箱样条曲面的单调性

引理1 设X_(s+1)={e~1,…,e~s,e~1+…+e~s},ξ∈X_(s+1),如果对于所有的i∈Z~s,都有C_(i+ξ)≥C_i,则箱样条曲面S(x)=■C_iΦ_i(x|X_(s+1))在ξ方向上是单调非降的。其中Φ_i(x|X_(s+1))是箱样条函数。定理1 设X_n={x~1,…,x~n}■Z~s■{0},对任意1≤i≤n,〈X_n■{x_i}〉=R~s,令I_k={j|Φ_j(x|X_n■{x~i})■0,x∈suppΦ_k(x|X_n■{x~i})},M_k=■(C_(j+x~i)+C_j)则箱样条曲面S(x)=∑C_jΦ_j(x|X_n),x∈R~S(1)在x~i方向上单调非降的必要条件是

Monotonicity of the box spline curye surfaces

The necessary conditions and sufficient conditions for the monotonicity of the box spline surfaces are presented.The sufficient conditions of the monotonicity of the box spline surfaces of the three and four-directional meshes on arbitrary direction are discussed.