Rational Bezier surface is a widely used surface fitting tool in CAD. When all the weights of a rational B@zier surface go to infinity in the form of power function, the limit of surface is the regular control surface...Rational Bezier surface is a widely used surface fitting tool in CAD. When all the weights of a rational B@zier surface go to infinity in the form of power function, the limit of surface is the regular control surface induced by some lifting function, which is called toric degenerations of rational Bezier surfaces. In this paper, we study on the degenerations of the rational Bezier surface with weights in the exponential function and indicate the difference of our result and the work of Garcia-Puente et al. Through the transformation of weights in the form of exponential function and power function, the regular control surface of rational Bezier surface with weights in the exponential function is defined, which is just the limit of the surface. Compared with the power function, the exponential function approaches infinity faster, which leads to surface with the weights in the form of exponential function degenerates faster.展开更多
This paper establishes an asymptotic formula with a power-saving error term for the number of rational points of bounded height on the singular cubic surface of P3 Qgiven by the following equation x0（x12+ x22）-x33= ...This paper establishes an asymptotic formula with a power-saving error term for the number of rational points of bounded height on the singular cubic surface of P3 Qgiven by the following equation x0（x12+ x22）-x33= 0 in agreement with the Manin-Peyre conjectures.展开更多
De Casteljau algorithm and degree elevation of Bézier and NURBS curves/surfaces are two important techniques in computer aided geometric design. This paper presents the de Casteljau algorithm and degree elevation...De Casteljau algorithm and degree elevation of Bézier and NURBS curves/surfaces are two important techniques in computer aided geometric design. This paper presents the de Casteljau algorithm and degree elevation of toric surface patches, which include tensor product and triangular rational Bézier surfaces as special cases. Some representative examples of toric surface patches with common shapes are illustrated to verify these two algorithms. Moreover, the authors also apply the degree elevation of toric surface patches to isogeometric analysis. And two more examples show the effectiveness of proposed method.展开更多
The optical windows used in aircrafts protect their imaging sensors from environmental effects. Considering the imaging performance, flat surfaces are traditionally used in the design of optical windows. For aircrafts...The optical windows used in aircrafts protect their imaging sensors from environmental effects. Considering the imaging performance, flat surfaces are traditionally used in the design of optical windows. For aircrafts operating at high speeds, the optical windows should be relatively aerodynamic, but a flat optical window may introduce unacceptably high drag to the airframes. The linear scanning infrared sensors used in aircrafts with, respectively, a flat window, a spherical window and a toric window in front of the aircraft sensors are designed and compared. Simulation results show that the optical design using a toric surface has the integrated advantages of field of regard, aerodynamic drag, narcissus effect, and imaging performance, so the optical window with a toric surface is demonstrated to be suited for this application.展开更多
This paper,largely written in 2009/2010,fits Landau-Ginzburg models into the mirror symmetry program pursued by the last author jointly with Mark Gross since 2001.This point of view transparently brings in tropical di...This paper,largely written in 2009/2010,fits Landau-Ginzburg models into the mirror symmetry program pursued by the last author jointly with Mark Gross since 2001.This point of view transparently brings in tropical disks of Maslov index 2 via the notion of broken lines,previously introduced in two dimensions by Mark Gross in his study of mirror symmetry for P2.A ma jor insight is the equivalence of properness of the Landau-Ginzburg potential with smoothness of the anticanonical divisor on the mirror side.We obtain proper superpotentials which agree on an open part with those classically known for toric varieties.Examples include mirror LG models for non-singular and singular del Pezzo surfaces,Hirzebruch surfaces and some Fano threefolds.展开更多
Inspired by the r-refinement method in isogeometric analysis,in this paper,the authors propose a curvature-based r-adaptive isogeometric method for planar multi-sided computational domains parameterized by toric surfa...Inspired by the r-refinement method in isogeometric analysis,in this paper,the authors propose a curvature-based r-adaptive isogeometric method for planar multi-sided computational domains parameterized by toric surface patches.The authors construct three absolute curvature metrics of isogeometric solution surface to characterize its gradient information,which is more straightforward and effective.The proposed method takes the internal weights as optimization variables and the resulting parameterization is analysis-suitable and injectivity-preserving with a theoretical guarantee.Several PDEs are solved over multi-sided computational domains parameterized by toric surface patches to demonstrate the effectiveness and efficiency of the proposed method.展开更多
基金Supported by the National Natural Science Foundation of China(11671068,11271060,11601064,11290143)Fundamental Research of Civil Aircraft(MJ-F-2012-04)the Fundamental Research Funds for the Central Universities(DUT16LK38)
文摘Rational Bezier surface is a widely used surface fitting tool in CAD. When all the weights of a rational B@zier surface go to infinity in the form of power function, the limit of surface is the regular control surface induced by some lifting function, which is called toric degenerations of rational Bezier surfaces. In this paper, we study on the degenerations of the rational Bezier surface with weights in the exponential function and indicate the difference of our result and the work of Garcia-Puente et al. Through the transformation of weights in the form of exponential function and power function, the regular control surface of rational Bezier surface with weights in the exponential function is defined, which is just the limit of the surface. Compared with the power function, the exponential function approaches infinity faster, which leads to surface with the weights in the form of exponential function degenerates faster.
基金supported by the program PRC 1457-Au For Di P(CNRS-NSFC)supported by National Natural Science Foundation of China(Grant No.11531008)+1 种基金the Ministry of Education of China(Grant No.IRT16R43)the Taishan Scholar Project of Shandong Province
文摘This paper establishes an asymptotic formula with a power-saving error term for the number of rational points of bounded height on the singular cubic surface of P3 Qgiven by the following equation x0（x12+ x22）-x33= 0 in agreement with the Manin-Peyre conjectures.
基金supported by the National Natural Science Foundation of China under Grant Nos.11671068 and 11801053。
文摘De Casteljau algorithm and degree elevation of Bézier and NURBS curves/surfaces are two important techniques in computer aided geometric design. This paper presents the de Casteljau algorithm and degree elevation of toric surface patches, which include tensor product and triangular rational Bézier surfaces as special cases. Some representative examples of toric surface patches with common shapes are illustrated to verify these two algorithms. Moreover, the authors also apply the degree elevation of toric surface patches to isogeometric analysis. And two more examples show the effectiveness of proposed method.
文摘The optical windows used in aircrafts protect their imaging sensors from environmental effects. Considering the imaging performance, flat surfaces are traditionally used in the design of optical windows. For aircrafts operating at high speeds, the optical windows should be relatively aerodynamic, but a flat optical window may introduce unacceptably high drag to the airframes. The linear scanning infrared sensors used in aircrafts with, respectively, a flat window, a spherical window and a toric window in front of the aircraft sensors are designed and compared. Simulation results show that the optical design using a toric surface has the integrated advantages of field of regard, aerodynamic drag, narcissus effect, and imaging performance, so the optical window with a toric surface is demonstrated to be suited for this application.
基金supported by the Studienstiftung des deutschen Volkessupported by NSF grant DMS-1903437。
文摘This paper,largely written in 2009/2010,fits Landau-Ginzburg models into the mirror symmetry program pursued by the last author jointly with Mark Gross since 2001.This point of view transparently brings in tropical disks of Maslov index 2 via the notion of broken lines,previously introduced in two dimensions by Mark Gross in his study of mirror symmetry for P2.A ma jor insight is the equivalence of properness of the Landau-Ginzburg potential with smoothness of the anticanonical divisor on the mirror side.We obtain proper superpotentials which agree on an open part with those classically known for toric varieties.Examples include mirror LG models for non-singular and singular del Pezzo surfaces,Hirzebruch surfaces and some Fano threefolds.
基金supported by the National Natural Science Foundation of China under Grant Nos.12071057,11671068,and 12001327.
文摘Inspired by the r-refinement method in isogeometric analysis,in this paper,the authors propose a curvature-based r-adaptive isogeometric method for planar multi-sided computational domains parameterized by toric surface patches.The authors construct three absolute curvature metrics of isogeometric solution surface to characterize its gradient information,which is more straightforward and effective.The proposed method takes the internal weights as optimization variables and the resulting parameterization is analysis-suitable and injectivity-preserving with a theoretical guarantee.Several PDEs are solved over multi-sided computational domains parameterized by toric surface patches to demonstrate the effectiveness and efficiency of the proposed method.
