
00975             {
00976                 dist = new_dist;
00977                 t = 1.0;
00978             }
00979         }
00980 
00981         /*  Return the point on the curve at parameter value t */
00982         return t;
00983     }
00984 };
00985 
00986 _ETL_END_NAMESPACE
00987 
00988 /* === E X T E R N S ======================================================= */
00989 
00990 /* === E N D =============================================================== */
00991 
00992 #endif

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