本文以共轭坐标方法讨论三角形几何学。相似变换在本文中起着重要的作用。本文得到三角形几何学的许多著名定理,其中有些定理用初等几何的综合法以及解析几何的卡氏坐标法或三线坐标法的证明并不简单。这个事实显示共轭坐标法的一个优点...本文以共轭坐标方法讨论三角形几何学。相似变换在本文中起着重要的作用。本文得到三角形几何学的许多著名定理,其中有些定理用初等几何的综合法以及解析几何的卡氏坐标法或三线坐标法的证明并不简单。这个事实显示共轭坐标法的一个优点。§1.相似变换及三角形的基本元素复数 x 及其共轭复数 y(或(?))组成一点的共轭坐标(x,y),其所代表之点亦用 x 表之。设 F 及 F′为两图形,其所有点间有一一对应的关系,以 x 及展开更多
Three transformation models （Bursa-Wolf, Molodensky, and WTUSM） are generally used between two data systems transformation. The linear models are used when the rotation angles are small; however, when the rotation a...Three transformation models （Bursa-Wolf, Molodensky, and WTUSM） are generally used between two data systems transformation. The linear models are used when the rotation angles are small; however, when the rotation angles get bigger, model errors will be produced. In this paper, we present a method with three main terms：① the traditional rotation angles θ,φ,ψ are substituted with a,b,c which are three respective values in the anti-symmetrical or Lodrigues matrix; ② directly and accurately calculating the formula of seven parameters in any value of rotation angles; and ③ a corresponding adjustment model is established. This method does not use the triangle function. Instead it uses addition, subtraction, multiplication and division, and the complexity of the equation is reduced, making the calculation easy and quick.展开更多
This paper deals with the computer modeling of structures starting from a point cloud. The CCTV （China Central Television） tower headquarters is the case for study because the shape of this building is non-stellar, ...This paper deals with the computer modeling of structures starting from a point cloud. The CCTV （China Central Television） tower headquarters is the case for study because the shape of this building is non-stellar, concave and multi-connected. It is composed of sowns and chains. The sown is the representation of a horizontal plane formed by dense points. The chain is a planar path modeled by rare points. The CCTV structure is defined only by the three orthogonal Cartesian coordinates of the points. The proposed computer modeling uses a sequence of procedures and the desired outputted 3D model is consistent. The first procedure is devoted to attributing points to their voxel and to estimating three values needed afterwards. The second procedure is devoted to analyzing clusters vertically and horizontally, to preliminarily distinguishing chains from sowns and to generating relational matching. The third procedure is devoted to building closed paths between all chains and all their projections on sowns. The fourth procedure is devoted to connecting points with triangles. The fifth procedure, still being implemented, is devoted to interpolating triangles with triangular splines. The results show it is possible to achieve the 3D model using the above mentioned procedures. These procedures are written, implemented and tested and they form a library of people＇s own software. The code is written using Matlab. It is not possible to obtain the required 3D model if the procedures are applied in the wrong order or one step is skipped. To conclude, it is possible to obtain the computer model of the CCTV using the provided sequence of procedures.展开更多
Currently, engineering processes require reduced manufacturing time and low cost, in addition to the growing demand for workpieces with high accuracy. Workpieces with complex geometries and free forms has been a commo...Currently, engineering processes require reduced manufacturing time and low cost, in addition to the growing demand for workpieces with high accuracy. Workpieces with complex geometries and free forms has been a common practice in industries from different sectors such as： automotive, aeronautics, bioengineering among others. One way to satisfy the market requirements satisfactorily is making measurements more efficient to make the production process faster, in other words, it is necessary to make the inspection system more accurate and flexible. The coordinate measuring evolved over the past three decades and today it is the technology that best meets the requirements of modem manufacturing through CMMs （coordinate measurement machines）. The CMMs are important tool for design, fabrication and inspection of manufactured products, also used in the application of reverse engineering. These machines are also used by engineers in order to produce an accurate digital model in a virtual space for later use in CAD （computer-aided design）/CAM （computer-aided manufacturing）. It is worth mentioning that the accuracy of the modeling process of given piece depends on the number of control points that are captured on the workpiece surface. Consequently, the laser inspection systems are the best tools for use in reverse engineering, but more expensive when compared to contact measurement systems that use the TTP （touch trigger probe）, also used by CMMs. In this case, this paper aims to present an approach based on NURBS （non-uniform rational B-splines） to obtain free form curves and surfaces from a group of points obtained by using a contact sensor, the touch trigger probe. NURBS is an important mathematical tool and consists of generalizations of Bezier curves and surfaces and B-splines. The approach proposed in this paper can be applied for obtaining free form curves and surfaces in spur and helical gears. Experimental results obtained by measuring spur gears showed that the NURBS technique contributes for application of CMMs with touch trigger probe in reverse engineering.展开更多
With extensive applications of space geodesy, three-dimensional datum transformation model has been necessarily used to transform the coordinates in the different coordinate systems.Its essence is to predict the coord...With extensive applications of space geodesy, three-dimensional datum transformation model has been necessarily used to transform the coordinates in the different coordinate systems.Its essence is to predict the coordinates of non-common points in the second coordinate system based on their coordinates in the first coordinate system and the coordinates of common points in two coordinate systems.Traditionally, the computation of seven transformation parameters and the transformation of noncommon points are individually implemented, in which the errors of coordinates are taken into account only in the second system although the coordinates in both two systems are inevitably contaminated by the random errors.Moreover, the coordinate errors of non-common points are disregarded when they are transformed using the solved transformation parameters.Here we propose the seamless (rigorous) datum transformation model to compute the transformation parameters and transform the non-common points integratively, considering the errors of all coordinates in both coordinate systems.As a result, a nonlinear coordinate transformation model is formulated.Based on the Gauss-Newton algorithm and the numerical characteristics of transformation parameters, two linear versions of the established nonlinear model are individually derived.Then the least-squares collocation (prediction) method is employed to trivially solve these linear models.Finally, the simulation experiment is carried out to demonstrate the performance and benefits of the presented method.The results show that the presented method can significantly improve the precision of the coordinate transformation, especially when the non-common points are strongly correlated with the common points used to compute the transformation parameters.展开更多
文摘本文以共轭坐标方法讨论三角形几何学。相似变换在本文中起着重要的作用。本文得到三角形几何学的许多著名定理,其中有些定理用初等几何的综合法以及解析几何的卡氏坐标法或三线坐标法的证明并不简单。这个事实显示共轭坐标法的一个优点。§1.相似变换及三角形的基本元素复数 x 及其共轭复数 y(或(?))组成一点的共轭坐标(x,y),其所代表之点亦用 x 表之。设 F 及 F′为两图形,其所有点间有一一对应的关系,以 x 及
文摘Three transformation models （Bursa-Wolf, Molodensky, and WTUSM） are generally used between two data systems transformation. The linear models are used when the rotation angles are small; however, when the rotation angles get bigger, model errors will be produced. In this paper, we present a method with three main terms：① the traditional rotation angles θ,φ,ψ are substituted with a,b,c which are three respective values in the anti-symmetrical or Lodrigues matrix; ② directly and accurately calculating the formula of seven parameters in any value of rotation angles; and ③ a corresponding adjustment model is established. This method does not use the triangle function. Instead it uses addition, subtraction, multiplication and division, and the complexity of the equation is reduced, making the calculation easy and quick.
文摘This paper deals with the computer modeling of structures starting from a point cloud. The CCTV （China Central Television） tower headquarters is the case for study because the shape of this building is non-stellar, concave and multi-connected. It is composed of sowns and chains. The sown is the representation of a horizontal plane formed by dense points. The chain is a planar path modeled by rare points. The CCTV structure is defined only by the three orthogonal Cartesian coordinates of the points. The proposed computer modeling uses a sequence of procedures and the desired outputted 3D model is consistent. The first procedure is devoted to attributing points to their voxel and to estimating three values needed afterwards. The second procedure is devoted to analyzing clusters vertically and horizontally, to preliminarily distinguishing chains from sowns and to generating relational matching. The third procedure is devoted to building closed paths between all chains and all their projections on sowns. The fourth procedure is devoted to connecting points with triangles. The fifth procedure, still being implemented, is devoted to interpolating triangles with triangular splines. The results show it is possible to achieve the 3D model using the above mentioned procedures. These procedures are written, implemented and tested and they form a library of people＇s own software. The code is written using Matlab. It is not possible to obtain the required 3D model if the procedures are applied in the wrong order or one step is skipped. To conclude, it is possible to obtain the computer model of the CCTV using the provided sequence of procedures.
文摘Currently, engineering processes require reduced manufacturing time and low cost, in addition to the growing demand for workpieces with high accuracy. Workpieces with complex geometries and free forms has been a common practice in industries from different sectors such as： automotive, aeronautics, bioengineering among others. One way to satisfy the market requirements satisfactorily is making measurements more efficient to make the production process faster, in other words, it is necessary to make the inspection system more accurate and flexible. The coordinate measuring evolved over the past three decades and today it is the technology that best meets the requirements of modem manufacturing through CMMs （coordinate measurement machines）. The CMMs are important tool for design, fabrication and inspection of manufactured products, also used in the application of reverse engineering. These machines are also used by engineers in order to produce an accurate digital model in a virtual space for later use in CAD （computer-aided design）/CAM （computer-aided manufacturing）. It is worth mentioning that the accuracy of the modeling process of given piece depends on the number of control points that are captured on the workpiece surface. Consequently, the laser inspection systems are the best tools for use in reverse engineering, but more expensive when compared to contact measurement systems that use the TTP （touch trigger probe）, also used by CMMs. In this case, this paper aims to present an approach based on NURBS （non-uniform rational B-splines） to obtain free form curves and surfaces from a group of points obtained by using a contact sensor, the touch trigger probe. NURBS is an important mathematical tool and consists of generalizations of Bezier curves and surfaces and B-splines. The approach proposed in this paper can be applied for obtaining free form curves and surfaces in spur and helical gears. Experimental results obtained by measuring spur gears showed that the NURBS technique contributes for application of CMMs with touch trigger probe in reverse engineering.
基金supported by National Basic Research Program of China(Grant No.2012CB957703)the National Natural Science Foundation of China(Grant Nos.41074018 and 41104002)
文摘With extensive applications of space geodesy, three-dimensional datum transformation model has been necessarily used to transform the coordinates in the different coordinate systems.Its essence is to predict the coordinates of non-common points in the second coordinate system based on their coordinates in the first coordinate system and the coordinates of common points in two coordinate systems.Traditionally, the computation of seven transformation parameters and the transformation of noncommon points are individually implemented, in which the errors of coordinates are taken into account only in the second system although the coordinates in both two systems are inevitably contaminated by the random errors.Moreover, the coordinate errors of non-common points are disregarded when they are transformed using the solved transformation parameters.Here we propose the seamless (rigorous) datum transformation model to compute the transformation parameters and transform the non-common points integratively, considering the errors of all coordinates in both coordinate systems.As a result, a nonlinear coordinate transformation model is formulated.Based on the Gauss-Newton algorithm and the numerical characteristics of transformation parameters, two linear versions of the established nonlinear model are individually derived.Then the least-squares collocation (prediction) method is employed to trivially solve these linear models.Finally, the simulation experiment is carried out to demonstrate the performance and benefits of the presented method.The results show that the presented method can significantly improve the precision of the coordinate transformation, especially when the non-common points are strongly correlated with the common points used to compute the transformation parameters.
