多变量埃尔米特样条

我知道如何从16个控制点(即4x4网格)做双立方补丁，如b样条，Catmull-Rom, Besier…

然而，我更喜欢类似于2D/3D的Hermite样条曲线，由4个控制点(p00,p01,p10,p11)和4个法线(n00,n01,n10,n11)在这些点上描述。

我试图对它进行编程(首先沿u做2x 1D插值，然后沿v插值结果)。

当我意识到我不知道如何使用表面法线而不是1D Hermite样条中的导数时，我陷入了困境。

如果我天真地只代入法线而不是导数，我会得到无意义的结果。请看这里的c++代码(我只取了相关的部分):

// basis function
template <class TYPE>  
inline void spline_hermite_basis( TYPE x, TYPE& c0, TYPE& c1, TYPE& d0, TYPE& d1 ){
    TYPE x2   = x*x;
    TYPE x3   = x*x2;
    c0        =  2*x3 - 3*x2 + 1;
    c1        = -2*x3 + 3*x2    ;
    d0        =    x3 - 2*x2 + x; 
    d1        =    x3 -   x2    ;
};
// derivative of basis functions
template <class TYPE>  
inline void dspline_hermite_basis( TYPE x, TYPE& c0, TYPE& c1, TYPE& d0, TYPE& d1 ){
    TYPE x2   = x*x;
    c0        =   6*x2 - 6*x    ;
    c1        =  -6*x2 + 6*x    ;
    d0        =   3*x2 - 4*x + 1; 
    d1        =   3*x2 - 2*x    ;
};
void cubicPatch_point( double u, double v, 
    const Vec3d& p00, const Vec3d& p01, const Vec3d& p10, const Vec3d& p11,  
    const Vec3d& n00, const Vec3d& n01, const Vec3d& n10, const Vec3d& n11, 
    Vec3d& p, Vec3d& n  
){
    double cc0,cc1,cd0,cd1;
    double dc0,dc1,dd0,dd1;
    // interpolation along u
     spline_hermite_basis<double>( u,  cc0, cc1, cd0, cd1 );
    dspline_hermite_basis<double>( u,  dc0, dc1, dd0, dd1 );
    Vec3d p0u,p1u,n0u,n1u;
    p0u.set_mul( p00, cc0 ); p0u.add_mul( p01, cc1 ); p0u.add_mul( n00, cd0 ); p0u.add_mul( n01, cd1 );   // p0u =  cc0*p00 + cc1*p01 + cd0*n00 + cd1*n01; 
    n0u.set_mul( p00, dc0 ); n0u.add_mul( p01, dc1 ); n0u.add_mul( n00, dd0 ); n0u.add_mul( n01, dd1 );   // n0u =  dc0*p00 + dc1*p01 + dd0*n00 + dd1*n01; 
    p1u.set_mul( p10, cc0 ); p1u.add_mul( p11, cc1 ); p1u.add_mul( n10, cd0 ); p1u.add_mul( n11, cd1 );   // p1u =  cc0*p10 + cc1*p11 + cd0*n10 + cd1*n11; 
    n1u.set_mul( p10, dc0 ); n1u.add_mul( p11, dc1 ); n1u.add_mul( n10, dd0 ); n1u.add_mul( n11, dd1 );   // n1u =  dc0*p10 + dc1*p11 + dd0*n10 + dd1*n11; 
    // interpolation along v
     spline_hermite_basis<double>( v,  cc0, cc1, cd0, cd1 );
    dspline_hermite_basis<double>( v,  dc0, dc1, dd0, dd1 );
    p.set_mul( p0u, cc0 );  p.add_mul( p1u, cc1 );  p.add_mul( n0u, cd0 );  p.add_mul( n1u, cd1 );        // p =  cc0*p0u + cc1*p1u + cd0*n0u + cd1*n1u; 
    n.set_mul( p0u, dc0 );  n.add_mul( p1u, dc1 );  n.add_mul( n0u, dd0 );  n.add_mul( n1u, dd1 );        // n =  dc0*p0u + dc1*p1u + dd0*n0u + dd1*n1u; 
}