建筑几何造型中的Lie群Lie代数方法

Lie Group and Lie Algebra Method in Architectural Geometric Modeling

Lie群Lie代数理论在机器人机构学中有着重要应用。E3中刚体的连续运动对应于Lie群SE(3)上的一条连续曲线。运动旋量^ξ∈se(3)作为Lie群SE(3)的Lie代数元素,将刚体运动与Lie群SE(3)联系起来了。根据se(3)与SE(3)之间的指数映射及刚体运动的叠加定理建立串联机器人正向运动指数积公式,从而解析表达出虚拟的刚体运动。伴随刚体的空间运动,刚体边缘曲线在E3空间扫掠出了一张曲面。用计算机图形学与几何造型的方法,获得与刚体运动相对应的,用nurbs表达的建筑几何造型曲面。

Lie group and Lie algebra theory have wide application in robot mechanism. Continuous motion of the rigid body in E3 corresponding to a continuous curve in SE （3） of Lie group. Motion screw is the Lie algebra element in SE （3） of Lie group and integrates rigid motion with SE （3） of Lie group. Constructing the forward mo- tion index product formula of serial robot based on exponential map and rigid body motion superposition theorem, thereby resolving expression of virtual rigid body motion. The edge of rigid body sweeps a curved surface in E3 space with space motion of the rigid body. Finally, obtaining building geometry swept surface which can be ex- pressed by nurbs and corresponding to rigid motion by using computer graphics and geometric modeling approaches.