四维欧氏空间中的广义常斜坡曲面

利用曲面位置向量的正交分解式研究四维欧氏空间中的一类广义常斜坡曲面(即曲面的单位位置向量和单位平均曲率向量的內积为常数的曲面). 首先将曲面的位置向量分解为切部分和法部分，然后对具有法平行和法部分是平均曲率向量特性的广义常斜坡曲面分别进行研究，得到了在这两类特殊情况下广义常斜坡曲面的存在性及其分类.讨论了当分解式的法部分完全位于曲面的平均曲率向量方向时，广义常斜坡曲面满足的表达式及相关性质，并证明了在这种情况下曲面可以选取曲率网作为参数，并且此时广义常斜坡面为一类特殊的曲面——陈氏面. 最后给出了这些曲面的一些例子，并画出了在三维空间上的投影.

Generalized Constant Slope Surfaces in E4

Generalized constant slope surfaces(GCSSs)， whose inner product of the unit position vector and the unit mean curvature vector is constant， are studied in four dimension Euclidean space.Firstly， the position vector is divided into the tangent and normal parts. Then， two special GCSSs， one is that the parallel normal vector field has the mean curvature vector characteristics and the other is that the normal parts has the mean curvature vector characteristics， are studied， respectively， with the existence and classification of the two special GCSSs obtained. The expression and performance of a GCSS of which the normal plane is parallel to the mean curvature vector are discussed， proving that the GCSS can be parameterized with the principal lines and defined as a Chen surface.Finally， some examples of these surfaces and 3-D projection of the surfaces are given by MATLAB.