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Revisão59 (tree)
Hora2020-01-02 00:46:03
Autormocchi_2012

Mensagem de Log

・最小二乗法で点列からNurbs曲線を作成する関数の追加
・ニュートン法によるNurbs曲線の最近点を求める関数の追加

Mudança Sumário

Diff

--- trunk/FitNurbsToPoints.cpp (revision 58)
+++ trunk/FitNurbsToPoints.cpp (revision 59)
@@ -1,6 +1,6 @@
11 /*
22 * FitNurbsToPoints
3- * Copylight (C) 2019 mocchi
3+ * Copylight (C) 2019,2020 mocchi
44 * mocchi_2003@yahoo.co.jp
55 * License: Boost ver.1
66 */
@@ -8,6 +8,7 @@
88 #include "ONGEO.h"
99 #include <cmath>
1010 #include <numeric>
11+#include <algorithm>
1112 #ifdef max
1213 #undef max
1314 #endif
@@ -93,7 +94,7 @@
9394 }
9495 };
9596
96-void InitializeSolver(int pt_cnt, int order, double *prm, double *knot, int knot_count, IEquationSolver *solver){
97+void InitializeSolver_ExactlyPassThroughPoints(int pt_cnt, int order, double *prm, double *knot, int knot_count, IEquationSolver *solver){
9798 int n = pt_cnt - 1;
9899 int m = knot_count;
99100 int p = order - 1;
@@ -108,32 +109,91 @@
108109 }
109110 knot[j+p-1] = s / static_cast<double>(p);
110111 }
111- ON_Matrix N(n+1, n+1);
112- for (int j = 0; j < n + 1; ++j){
113- for (int i = 0; i < n + 1; ++i){
114- N[j][i] = 0;
112+ if (order > 2){
113+ ON_Matrix N(n+1, n+1);
114+ for (int j = 0; j < n + 1; ++j){
115+ for (int i = 0; i < n + 1; ++i){
116+ N[j][i] = 0;
117+ }
115118 }
119+ N[0][0] = 1; N[n][n] = 1;
120+
121+ ON_SimpleArray<double> basis(order * order); basis.SetCount(basis.Capacity());
122+ for (int j = 1; j < n; ++j){
123+ double t = prm[j];
124+ int span_index = ON_NurbsSpanIndex(order, n+1, knot, t, 0, 0);
125+ for (int i = 0; i < span_index; ++i){
126+ N[j][i] = 0;
127+ }
128+ ON_EvaluateNurbsBasis(order, knot + span_index, t, basis.First());
129+ for (int i = span_index; i < span_index + order; ++i){
130+ N[j][i] = basis[i-span_index];
131+ }
132+ for (int i = span_index + order; i <= n; ++i){
133+ N[j][i] = 0;
134+ }
135+ }
136+
137+ solver->Initialize(N);
116138 }
117- N[0][0] = 1; N[n][n] = 1;
139+}
118140
141+// R のサイズは cpt_cnt - 2
142+void InitializeSolver_PassThroughPoints_LSQ_EndPointsConstraint(int pt_cnt, int order, int cpt_cnt, double *prm, double *knot, int knot_count, const ON_3dPoint *pts, ON_3dPoint *R, IEquationSolver *solver){
143+ int n = cpt_cnt - 1;
144+ int m = pt_cnt - 1;
145+ int p = order - 1;
146+ for (int i = 0; i < p; ++i){
147+ knot[i] = 0;
148+ knot[knot_count-i-1] = 1;
149+ }
150+
151+ double d = static_cast<double>(m + 1) / static_cast<double>(n - p + 1);
152+ for (double j = 1.0; j <= n - p; j += 1.0){
153+ double i = std::floor(j * d);
154+ int ii = static_cast<int>(i);
155+ double a = j * d - i;
156+ knot[static_cast<int>(j)+p-1] = (1.0 - a) * prm[ii-1] + a * prm[ii];
157+ }
158+
159+ ON_Matrix N(m-1, n-1);
119160 ON_SimpleArray<double> basis(order * order); basis.SetCount(basis.Capacity());
120- for (int j = 1; j < n; ++j){
121- double t = prm[j];
161+ ON_ClassArray<ON_3dPoint> rpt(m-1);
162+
163+ for (int k = 1; k <= m-1; ++k){
164+ int r = k;
165+ double t = prm[k];
166+
122167 int span_index = ON_NurbsSpanIndex(order, n+1, knot, t, 0, 0);
123- for (int i = 0; i < span_index; ++i){
124- N[j][i] = 0;
125- }
126168 ON_EvaluateNurbsBasis(order, knot + span_index, t, basis.First());
127- for (int i = span_index; i < span_index + order; ++i){
128- N[j][i] = basis[i-span_index];
169+ double basis_0 = 0, basis_n = 0;
170+ for (int c = 0; c <= n; ++c){
171+ double &dest = (c == 0) ? basis_0 : ((c == n) ? basis_n : N[r-1][c-1]);
172+ int basis_idx = c - span_index;
173+ if (basis_idx < 0 || basis_idx > order){
174+ dest = 0;
175+ }else{
176+ dest = basis[basis_idx];
177+ }
129178 }
130- for (int i = span_index + order; i <= n; ++i){
131- N[j][i] = 0;
179+ rpt.Append(pts[k] - pts[0] * basis_0 - pts[m] * basis_n);
180+ }
181+
182+ for (int c = 1; c <= n-1; ++c){
183+ ON_3dPoint Rc(0,0,0);
184+ for (int r = 1; r <= m-1; ++r){
185+ Rc += rpt[r-1] * N[r-1][c-1];
132186 }
187+ R[c-1] = Rc;
133188 }
134189
135- solver->Initialize(N);
190+ ON_Matrix NT = N;
191+ NT.Transpose();
192+ ON_Matrix NN;
193+ NN.Multiply(NT, N);
194+ solver->Initialize(NN);
136195 }
196+
137197 }
138198
139199 bool ONGEO_FitNurbsCurveToPointArray(const ON_3dPoint *pts, int pt_cnt, int order, ONGEO_NI_ParameterMethod pmethod, ON_NurbsCurve &nc, ONGEO_NI_Solver solver_id){
@@ -151,11 +211,14 @@
151211 ON_SimpleArray<double> prm(n+1); prm.SetCount(n+1);
152212 CreateParameter(pts, pt_cnt, 1, pmethod, prm);
153213
214+ InitializeSolver_ExactlyPassThroughPoints(pt_cnt, order, prm.First(), nc.m_knot, nc.KnotCount(), solver);
154215
155- InitializeSolver(pt_cnt, order, prm.First(), nc.m_knot, nc.KnotCount(), solver);
156-
157216 ON_ClassArray<ON_3dPoint> Q(n+1); Q.SetCount(n+1);
158- solver->Solve(n+1, pts, Q.First(), 1);
217+ if (order == 2){
218+ for (int i = 0; i < pt_cnt; ++i) Q[i] = pts[i];
219+ }else{
220+ solver->Solve(n+1, pts, Q.First(), 1);
221+ }
159222 for (int i = 0; i <= n; ++i){
160223 nc.SetCV(i, Q[i]);
161224 }
@@ -163,6 +226,217 @@
163226 return true;
164227 }
165228
229+double ONGEO_FitNurbsCurveToPointArray_LeastSquare(const ON_3dPoint *pts, int pt_cnt, int order, int cpt_cnt, ONGEO_NI_ParameterMethod pmethod, ON_NurbsCurve &nc, double *prm_, double *err_, ONGEO_NI_Solver solver_id){
230+ Solver_ON_Matrix_Inverse solver_omi;
231+ IEquationSolver *solver = 0;
232+ if (solver_id == ONGEO_NI_Solver_ON_Matrix_Inverse){
233+ solver = &solver_omi;
234+ }
235+
236+ ON_NurbsCurve pol;
237+ ONGEO_FitNurbsCurveToPointArray(pts, pt_cnt, 2, pmethod, pol, solver_id);
238+
239+ int n = pt_cnt - 1;
240+ ON_ClassArray<double> prm_buf;
241+ double *prm = prm_;
242+ if (!prm){
243+ prm_buf.SetCapacity(pt_cnt);
244+ prm_buf.SetCount(pt_cnt);
245+ prm = prm_buf.First();
246+ }
247+ for (int i = 0; i < n+1; ++i){
248+ prm[i] = pol.Knot(i);
249+ }
250+
251+ nc.Create(3, false, order, cpt_cnt);
252+ ON_ClassArray<ON_3dPoint> R(nc.CVCount()-2); R.SetCount(R.Capacity());
253+ InitializeSolver_PassThroughPoints_LSQ_EndPointsConstraint(pt_cnt, order, nc.CVCount(), prm, nc.m_knot, nc.KnotCount(), pts, R, solver);
254+ ON_ClassArray<ON_3dPoint> P(nc.CVCount()); P.SetCount(P.Capacity());
255+ P[0] = pts[0];
256+ P[P.Count()-1] = pts[pt_cnt-1];
257+ solver->Solve(R.Count(), R.First(), P.First()+1, 1);
258+ for (int i = 0; i < nc.CVCount(); ++i) nc.SetCV(i, P[i]);
259+
260+ double err_max = 0;
261+ for (int i = 1; i < pt_cnt - 1; ++i){
262+ double err = -1;
263+ double t = prm[i];
264+ if (ONGEO_NearestPointNurbsCurve_Newton(nc, pts[i], t)){
265+ if (prm_) prm_[i] = t;
266+ err = pts[i].DistanceTo(nc.PointAt(t));
267+ if (err_) err_[i] = err;
268+ }
269+ if (err_max < err) err_max = err;
270+ }
271+ return err_max;
272+}
273+
274+#if 0
275+double CalculateBr(const ON_NurbsCurve &nc, int r, int s){
276+ int ord = nc.Order();
277+ int p = ord - 1;
278+ int last = r - s;
279+ int first = r - p;
280+ int off = first - 1;
281+ const double *knot = nc.Knot();
282+
283+ ON_ClassArray<ON_3dPoint> temp(last + 2 - off);
284+ temp.SetCount(temp.Capacity());
285+ nc.GetCV(off, temp[0]);
286+ nc.GetCV(last+1, temp[last + 1 - off]);
287+ int i = first, j = last;
288+ int ii = 1, jj = last - off;
289+ while(j-i > 0){
290+ double alfi = (knot[r] - knot[i]) / (knot[i+ord] - knot[i]);
291+ double alfj = (knot[r] - knot[j]) / (knot[j+ord] - knot[j]);
292+ ON_3dPoint Pi, Pj;
293+ nc.GetCV(i, Pi); nc.GetCV(j, Pj);
294+ temp[ii] = (Pi - temp[ii-1] * (1.0 - alfi)) / alfi;
295+ temp[jj] = (Pj - temp[jj+1] * alfj) / (1.0 - alfj);
296+ i = i + 1, ii = ii + 1;
297+ j = j - 1, jj = jj - 1;
298+ }
299+ if (j - 1 < 0){
300+ return temp[ii-1].DistanceTo(temp[jj+1]);
301+ }else{
302+ double alfi = (knot[r] - knot[i]) / (knot[i+ord] - knot[i]);
303+ ON_3dPoint Pi;
304+ nc.GetCV(i, Pi);
305+ return Pi.DistanceTo(temp[ii+1] * alfi + (1.0 - alfi) * temp[ii-1]);
306+ }
307+}
308+
309+bool ONGEO_FitNurbsCurveToPointArray(const ON_3dPoint *pts, int pt_cnt, int order, double tolerance, ONGEO_NI_ParameterMethod pmethod, ON_NurbsCurve &nc, ONGEO_NI_Solver solver_id){
310+ if (tolerance <= 0) return ONGEO_FitNurbsCurveToPointArray(pts, pt_cnt, order, pmethod, nc, solver_id);
311+
312+ Solver_ON_Matrix_Inverse solver_omi;
313+ IEquationSolver *solver = 0;
314+ if (solver_id == ONGEO_NI_Solver_ON_Matrix_Inverse){
315+ solver = &solver_omi;
316+ }
317+
318+ ON_SimpleArray<double> basis(order * order); basis.SetCount(basis.Capacity());
319+
320+ ON_NurbsCurve nc_tmp;
321+ ONGEO_FitNurbsCurveToPointArray(pts, pt_cnt, 2, pmethod, nc_tmp, solver_id);
322+
323+ int n = pt_cnt - 1;
324+ ON_SimpleArray<double> prm(n+1); prm.SetCount(n+1);
325+ ON_SimpleArray<double> err(n+1); err.SetCount(n+1);
326+ for (int i = 0; i < n+1; ++i){
327+ prm[i] = nc_tmp.Knot(i);
328+ err[i] = 0;
329+ }
330+
331+ int p = order - 1;
332+ // ノット除去
333+ for (int deg = 1; deg <= p; ++deg){
334+ ON_NurbsCurve nc_tmp2;
335+ nc_tmp2.Create(3, false, deg+1, nc_tmp.CVCount());
336+ // ノット除去
337+ while(true){
338+ // 最小の Br 値と、そのBrに対応するノット番号を取得
339+ double Br_min = -1, err_min = -1;
340+ const double *knot = nc_tmp.Knot();
341+ int s = nc_tmp.KnotMultiplicity(0);
342+ int r_min = -1, r = s-1;
343+ s = nc_tmp.KnotMultiplicity(r+1);
344+ r += s;
345+ while(s > 0){
346+ double Br = CalculateBr(nc_tmp, r, s);
347+
348+ int p_s = deg + s;
349+ s = nc_tmp.KnotMultiplicity(r+1);
350+ if (r + s >= nc_tmp.KnotCount()-1) break;
351+#if 0
352+ nc_tmp2 = nc_tmp;
353+ nc_tmp2.RemoveSpan(r-(p-1));
354+ double Br;
355+ if ((p_s & 1) == 0){
356+ int k = p_s / 2;
357+ double a = (knot[r] - knot[r-k]) / (knot[r-k+p+1] - knot[r-k]);
358+ ON_3dPoint P, P1r, P1l;
359+ nc_tmp.GetCV(r-k, P);
360+ nc_tmp2.GetCV(r-k+1, P1r);
361+ int rk_1 = r-k-1;
362+ nc_tmp2.GetCV(rk_1 >= 0 ? rk_1 : 0, P1l);
363+ Br = P.DistanceTo(P1r * a + P1l * (1.0-a));
364+ }else{
365+ int k = (p_s + 1) / 2;
366+ ON_3dPoint P1r, P1l;
367+ nc_tmp2.GetCV(r-k+1, P1r);
368+ nc_tmp2.GetCV(r-k, P1l);
369+ Br = P1r.DistanceTo(P1l);
370+ }
371+#endif
372+ if (Br_min < 0 || Br_min > Br) Br_min = Br, r_min = r;
373+ r += s;
374+ }
375+ if (Br_min == -1) break;
376+ {
377+ int span_index = ON_NurbsSpanIndex(order, nc_tmp.CVCount(), knot, knot[r_min], 0, 0);
378+ ON_EvaluateNurbsBasis(order, knot + span_index, knot[r_min], basis.First());
379+ int s_min = nc_tmp.KnotMultiplicity(r_min);
380+ int p_s = deg + s_min;
381+ if ((p_s & 1) == 0){
382+ int k = p_s / 2;
383+ err_min = basis[r_min - k - span_index+1] * Br_min;
384+ }else{
385+ int k = (p_s + 1) / 2;
386+ double a = (knot[r_min] - knot[r_min-k+1])/(knot[r_min-k+deg+2] - knot[r_min-k+1]);
387+ err_min = (1.0 - a) * basis[r_min - k + 1 - span_index+1] * Br_min;
388+ }
389+ }
390+ if (err_min > tolerance) break;
391+ int idx_s = ON_NurbsSpanIndex(order, nc_tmp.CVCount(), knot, knot[r_min], 0, 0);
392+ int idx_n = idx_s + 2 * deg - 1;
393+
394+ ptrdiff_t diff_s = std::lower_bound(prm.First(), prm.Last()+1, knot[idx_s]) - prm.First();
395+ ptrdiff_t diff_e = std::upper_bound(prm.First(), prm.Last()+1, knot[idx_n]) - prm.First();
396+ double *es = err.First() + diff_s, *ee = err.First() + diff_e;
397+
398+ bool knot_removable = true;
399+ for (double *e = es; e != ee; ++e){
400+ if (*e + err_min < tolerance) continue;
401+ knot_removable = false;
402+ break;
403+ }
404+ if (knot_removable){
405+ for (double *e = es; e != ee; ++e) *e += err_min;
406+ nc_tmp.RemoveSpan(r_min-(order-2));
407+ }else break;
408+ }
409+ if (deg == p){
410+ nc = nc_tmp;
411+ break;
412+ }
413+
414+ // 次数を1上げて最小二乗法で曲線を再生成
415+ nc_tmp.Create(3, false, deg+2, nc_tmp.CVCount());
416+ ON_ClassArray<ON_3dPoint> R(nc_tmp.CVCount()-2); R.SetCount(R.Capacity());
417+ InitializeSolver_PassThroughPoints_LSQ_EndPointsConstraint(pt_cnt, deg+2, nc_tmp.CVCount(), prm, nc_tmp.m_knot, nc_tmp.KnotCount(), pts, R, solver);
418+ ON_ClassArray<ON_3dPoint> P(nc_tmp.CVCount()); P.SetCount(P.Capacity());
419+ P[0] = pts[0];
420+ P[P.Count()-1] = pts[pt_cnt-1];
421+ solver->Solve(R.Count(), R.First(), P.First()+1, 1);
422+ for (int i = 0; i < nc_tmp.CVCount(); ++i) nc_tmp.SetCV(i, P[i]);
423+
424+ double err_max = 0;
425+ for (int i = 1; i < pt_cnt - 1; ++i){
426+ double t = prm[i];
427+ if (ONGEO_NearestPointNurbsCurve_Newton(nc_tmp, pts[i], t)){
428+ prm[i] = t;
429+ err[i] = pts[i].DistanceTo(nc_tmp.PointAt(t));
430+ }
431+ if (err_max < err[i]) err_max = err[i];
432+ }
433+ }
434+
435+ return true;
436+}
437+#endif
438+
439+
166440 bool ONGEO_FitNurbsSurfaceToPointGrid(const ON_3dPoint *pts, int u_count, int v_count, int u_order, int v_order, ONGEO_NI_ParameterMethod pmethod, ON_NurbsSurface &nf, ONGEO_NI_Solver solver_id){
167441 if (u_count < u_order || v_count < v_order || !pts) return false;
168442 int n = u_count - 1, m = v_count - 1;
@@ -187,7 +461,7 @@
187461 }
188462 for (int i = 0; i <= n; ++i) prm_u[i] /= static_cast<double>(m+1);
189463
190- InitializeSolver(n + 1, u_order, prm_u.First(), nf.m_knot[0], nf.KnotCount(0), solver);
464+ InitializeSolver_ExactlyPassThroughPoints(n + 1, u_order, prm_u.First(), nf.m_knot[0], nf.KnotCount(0), solver);
191465
192466 ON_ClassArray<ON_3dPoint> R((n+1) * (m+1)); R.SetCount(R.Capacity());
193467 for (int j = 0; j <= m; ++j){
@@ -203,7 +477,7 @@
203477 }
204478 for (int j = 0; j <= m; ++j) prm_v[j] /= static_cast<double>(n+1);
205479
206- InitializeSolver(m + 1, v_order, prm_v.First(), nf.m_knot[1], nf.KnotCount(1), solver);
480+ InitializeSolver_ExactlyPassThroughPoints(m + 1, v_order, prm_v.First(), nf.m_knot[1], nf.KnotCount(1), solver);
207481
208482 ON_ClassArray<ON_3dPoint> Q(m+1); Q.SetCount(m+1);
209483 for (int i = 0; i <= n; ++i){
--- trunk/NearestPointCurve_Newton.cpp (nonexistent)
+++ trunk/NearestPointCurve_Newton.cpp (revision 59)
@@ -0,0 +1,54 @@
1+/*
2+ * FitNurbsToPoints
3+ * Copylight (C) 2020 mocchi
4+ * mocchi_2003@yahoo.co.jp
5+ * License: Boost ver.1
6+ */
7+
8+#define NOMINMAX
9+#include "opennurbs.h"
10+#include "ONGEO.h"
11+#include <cmath>
12+
13+// Reference : Piegl, L. and Tiller, W.
14+// "The NURBS Book", Second edition.
15+// Springer-Verlag Berlin, Heidelberg, 1997.
16+
17+// pp.229 - 234 : Chapter 6.1, "Point Inversion and Projection for Curves and Surfaces"
18+
19+bool ONGEO_NearestPointNurbsCurve_Newton(const ON_NurbsCurve &nc, const ON_3dPoint &pt_query, double &t, ON_Interval *range_, int iteration_max){
20+ ON_3dPoint pt;
21+ ON_3dVector der1, der2;
22+ ON_Interval range;
23+ if (range_) range = *range_;
24+ else range = nc.Domain();
25+ for(int i = 0; i < iteration_max; ++i){
26+ nc.Ev2Der(t, pt, der1, der2);
27+ double der1_len = der1.Length();
28+ ON_3dVector diff = pt - pt_query;
29+ double diff_len = diff.Length();
30+ double numerator = ON_DotProduct(der1, diff);
31+
32+ bool cond_1 = diff_len <= ON_ZERO_TOLERANCE;
33+ bool cond_2 = std::abs(numerator / (der1_len * diff_len)) <= ON_ZERO_TOLERANCE;
34+ bool cond_4 = false;
35+
36+ if (!cond_1 || !cond_2){
37+ double tn = t - numerator / (ON_DotProduct(der2, diff) + der1_len * der1_len);
38+ if (!nc.IsClosed()){
39+ if (tn < range.Min()) tn = range.Min();
40+ else if (tn > range.Max()) tn = range.Max();
41+ }else{
42+ if (tn < range.Min()) tn = range.Max() - (range.Min() - tn);
43+ else if (tn > range.Max()) tn = range.Min() - (tn - range.Max());
44+ }
45+ cond_4 = std::abs((tn - t) * der1_len) <= ON_ZERO_TOLERANCE;
46+ t = tn;
47+ }
48+ if (cond_1 || cond_2 || cond_4){
49+ return true;
50+ }
51+ }
52+ return false;
53+}
54+
--- trunk/ONGEO.h (revision 58)
+++ trunk/ONGEO.h (revision 59)
@@ -79,6 +79,15 @@
7979 /// @return クエリ点と最近点との距離
8080 ONGEO_DECL double ONGEO_NearestPointBezierCurve_BBClipping(const ON_BezierCurve *bc_begin, const ON_BezierCurve *bc_end, double tolerance, const ON_3dPoint &pt_query, const ON_BezierCurve *&bc_nearest, double &t, ON_3dPoint &pt_nearest);
8181
82+/// 指定した点に最も近いNurbs曲線上のパラメータをニュートン法による収束計算で求める。
83+/// @param [in] nc Nurbs曲線
84+/// @param [in] pt_query クエリ点
85+/// @param [in/out] t クエリ点に近いNurbs曲線上に位置を与えると、クエリ点に最も近いNurbs曲線上の位置を返す。
86+/// @param [in] range_ 解が有りうる範囲を指定する。null_ptr を与えた場合は、曲線のパラメータ範囲全域を対象にする。
87+/// @param [in] iteration_max 最も近い点の三次元座標
88+/// @return true: 収束成功、 false: 収束失敗
89+ONGEO_DECL bool ONGEO_NearestPointNurbsCurve_Newton(const ON_NurbsCurve &nc, const ON_3dPoint &pt_query, double &t, ON_Interval *range_ = 0, int iteration_max = 100000);
90+
8291 /// 有理ベジエ曲線の曲線長を再分割による収束計算で求める(制御点列長と始点-終点距離の差がトレランス以下になるまで)。
8392 /// @param [in] bc ベジエ曲線
8493 /// @param [in] tolerance トレランス
@@ -452,11 +461,37 @@
452461 /// @param [in] pt_cnt 点列の個数
453462 /// @param [in] order Nurbs曲線の階数
454463 /// @param [in] pmethod 曲線パラメータの計算方法
455-/// @param [put] nc 作成した Nurbs曲線
464+/// @param [out] nc 作成した Nurbs曲線
456465 /// @param [in] solver 連立一次方程式をどの方法で解くか？
457466 /// @return true:成功、false:失敗
458467 ONGEO_DECL bool ONGEO_FitNurbsCurveToPointArray(const ON_3dPoint *pts, int pt_cnt, int order, ONGEO_NI_ParameterMethod pmethod, ON_NurbsCurve &nc, ONGEO_NI_Solver solver = ONGEO_NI_Solver_ON_Matrix_Inverse);
459468
469+/// 与えられた点列から最小二乗法を用いて、指定した制御点数、階数の Nurbs曲線を作成する。
470+/// @param [in] pts 点列
471+/// @param [in] pt_cnt 点列の個数
472+/// @param [in] order Nurbs曲線の階数
473+/// @param [in] cpt_cnt 制御点数
474+/// @param [in] pmethod 曲線パラメータの計算方法
475+/// @param [out] nc 作成した Nurbs曲線
476+/// @param [out] prm 各点を曲線に投影した際のパラメータ。呼び出し元で pt_cnt 分の領域を確保すること。null_ptr の時は格納しない。
477+/// @param [out] err 各点と曲線との誤差。呼び出し元で pt_cnt 分の領域を確保すること。null_ptr の時は格納しない。
478+/// @param [in] solver 連立一次方程式をどの方法で解くか？
479+/// @return 誤差の最大値
480+ONGEO_DECL double ONGEO_FitNurbsCurveToPointArray_LeastSquare(const ON_3dPoint *pts, int pt_cnt, int order, int cpt_cnt, ONGEO_NI_ParameterMethod pmethod, ON_NurbsCurve &nc, double *prm = 0, double *err = 0, ONGEO_NI_Solver solver_id = ONGEO_NI_Solver_ON_Matrix_Inverse);
481+
482+#if 0
483+/// 与えられた点列から、許容誤差 tolerance 以内を通るNurbs曲線を作成する。
484+/// @param [in] pts 点列
485+/// @param [in] pt_cnt 点列の個数
486+/// @param [in] order Nurbs曲線の階数
487+/// @param [in] tolerance 許容誤差
488+/// @param [in] pmethod 曲線パラメータの計算方法
489+/// @param [out] nc 作成した Nurbs曲線
490+/// @param [in] solver 連立一次方程式をどの方法で解くか？
491+/// @return true:成功、false:失敗
492+ONGEO_DECL bool ONGEO_FitNurbsCurveToPointArray(const ON_3dPoint *pts, int pt_cnt, int order, double tolerance, ONGEO_NI_ParameterMethod pmethod, ON_NurbsCurve &nc, ONGEO_NI_Solver solver = ONGEO_NI_Solver_ON_Matrix_Inverse);
493+#endif
494+
460495 /// 与えられた点グリッドを通るNurbs曲面を作成する。
461496 /// @param [in] pts 点グリッド (P_u0v0、 P_u1v0、P_u2v0、...、P_u0v1、P_u1v1、... の順に並べる)
462497 /// @param [in] u_count 点グリッドのU方向の個数
@@ -464,7 +499,7 @@
464499 /// @param [in] u_order Nurbs曲面のU方向階数
465500 /// @param [in] v_order Nurbs曲面のV方向階数
466501 /// @param [in] pmethod 曲線パラメータの計算方法
467-/// @param [put] nf 作成した Nurbs曲面
502+/// @param [out] nf 作成した Nurbs曲面
468503 /// @param [in] solver 連立一次方程式をどの方法で解くか？
469504 /// @return true:成功、false:失敗
470505 ONGEO_DECL bool ONGEO_FitNurbsSurfaceToPointGrid(const ON_3dPoint *pts, int u_count, int v_count, int u_order, int v_order, ONGEO_NI_ParameterMethod pmethod, ON_NurbsSurface &nf, ONGEO_NI_Solver solver = ONGEO_NI_Solver_ON_Matrix_Inverse);
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