NURBS曲线
相关文件：
- NURBSCurve.h
- NURBSCurve.cpp
设NURBS曲线 的阶为，控制顶点为，对应的权重为，节点向量为，则曲线方程为：
将表示为齐次坐标形式，即，则
在NURBSCurve类中controlPw即表示了齐次形式的控制顶点。
NURBSCurve类
下面是NURBSCurve的类表示
#ifndef NURBSCURVE_H
#define NURBSCURVE_H
#include <igl/opengl/glfw/Viewer.h>
using namespace Eigen;
using namespace std;
struct NURBSCurve
{
NURBSCurve(){}
/*input format:
_n : P_0,P_1,...,P_n; _n is the final index
_k : order of BSpline
_controlP: P_0,P_1,...,P_n; (n+1) by 2 or 3
_knots : t_0,t_1,...,t_(n+k); */
NURBSCurve(int _n, int _k, MatrixXd _controlP, VectorXd _knots, bool _isRational = false);
bool isRational = false;
int n; // P_0,P_1,...,P_n; _n is the final index
int k; // order of BSpline
VectorXd knots;
MatrixXd controlP;
MatrixXd controlPw;
};
#endif // !NURBSCURVE_H
NURBSCurve文件格式
以下两个函数用于读取和保存NURBSCurve
// load
bool loadNURBS(string);
// save
bool saveNURBS(string);
其中NRUBSCurve的格式的一个例子如下：
# NURBSCurve 非有理格式示例
0 # isRational，0表示非有理
6 4 # n k
3 # 控制点坐标维数
2 3 0 # 控制点坐标
1.5 1 0
- 1 1 0
- 2 0 0
- 1 - 1 0
1 - 1 0
2 0 0
0 0 0 0 0.25 0.5 0.75 1 1 1 1 # 节点向量
----------------------------------------------------------
1 # isRational, 1表示有理
6 3 # n k
4 # 齐次坐标形式的维数
1 0 0 1 # 齐次坐标
0.5 0.5 0 0.5
-0.5 0.5 0 0.5
-1 0 0 1
-0.5 -0.5 0 0.5
0.5 -0.5 0 0.5
1 0 0 1
0 0 0 0.25 0.5 0.5 0.75 1 1 1 # 节点向量
NURBSCurve显示
// display by libigl
void draw(igl::opengl::glfw::Viewer& viewer, bool showpolygon=true,bool showsurface=true,double resolution = 0.01, Eigen::RowVector3d color = Eigen::RowVector3d(1, 0, 0));
// draw controlpolygon
void drawControlPolygon(igl::opengl::glfw::Viewer &viewer);
// draw NURBS surface
void drawSurface(igl::opengl::glfw::Viewer &viewer, double resolution = 0.01, Eigen::RowVector3d color = Eigen::RowVector3d(1, 0, 0));
如上代码所示，调用draw函数可以通过showpolygon,showsurface两个布尔变量控制显示控制网格和NURBS曲线。
API
NURBSCurve的主要函数如下，具体实现细节见代码
// This file is part of NURBS, a simple NURBS library.
// github repo: https://github.com/aijm/NURBS
// Copyright (C) 2018 Jiaming Ai <aichangeworld@gmail.com>
#ifndef NURBSCURVE_H
#define NURBSCURVE_H
#include <igl/opengl/glfw/Viewer.h>
using namespace Eigen;
using namespace std;
struct NURBSCurve
{
NURBSCurve(){}
/*input format:
_n : P_0,P_1,...,P_n; _n is the final index
_k : order of BSpline
_controlP: P_0,P_1,...,P_n; (n+1) by 2 or 3
_knots : t_0,t_1,...,t_(n+k); */
NURBSCurve(int _n, int _k, MatrixXd _controlP, VectorXd _knots, bool _isRational = false);
NURBSCurve(const NURBSCurve &curve){
isRational = curve.isRational;
n = curve.n;
k = curve.k;
knots = curve.knots;
controlPw = curve.controlPw;
}
NURBSCurve& operator=(const NURBSCurve &curve){
isRational = curve.isRational;
n = curve.n;
k = curve.k;
knots = curve.knots;
controlPw = curve.controlPw;
return *this;
}
// load
bool loadNURBS(string);
// save
bool saveNURBS(string);
// find the knot interval of t by binary searching
int find_ind(double t)const;
// evaluate the coordinate of curvePoint with parameter t
MatrixXd eval(double t)const;
MatrixXd eval_tangent(double t) const;
// chord length parameterization
static VectorXd parameterize(const MatrixXd &points);
// basis function N_(i,p)(t)
static double Basis(const VectorXd &_knots, double _t, int _i = 0, int _p = 3);
// Berivative of Blending function N[s0,s1,s2,s3,s4](t)
static Eigen::RowVectorXd DersBasis(const Eigen::MatrixXd &knots, double t, int i = 0, int p = 3);
// interpolate by bspline of degree 3
void interpolate(const MatrixXd &points);
// interpolate with appointed knot vector
void interpolate(const MatrixXd &points, const VectorXd &knotvector);
// interpolate with tangent constraint
void interpolate_tangent(const MatrixXd &points, const MatrixXd &tangent);
// interpolate with tangent constraint
void interpolate_tangent(const MatrixXd &points, const MatrixXd &tangent, const VectorXd ¶ms);
// interpolate with tangent constraint
void interpolate_tangent_improve(const MatrixXd &points, const MatrixXd &tangent, const VectorXd ¶ms);
// interpolate with tangent constraint
void interpolate_optimize(const MatrixXd &points, const MatrixXd &tangent, double learning_rate = 0.01);
// interpolate with tangent constraint
void interpolate_optimize(const MatrixXd &points, const MatrixXd &tangent, const VectorXd ¶ms, double learning_rate = 0.01);
// interpolate with tangent constraint
void interpolate_optimize1(const MatrixXd &points, const MatrixXd &tangent, const VectorXd ¶ms, double learning_rate = 0.01);
// pia fit by B-spline of degree 3
void piafit(const MatrixXd &points,int max_iter_num=100, double eps=1e-5);
// pia fit with appointed knot vector
void piafit(const MatrixXd &points, const VectorXd &knotvector, int max_iter_num = 100, double eps = 1e-5);
// given Q_0,...,Q_m, fit by B-spline with control points P_0,...,P_n
void lspiafit(const MatrixXd & points, int n_cpts, int max_iter_num = 100, double eps = 1e-5);
// lspia fit with appointed knot vector
void lspiafit(const MatrixXd & points, const VectorXd& params, int n_cpts, const VectorXd & knotvector, int max_iter_num = 100, double eps = 1e-5);
// lspia fit with appointed knot vector
void lspiafit(const MatrixXd & points, const VectorXd& params, const MatrixXd& cpts, const VectorXd & knotvector, int max_iter_num = 100, double eps = 1e-5);
// kont insertion
bool insert(double t);
// display by libigl
void draw(igl::opengl::glfw::Viewer& viewer, bool showpolygon=true,bool showsurface=true,double resolution = 0.01, Eigen::RowVector3d color = Eigen::RowVector3d(1, 0, 0));
// draw controlpolygon
void drawControlPolygon(igl::opengl::glfw::Viewer &viewer);
// draw NURBS surface
void drawSurface(igl::opengl::glfw::Viewer &viewer, double resolution = 0.01, Eigen::RowVector3d color = Eigen::RowVector3d(1, 0, 0));
bool isRational = false;
int n; // P_0,P_1,...,P_n; _n is the final index
int k; // order of BSpline
VectorXd knots;
MatrixXd controlP;
MatrixXd controlPw;
};
#endif // !NURBSCURVE_H