Biharmonic coordinates have become a powerful tool for cage-based deformation, owing to their inherent interpolation properties. However, their derivation for polynomial cages in 3D has remained unsolved. To address this, we propose closed-form expressions for polynomial 3D biharmonic coordinates and their derivatives when deformed from polygonal cages using the high-order boundary element method. Our primary contribution lies in the analytical derivation of the kernel integration using recursive differentiation techniques. Due to the enriched deformation space of biharmonic coordinates and the flexibility of polynomial cages, our method supports a broad range of deformations, as demonstrated through extensive experiments.

We propose closed-form Cauchy coordinates and their derivatives for 2D closed high-order input cages composed of arbitrary-order polynomial curves. Our coordinates facilitate the transformation of input polynomial curves into output curves of any desired polynomial order. Central to our derivation is the creative use of the residue theorem with the logarithmic function to obtain the integral of a rational polynomial required for extending the classical 2D Cauchy coordinates to high-order input cages. Our coordinates enable smooth cage-aware angle-preserving deformations, and the derivatives allow for point-to-point deformation. Moreover, our derivation can be extended to the input cages with rational polynomial curves. Through various 2D deformations, we demonstrate how users can intuitively manipulate Bézier control points to achieve desired deformations easily.

We derive closed-form expressions of biharmonic coordinates for 2D high-order cages, enabling the transformation of the input polynomial curves into polynomial curves of any order. Central to our derivation is the use of the high-order boundary element method. We demonstrate the practicality and effectiveness of our method on various 2D deformations. In practice, users can easily manipulate the Bézier control points to perform the desired intuitive deformation, as the biharmonic coordinates provide an enriched deformation space and encourage the alignment between the boundary cage and its interior geometry.

We derive closed-form expressions for generalized winding numbers of rational parametric curves for robust containment queries. Given an oriented rational parametric curve and a query point, the generalized winding number can be reformulated to an integral of a rational polynomial. The key to computing the integral lies in using the residue theorem. Then, add up the contributions of each curve to obtain the generalized winding numbers of a set of rational parametric curves. Furthermore, the derivatives of generalized winding numbers are easily derived. Consequently, the expressions for generalized winding numbers are concise and computationally efficient, becoming faster than state-of-the-art methods. Moreover, the computational costs for various query points are almost the same.

We propose a novel method to robustly and efficiently compute the maximum allowable step sizes so that the 3D high-order finite elements continuously preserve geometric validity when moving along the given directions with positive step sizes smaller than the computed ones. We transform the problem of finding the maximum allowable step sizes to one of solving roots of cubic polynomials. To use interval arithmetic to avoid numerical issues in cubic equation solving, we completely enumerate the roots of cubic polynomials and apply the interval version of the Newton-Raphson iteration. The effectiveness of our algorithm is demonstrated through extensive testing. Compared to the state-of-the-art method, our algorithm achieves higher efficiency.

We propose a new structure called a higher-order shell, which is composed of a set of triangular prisms. Each triangular prism is enveloped by three Bézier triangles (top, middle, and bottom) and three side surfaces, each of which is trimmed from a bilinear surface. Moreover, we define a continuous vector field to smoothly and bijectively transfer attributes between two surfaces inside the shell. Since the higher-order shell has several hard construction constraints, we apply an interior-point strategy to robustly and automatically construct a high-order shell for an input mesh. Specifically, the strategy starts from a valid linear shell with a small thickness. Then, the shell is optimized until the specified thickness is reached, where explicit checks ensure that the constraints are always satisfied. We extensively test our method on more than 8300 models, demonstrating its robustness and performance. Compared to state-of-the-art methods, our bijective projection is smoother, and the space between the shell and input mesh is more uniform.

Abstract We propose a novel method to generate a high‐order triangular mesh for an input 2D domain with two key characteristics: (1) the mesh precisely conforms to a set of input piecewise rational domain curves, and (2) the geometric map on each curved triangle is injective. Central to the algorithm is a new sufficient condition for placing control points of a rational Bézier triangle to guarantee that the conformance and injectivity constraints are theoretically satisfied. Taking advantage of this condition, we provide an explicit construct that robustly creates higher‐order 2D meshes satisfying the two characteristics. We demonstrate the robustness and effectiveness of our algorithm over a data set containing 2200 examples.