A field (2-parameter family of similarities) $\mathbf{S}(x,y)$ is Tran-Similar (TS) if there exist similarities $\mathbf{U}$ and $\mathbf{V}$ such that $\mathbf{S}(x,y)={\mathbf{U}}^x\cdot {\mathbf{V}}^y$ and $\mathbf{U}\cdot \mathbf{V}=\mathbf{V}\cdot \mathbf{U}$. The recently proposed COTS map, $\mathbf{M}$, is defined in terms of a planar TS field, $\mathbf{S}$, as $\mathbf{M}(x,y)=\mathbf{S}(x,y)\cdot A$ and is Corner-Operated (CO), i.e., controlled by the images $A$, $B$, $C$, and $D$ of the four corners of the unit-square in parameter space. The CO property enables intuitive design of planar warps. The TS property has several benefits: (1) The COTS map distributes distortion evenly; (2) It may be used to create pleasing patterns in which consecutive instances along any given row or column are related by the same similarity; and (3) Point-Membership Classification and Integral-Property Computation may be performed in constant time. However, the COTS map is only defined when the four corners are coplanar. To remove this restriction, we propose the Bent COTS (BeCOTS) generalization of the COTS map to three dimensions. It is CO (operated by its four non-coplanar corners) and TS (inheriting the properties of COTS mentioned above). We discuss potential benefits of BeCOTS for supporting the design and parameterization of highly-complex, bent lattices, for accelerating their processing, and for reducing the cost of manufacturing a class of large architectural structures.

字段（相似性的2参数系列） $\mathbf{小号}（X，ÿ）$ 如果存在相似之处，则为Tran-Similar（TS） $\mathbf{ü}$ 和 $\mathbf{V}$ 这样 $\mathbf{小号}（X，ÿ）={\mathbf{ü}}^X\cdot {\mathbf{V}}^{ÿ}$ 和 $\mathbf{ü}\cdot \mathbf{V}=\mathbf{V}\cdot \mathbf{ü}$。最近提出的COTS地图，$\mathbf{中号}$是根据平面TS场定义的， $\mathbf{小号}$，作为 $\mathbf{中号}（X，ÿ）=\mathbf{小号}（X，ÿ）\cdot \mathrm{一种}$ 并且是角落操作（CO），即由图像控制 $\mathrm{一种}$， $乙$， $C$和 $d$参数空间中单位正方形的四个角的角。CO属性可实现平面经纱的直观设计。TS属性具有以下优点：（1）COTS映射均匀分布失真；（2）可用于创建令人愉悦的模式，其中沿给定行或列的连续实例以相同的相似性关联；（3）可以按固定时间进行会员资格分类和积分计算。但是，仅当四个角共面时才定义COTS贴图。为了消除此限制，我们建议将COTS映射的Bent COTS（BeCOTS）归纳为三个维度。它是CO（由其四个非共面的角操作）和TS（继承了上述COTS的属性）。