Some relations in a complete quadrilateral are derived. In connection with these relations some special conics related to the angular points and sides of the quadrilateral are discussed. A theorem of Carnot valid for a triangle is extended to a quadrilateral.

This paper presents a triangular-to-quadrilateral mesh conversion method that can control the directionality of the output quadrilateral mesh according to a user-specified vector field. Given a triangular mesh and a vector field, the method first scores all possible quadrilaterals that can be formed by pairs of adjacent triangles, according to their shape and directionality. It then converts th...

This work presents a new approach to conformal all-quadrilateral mesh adaptation. Most current quadrilateral adaptivity techniques rely on mesh refinement or a complete remesh of the domain. In contrast, we introduce a new method that incorporates both conformal refinement and coarsening strategies on an existing mesh of any density or configuration. Given a sizing function, this method modifie...

Asymptote is widely used in astronomy, mechanics, architecture and relevant subjects. In this paper, by analyzing the Frenet frame and the Darboux frame of a curve on the surface, the necessary and sufficient conditions for a quadrilateral boundary being asymptotic of a surface are derived. This quadrilateral is called asymptotic quadrilateral. Given corner data including positions, tangents an...

This paper proposes an automated quadrilateral meshing technique based on the conversion of triangular meshes. Given a triangular mesh and a vector eld representing the desired directionality of quadrilateral elements, our new approach generates a well-shaped and well-aligned quadrilateral mesh. In the approach, the values of three scalar functions are rst calculated for each pair of adjacent t...

An automatic adaptive quadrilateral mesh conversion scheme for the generation of adaptive refinement meshes over analytical curved surfaces is proposed. The starting point of the quadrilateral mesh generator is a background triangular mesh of the curved surface. By a carefully controlled process to merge two triangles at a time the triangular mesh can be completely converted to quadrilaterals. ...

We prove five necessary and sufficient conditions for a convex quadrilateral to have an excircle and compare them to similar conditions for a quadrilateral to have an incircle.

Abstract Multistable origami and its snapping behaviors between the folded states have attracted scientists’ engineers’ attention as building block for design of mechanical devices metamaterials. We propose a novel method designing origami-based multistable structures, by which we mean (1) to obtain prescribed overall motion (2) control stiffness provided elastic strain. solve this problem firs...

This paper presents a computational method for quadrilateral meshing of a thin, or narrow, two-dimensional domain for finite element analysis. The proposed method creates a well-shaped single-layered, multi-layered, or partially multi-layered quadrilateral mesh. Element sizes can be uniform or graded. A high quality, layered quadrilateral mesh is often required for finite element analysis of a ...

Martin Josefsson [1] has coined the term ‘bisect-diagonal quadrilateral’ for a quadrilateral with at least one diagonal bisected by other diagonal, and extensively explored some of its properties. This also been called ‘bisecting [2], ‘sloping-kite’ or ‘sliding-kite’ [3], ‘slant kite’ [4]. The purpose this paper is to explore more properties quadrilateral.