A new algorithm is presented that generates developable Bézier surfaces through a Bézier curve called a directrix. The algorithm is based on differential geometry theory on necessary and sufficient condition...A new algorithm is presented that generates developable Bézier surfaces through a Bézier curve called a directrix. The algorithm is based on differential geometry theory on necessary and sufficient conditions for a surface which is developable, and on degree evaluation formula for parameter curves and linear independence for Bernstein basis. No nonlinear characteristic equations have to be solved. Moreover the vertex for a cone and the edge of regression for a tangent surface can be obtained easily. Aumann’s algorithm for developable surfaces is a special case of this paper.展开更多
Curvature lines are special and important curves on surfaces.It is of great significance to construct developable surface interpolated on curvature lines in engineering applications.In this paper,the shape optimizatio...Curvature lines are special and important curves on surfaces.It is of great significance to construct developable surface interpolated on curvature lines in engineering applications.In this paper,the shape optimization of generalized cubic ball developable surface interpolated on the curvature line is studied by using the improved reptile search algorithm.Firstly,based on the curvature line of generalized cubic ball curve with shape adjustable,this paper gives the construction method of SGC-Ball developable surface interpolated on the curve.Secondly,the feedback mechanism,adaptive parameters and mutation strategy are introduced into the reptile search algorithm,and the Feedback mechanism-driven improved reptile search algorithm effectively improves the solving precision.On IEEE congress on evolutionary computation 2014,2017,2019 and four engineering design problems,the feedback mechanism-driven improved reptile search algorithm is compared with other representative methods,and the result indicates that the solution performance of the feedback mechanism-driven improved reptile search algorithm is competitive.At last,taking the minimum energy as the evaluation index,the shape optimization model of SGC-Ball interpolation developable surface is established.The developable surface with the minimum energy is achieved with the help of the feedback mechanism-driven improved reptile search algorithm,and the comparison experiment verifies the superiority of the feedback mechanism-driven improved reptile search algorithm for the shape optimization problem.展开更多
The paper deals with the principle of envelope of a one-parameter plane family to design developable surfaces. Three methods of designing developable surfaces are presented. They are designing a developable surface ba...The paper deals with the principle of envelope of a one-parameter plane family to design developable surfaces. Three methods of designing developable surfaces are presented. They are designing a developable surface based on one curve on it and its normal line, designing a developable surface based on two curves on it and designing a developable surface based on one curve and one surface. They meet the requirements of engineering fields.展开更多
The developable surface is an important surface in computer aided design, geometric modeling and industrial manufactory. It is often given in the standard parametric form, but it can also be in the implicit form which...The developable surface is an important surface in computer aided design, geometric modeling and industrial manufactory. It is often given in the standard parametric form, but it can also be in the implicit form which is commonly used in algebraic geometry. Not all algebraic developable surfaces have rational parametrizations. In this paper, the authors focus on the rational developable surfaces. For a given algebraic surface, the authors first determine whether it is developable by geometric inspection, and then give a rational proper parametrization in the affrmative case. For a rational parametric surface, the authors also determine the developability and give a proper reparametrization for the developable surface.展开更多
In this paper we address the problem of interpolating a spline developable patch bounded by a given spline curve and the first and the last rulings of the developable surface. To complete the boundary of the patch, a ...In this paper we address the problem of interpolating a spline developable patch bounded by a given spline curve and the first and the last rulings of the developable surface. To complete the boundary of the patch, a second spline curve is to be given. Up to now this interpolation problem could be solved, but without the possibility of choosing both endpoints for the rulings. We circumvent such difficulty by resorting to degree elevation of the developable surface. This is useful for solving not only this problem, but also other problems dealing with triangular developable patches.展开更多
In this paper we construct developable surface patches which are bounded by two rational or NURBS curves,though the resulting patch is not a rational or NURBS surface in general.This is accomplished by reparameterizin...In this paper we construct developable surface patches which are bounded by two rational or NURBS curves,though the resulting patch is not a rational or NURBS surface in general.This is accomplished by reparameterizing one of the boundary curves.The reparameterization function is the solution of an algebraic equation.For the relevant case of cubic or cubic spline curves,this equation is quartic at most,quadratic if the curves are B´ezier or splines and lie on parallel planes,and hence it may be solved either by standard analytical or numerical methods.展开更多
Over the next 20 years,China's urban rail transit(hereinafter referred to as'urban rail')will face large-scalerenovation of existing line facilities and equipment,with more than 1000 km of renovated lines ...Over the next 20 years,China's urban rail transit(hereinafter referred to as'urban rail')will face large-scalerenovation of existing line facilities and equipment,with more than 1000 km of renovated lines to be added eachyear.In 2024,the China Association of Metros issued the Guiding Opinions on the Renovation of Existing UrbanRail Transit Lines in China,providing guiding opinions on norms,standards,and implementation approaches forthe renovation of existing lines in the coming period.In the practical work of renovating existing urban rail lines,it is necessary to continuously explore and refine relevant theoretical methods in line with industry developmenttrends and urban development requirements.The following are the author's recent reflections on theoreticalinnovation in this field.展开更多
In this paper we provide a characterisation of rational developable surfaces in terms of the blossoms of the bounding curves and three rational functions∧,M,ν.Properties of developable surfaces are revised in this f...In this paper we provide a characterisation of rational developable surfaces in terms of the blossoms of the bounding curves and three rational functions∧,M,ν.Properties of developable surfaces are revised in this framework.In particular,a closed algebraic formula for the edge of regression of the surface is obtained in terms of the functions∧,M,ν,which are closely related to the ones that appear in the standard decomposition of the derivative of the parametrisation of one of the bounding curves in terms of the director vector of the rulings and its derivative.It is also shown that all rational developable surfaces can be described as the set of developable surfaces which can be constructed with a constant∧,M,ν.The results are readily extended to rational spline developable surfaces.展开更多
We study the real Bonnet surfaces which accept one unique nontrivial isometry that preserves the mean curvature, in the three-dimensional Euclidean space. We give a general criterion for these surfaces and use it to d...We study the real Bonnet surfaces which accept one unique nontrivial isometry that preserves the mean curvature, in the three-dimensional Euclidean space. We give a general criterion for these surfaces and use it to determine the tangential developable surfaces of this kind. They are determined implicitly by elliptic integrals of the third kind. Only the tangential developable surfaces of circular helices are explicit examples for which we completely determine the above unique nontrivial isometry.展开更多
We propose a method for generating a ruled B-spline surface fitting to a sequence of pre-defined ruling lines and the generated surface is required to be as-developable-as-possible.Specifically,the terminal ruling lin...We propose a method for generating a ruled B-spline surface fitting to a sequence of pre-defined ruling lines and the generated surface is required to be as-developable-as-possible.Specifically,the terminal ruling lines are treated as hard constraints.Different from existing methods that compute a quasi-developable surface from two boundary curves and cannot achieve explicit ruling control,our method controls ruling lines in an intuitive way and serves as an effective tool for computing quasi-developable surfaces from freely-designed rulings.We treat this problem from the point of view of numerical optimization and solve for surfaces meeting the distance error tolerance allowed in applications.The performance and the efficacy of the proposed method are demonstrated by the experiments on a variety of models including an application of the method for path planning in 5-axis computer numerical control(CNC)flank milling.展开更多
While considering a mirror and light rays coming either from a point source or from infinity,the reflected light rays may have an envelope,called a caustic curve.In this paper,we study developable surfaces as mirrors....While considering a mirror and light rays coming either from a point source or from infinity,the reflected light rays may have an envelope,called a caustic curve.In this paper,we study developable surfaces as mirrors.These caustic surfaces,described in a closed form,are also developable surfaces of the same type as the original mirror surface.We provide efficient,algorithmic computation to find the caustic surface of each of the three types of developable surfaces(cone,cylinder,and tangent surface of a spatial curve).We also provide a potential application of the results in contemporary free-form architecture design.展开更多
Discussions on Chinese modernization are offering African countries both conceptual inspiration and practical references as they explore their own sustainable development paths.
In the lush heart of Uganda’s Busoga sub-region,Isaac Imaka is charting a new course for rural development.After seven years in national media,he left the newsroom and stepped into the soil.The former reporter with t...In the lush heart of Uganda’s Busoga sub-region,Isaac Imaka is charting a new course for rural development.After seven years in national media,he left the newsroom and stepped into the soil.The former reporter with the Daily Monitor was driven by the belief that communities like his in Jinja North deserved more than chronic poverty and hand-to-mouth survival.展开更多
China is carving out a distinctive development path which features urban-rural integration.This approach has not only yielded tangible results domestically but also drawn the attention of other countries.
China aims to build on its achievements of the past five years to accelerate rural revitalization during the 15th Five-Year Plan period.IN China,issues relating to agriculture,rural areas,and rural residents have alwa...China aims to build on its achievements of the past five years to accelerate rural revitalization during the 15th Five-Year Plan period.IN China,issues relating to agriculture,rural areas,and rural residents have always been a priority in national development.At its fourth plenary session held in October 2025,the 20th Central Committee of the Communist Party of China(CPC)once again emphasized the need to accelerate agricultural and rural modernization,and advance all-around rural revitalization over the next five years.展开更多
The CPC Central Committee’s Recommendations for Formulating the 15th Five-Year Plan for National Economic and Social Development place rural revitalization as a priority within the overall framework of modernization,...The CPC Central Committee’s Recommendations for Formulating the 15th Five-Year Plan for National Economic and Social Development place rural revitalization as a priority within the overall framework of modernization,clearly stating that agricultural and rural modernization has a major bearing on Chinese modernization as a whole and the outcomes it delivers.展开更多
Neurodevelopmental processes represent a finely tuned interplay between genetic and environmental factors,shaping the dynamic landscape of the developing brain.A major component of the developing brain that enables th...Neurodevelopmental processes represent a finely tuned interplay between genetic and environmental factors,shaping the dynamic landscape of the developing brain.A major component of the developing brain that enables this dynamic is the white matter(WM),known to be affected in neurodevelopmental disorders(NDDs)(Rokach et al.,2024).WM formation is mediated by myelination,a multifactorial process driven by neuro-glia interactions dependent on proper neuronal functionality(Simons and Trajkovic,2006).Another key aspect of neurodevelopmental abnormalities involves neuronal dynamics and function,with recent advances significantly enhancing our understanding of both neuronal and glial mitochondrial function(Devine and Kittler,2018;Rojas-Charry et al.,2021).Energy homeostasis in neurons,attributed largely to mitochondrial function,is critical for proper functionality and interactions with oligodendrocytes(OLs),the cells forming myelin in the brain’s WM.We herein discuss the interplay between these processes and speculate on potential dysfunction in NDDs.展开更多
Common mental disorders(CMDs),such as depressive and anxiety disorders,constitute a significant public health problem in low-and middle-income countries,such as India,where they rank among the leading causes of disabi...Common mental disorders(CMDs),such as depressive and anxiety disorders,constitute a significant public health problem in low-and middle-income countries,such as India,where they rank among the leading causes of disability and impaired quality of life.Outcomes are further compromised by a large treatment gap,poor adherence to therapeutic regimens,and high attrition rates.The prevalence and severity of CMDs are disproportionately higher in women.Additionally,structural factors influencing healthcare access,along with sociocultural factors,such as gender-based violence,limited autonomy in healthcare decisions,and greater levels of discrimination and stigma,result in poorer outcomes among women with CMDs.Therefore,there is a pressing need for care packages that are culturally sensitive,gender-responsive,and designed to address these structural and sociocultural factors,as highlighted in the literature from India.展开更多
Governance debates gained strong momentum in Africa in early December 2025 as the China-Kenya Readers Forum on Xi Jinping:The Governance of China convened in Nairobi on 1 December 2025,followed by a promotional event ...Governance debates gained strong momentum in Africa in early December 2025 as the China-Kenya Readers Forum on Xi Jinping:The Governance of China convened in Nairobi on 1 December 2025,followed by a promotional event for the English edition of the book’s fifth volume on 3 December 2025 in Johannesburg,South Africa.展开更多
Nanjing’s determination to transform itself from a production base to a research center reflects China’s evolution toward higher-quality development.A refrigerator that thaws frozen meat in 10 minutes and then keeps...Nanjing’s determination to transform itself from a production base to a research center reflects China’s evolution toward higher-quality development.A refrigerator that thaws frozen meat in 10 minutes and then keeps it fresh,a cooker hood that remains clean even after 10 years without disassembling it for cleaning.展开更多
基金Project supported by the National Basic Research Program (973) of China (No. 2004CB719400), the National Natural Science Founda-tion of China (Nos. 60373033 and 60333010) and the National Natural Science Foundation for Innovative Research Groups (No. 60021201), China
文摘A new algorithm is presented that generates developable Bézier surfaces through a Bézier curve called a directrix. The algorithm is based on differential geometry theory on necessary and sufficient conditions for a surface which is developable, and on degree evaluation formula for parameter curves and linear independence for Bernstein basis. No nonlinear characteristic equations have to be solved. Moreover the vertex for a cone and the edge of regression for a tangent surface can be obtained easily. Aumann’s algorithm for developable surfaces is a special case of this paper.
基金supported by the National Natural Science Foundation of China(Grant No.52375264).
文摘Curvature lines are special and important curves on surfaces.It is of great significance to construct developable surface interpolated on curvature lines in engineering applications.In this paper,the shape optimization of generalized cubic ball developable surface interpolated on the curvature line is studied by using the improved reptile search algorithm.Firstly,based on the curvature line of generalized cubic ball curve with shape adjustable,this paper gives the construction method of SGC-Ball developable surface interpolated on the curve.Secondly,the feedback mechanism,adaptive parameters and mutation strategy are introduced into the reptile search algorithm,and the Feedback mechanism-driven improved reptile search algorithm effectively improves the solving precision.On IEEE congress on evolutionary computation 2014,2017,2019 and four engineering design problems,the feedback mechanism-driven improved reptile search algorithm is compared with other representative methods,and the result indicates that the solution performance of the feedback mechanism-driven improved reptile search algorithm is competitive.At last,taking the minimum energy as the evaluation index,the shape optimization model of SGC-Ball interpolation developable surface is established.The developable surface with the minimum energy is achieved with the help of the feedback mechanism-driven improved reptile search algorithm,and the comparison experiment verifies the superiority of the feedback mechanism-driven improved reptile search algorithm for the shape optimization problem.
文摘The paper deals with the principle of envelope of a one-parameter plane family to design developable surfaces. Three methods of designing developable surfaces are presented. They are designing a developable surface based on one curve on it and its normal line, designing a developable surface based on two curves on it and designing a developable surface based on one curve and one surface. They meet the requirements of engineering fields.
基金supported by Beijing Nova Program under Grant No.Z121104002512065The author PerezDíaz S is a member of the Research Group ASYNACS(Ref.CCEE2011/R34)
文摘The developable surface is an important surface in computer aided design, geometric modeling and industrial manufactory. It is often given in the standard parametric form, but it can also be in the implicit form which is commonly used in algebraic geometry. Not all algebraic developable surfaces have rational parametrizations. In this paper, the authors focus on the rational developable surfaces. For a given algebraic surface, the authors first determine whether it is developable by geometric inspection, and then give a rational proper parametrization in the affrmative case. For a rational parametric surface, the authors also determine the developability and give a proper reparametrization for the developable surface.
基金partially supported by the Spanish Ministerio de Economía y Competividad(No.TRA2014-56792-P)
文摘In this paper we address the problem of interpolating a spline developable patch bounded by a given spline curve and the first and the last rulings of the developable surface. To complete the boundary of the patch, a second spline curve is to be given. Up to now this interpolation problem could be solved, but without the possibility of choosing both endpoints for the rulings. We circumvent such difficulty by resorting to degree elevation of the developable surface. This is useful for solving not only this problem, but also other problems dealing with triangular developable patches.
基金This work is partially supported by the Spanish Ministerio de Economiay Competitividad through research grant TRA2015-67788-P.
文摘In this paper we construct developable surface patches which are bounded by two rational or NURBS curves,though the resulting patch is not a rational or NURBS surface in general.This is accomplished by reparameterizing one of the boundary curves.The reparameterization function is the solution of an algebraic equation.For the relevant case of cubic or cubic spline curves,this equation is quartic at most,quadratic if the curves are B´ezier or splines and lie on parallel planes,and hence it may be solved either by standard analytical or numerical methods.
文摘Over the next 20 years,China's urban rail transit(hereinafter referred to as'urban rail')will face large-scalerenovation of existing line facilities and equipment,with more than 1000 km of renovated lines to be added eachyear.In 2024,the China Association of Metros issued the Guiding Opinions on the Renovation of Existing UrbanRail Transit Lines in China,providing guiding opinions on norms,standards,and implementation approaches forthe renovation of existing lines in the coming period.In the practical work of renovating existing urban rail lines,it is necessary to continuously explore and refine relevant theoretical methods in line with industry developmenttrends and urban development requirements.The following are the author's recent reflections on theoreticalinnovation in this field.
基金This work is partially supported by the Spanish Ministerio de Economia y Competitividad through research grant TRA2015-67788-P.
文摘In this paper we provide a characterisation of rational developable surfaces in terms of the blossoms of the bounding curves and three rational functions∧,M,ν.Properties of developable surfaces are revised in this framework.In particular,a closed algebraic formula for the edge of regression of the surface is obtained in terms of the functions∧,M,ν,which are closely related to the ones that appear in the standard decomposition of the derivative of the parametrisation of one of the bounding curves in terms of the director vector of the rulings and its derivative.It is also shown that all rational developable surfaces can be described as the set of developable surfaces which can be constructed with a constant∧,M,ν.The results are readily extended to rational spline developable surfaces.
文摘We study the real Bonnet surfaces which accept one unique nontrivial isometry that preserves the mean curvature, in the three-dimensional Euclidean space. We give a general criterion for these surfaces and use it to determine the tangential developable surfaces of this kind. They are determined implicitly by elliptic integrals of the third kind. Only the tangential developable surfaces of circular helices are explicit examples for which we completely determine the above unique nontrivial isometry.
基金This work was supported by the National Key Research and Development Program of China under Grant No.2018YFB1702900the National Natural Science Foundation of China under Grant No.62072139the Joint Funds of the National Natural Science Foundation of China with Zhejiang Integration of Informatization and Industrialization Key Project under Grant No.U1609218.
文摘We propose a method for generating a ruled B-spline surface fitting to a sequence of pre-defined ruling lines and the generated surface is required to be as-developable-as-possible.Specifically,the terminal ruling lines are treated as hard constraints.Different from existing methods that compute a quasi-developable surface from two boundary curves and cannot achieve explicit ruling control,our method controls ruling lines in an intuitive way and serves as an effective tool for computing quasi-developable surfaces from freely-designed rulings.We treat this problem from the point of view of numerical optimization and solve for surfaces meeting the distance error tolerance allowed in applications.The performance and the efficacy of the proposed method are demonstrated by the experiments on a variety of models including an application of the method for path planning in 5-axis computer numerical control(CNC)flank milling.
基金Project supported by the European Union and the European Social Fund(No.EFOP-3.6.3-VEKOP-16-2017-00002).Open Access funding provided by European Union and the European Social Fund。
文摘While considering a mirror and light rays coming either from a point source or from infinity,the reflected light rays may have an envelope,called a caustic curve.In this paper,we study developable surfaces as mirrors.These caustic surfaces,described in a closed form,are also developable surfaces of the same type as the original mirror surface.We provide efficient,algorithmic computation to find the caustic surface of each of the three types of developable surfaces(cone,cylinder,and tangent surface of a spatial curve).We also provide a potential application of the results in contemporary free-form architecture design.
文摘Discussions on Chinese modernization are offering African countries both conceptual inspiration and practical references as they explore their own sustainable development paths.
文摘In the lush heart of Uganda’s Busoga sub-region,Isaac Imaka is charting a new course for rural development.After seven years in national media,he left the newsroom and stepped into the soil.The former reporter with the Daily Monitor was driven by the belief that communities like his in Jinja North deserved more than chronic poverty and hand-to-mouth survival.
文摘China is carving out a distinctive development path which features urban-rural integration.This approach has not only yielded tangible results domestically but also drawn the attention of other countries.
文摘China aims to build on its achievements of the past five years to accelerate rural revitalization during the 15th Five-Year Plan period.IN China,issues relating to agriculture,rural areas,and rural residents have always been a priority in national development.At its fourth plenary session held in October 2025,the 20th Central Committee of the Communist Party of China(CPC)once again emphasized the need to accelerate agricultural and rural modernization,and advance all-around rural revitalization over the next five years.
文摘The CPC Central Committee’s Recommendations for Formulating the 15th Five-Year Plan for National Economic and Social Development place rural revitalization as a priority within the overall framework of modernization,clearly stating that agricultural and rural modernization has a major bearing on Chinese modernization as a whole and the outcomes it delivers.
文摘Neurodevelopmental processes represent a finely tuned interplay between genetic and environmental factors,shaping the dynamic landscape of the developing brain.A major component of the developing brain that enables this dynamic is the white matter(WM),known to be affected in neurodevelopmental disorders(NDDs)(Rokach et al.,2024).WM formation is mediated by myelination,a multifactorial process driven by neuro-glia interactions dependent on proper neuronal functionality(Simons and Trajkovic,2006).Another key aspect of neurodevelopmental abnormalities involves neuronal dynamics and function,with recent advances significantly enhancing our understanding of both neuronal and glial mitochondrial function(Devine and Kittler,2018;Rojas-Charry et al.,2021).Energy homeostasis in neurons,attributed largely to mitochondrial function,is critical for proper functionality and interactions with oligodendrocytes(OLs),the cells forming myelin in the brain’s WM.We herein discuss the interplay between these processes and speculate on potential dysfunction in NDDs.
文摘Common mental disorders(CMDs),such as depressive and anxiety disorders,constitute a significant public health problem in low-and middle-income countries,such as India,where they rank among the leading causes of disability and impaired quality of life.Outcomes are further compromised by a large treatment gap,poor adherence to therapeutic regimens,and high attrition rates.The prevalence and severity of CMDs are disproportionately higher in women.Additionally,structural factors influencing healthcare access,along with sociocultural factors,such as gender-based violence,limited autonomy in healthcare decisions,and greater levels of discrimination and stigma,result in poorer outcomes among women with CMDs.Therefore,there is a pressing need for care packages that are culturally sensitive,gender-responsive,and designed to address these structural and sociocultural factors,as highlighted in the literature from India.
文摘Governance debates gained strong momentum in Africa in early December 2025 as the China-Kenya Readers Forum on Xi Jinping:The Governance of China convened in Nairobi on 1 December 2025,followed by a promotional event for the English edition of the book’s fifth volume on 3 December 2025 in Johannesburg,South Africa.
文摘Nanjing’s determination to transform itself from a production base to a research center reflects China’s evolution toward higher-quality development.A refrigerator that thaws frozen meat in 10 minutes and then keeps it fresh,a cooker hood that remains clean even after 10 years without disassembling it for cleaning.
