The aim of this paper is to study numerical realization of the conditions of Max Nther's residual intersection theorem. The numerical realization relies on obtaining the intersection of two algebraic curves by homotopy continuation method, computing the approximate places of an algebraic curve, getting the exact orders of a polynomial at the places, and determining the multiplicity and character of a point of an algebraic curve. The numerical experiments show that our method is accurate, effective and robust without using multiprecision arithmetic,even if the coefficients of algebraic curves are inexact. We also conclude that the computational complexity of the numerical realization is polynomial time.
Mesh parameterization is one of the fundamental operations in computer graphics(CG) and computeraided design(CAD). In this paper, we propose a novel local/global parameterization approach, ARAP++, for singleand multi-boundary triangular meshes. It is an extension of the as-rigid-as-possible(ARAP) approach, which stitches together 1-ring patches instead of individual triangles. To optimize the spring energy, we introduce a linear iterative scheme which employs convex combination weights and a fitting Jacobian matrix corresponding to a prescribed family of transformations. Our algorithm is simple, efficient, and robust. The geometric properties(angle and area)of the original model can also be preserved by appropriately prescribing the singular values of the fitting matrix. To reduce the area and stretch distortions for high-curvature models, a stretch operator is introduced. Numerical results demonstrate that ARAP++ outperforms several state-of-the-art methods in terms of controlling the distortions of angle, area, and stretch. Furthermore, it achieves a better visualization performance for several applications, such as texture mapping and surface remeshing.
Projective invariants are not only important objects in mathematics especially in geometry,but also widely used in many practical applications such as in computer vision and object recognition. In this work,we show a projective invariant named as characteristic number,from which we obtain an intrinsic property of an algebraic hypersurface involving the intersections of the hypersurface and some lines that constitute a closed loop. From this property,two high-dimensional generalizations of Pascal's theorem are given,one establishing the connection of hypersurfaces of distinct degrees,and the other concerned with the intersections of a hypersurface and a simplex.
In this paper, the quadratic nonconforming brick element(MSLK element) introduced in [10] is used for the 3D Stokes equations. The instability for the mixed element pair MSLK-P_1 is analyzed, where the vector-valued MSLK element approximates the velocity and the piecewise P_1 element approximates the pressure. As a cure, we adopt the piecewise P_1 macroelement to discretize the pressure instead of the standard piecewise P_1 element on cuboid meshes. This new pair is stable and the optimal error estimate is achieved. Numerical examples verify our theoretical analysis.
