二次B样条曲线曲面的扩展

构造了一组由三个含参数m的函数构成的函数组, 该函数组线性无关, 称之为mB基。mB基具有非负性、规范性、对称性等良好的性质, 而且具有非常特殊的端点性质。基于mB基定义了一种新的样条曲线, 称之为mB曲线。mB曲线段可以转化为Bézier曲线的形式, 借助Bézier曲线的de Casteljau算法, 给出了mB曲线段的递推求值算法。mB曲线具有与二次均匀B样条曲线相同的端点行为, 即插值于控制多边形首末边的中点, 与控制多边形的首末边相切。另外, mB曲线的形状和连续性均可以通过参数m进行自由调节, 而且调节方式既可以是整体的, 又可以是局部的。利用张量积方法, 将mB曲线推广到了曲面, 称之为mB曲面。mB曲面具有与mB曲线类似的性质。

Extension of quadratic B spline curve and surface

A group of function containing three functions with parameter m is constructed. This group of function is linearly independent,which is named mB basis. The mB basis has non negativity, normalization, symmetry and special endpoint property. Based on the mB basis, a new kind of spline curve is defined, which is called mB curve. The mB curve segment can be converted into the form of Bézier curve. By virtue of the de Casteljau algorithm of Bézier curve, the recursive evaluation algorithm of the mB curve is given. The mB curve has the same endpoint behavior with the quadratic uniform B-spline curve, that is to say, the mB curve interpolates the midpoints of the first and the last sides of the control polygon, and tangent to the first and the last sides of the control polygon. Besides, the shape and continuity of the mB curve can be freely adjusted by the parameter m. And the adjustment can be either integral, or local. Using the tensor product method, the mB curve is extended to mB surface. The mB surface has similar properties with the mB curve.